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Вопросы, страница 74 - гдз по геометрии 7 класс учебник Смирнов, Туяков
Авторы: Смирнов В. А., Туяков Е. А.
Тип: Учебник
Издательство: Мектеп
Год издания: 2017 - 2026
ISBN: 978-601-07-0873-0
Утверждено Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 7 классе
Глава 2. Треугольники. Параграф 13. Прямоугольные треугольники - страница 74.
App\Models\Task {#1030 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1129679 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "74" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/vopr-13" "field_display_title" => "Вопросы" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1036 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1034 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1045 #items: array:1 [ 0 => App\Models\Branch {#1044 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1129366 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Треугольники" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1049 #items: array:1 [ 0 => App\Models\Term {#1046 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "field_page_check_not_needed" => null "field_short_name" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "39" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1057 #items: array:1 [ 0 => App\Models\Book {#1056 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 2907 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1061 #items: array:1 [ 0 => App\Models\Term {#1058 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6477 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Геометрия" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "геометрию" "field_creative_case" => "геометрией" "field_dative_case" => "геометрии" "field_genitive_case" => "геометрии" "field_nominative_case" => "геометрия" "field_prepositional_case" => "геометрии" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "geometrija" ] #original: array:10 [ "id" => 6477 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Геометрия" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "геометрию" "field_creative_case" => "геометрией" "field_dative_case" => "геометрии" "field_genitive_case" => "геометрии" "field_nominative_case" => "геометрия" "field_prepositional_case" => "геометрии" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "geometrija" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1063 #items: array:1 [ 0 => App\Models\Term {#1060 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5456 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "7" "field_cases" => array:6 [ "field_accusative_case" => "седьмой" "field_creative_case" => "седьмым" "field_dative_case" => "седьмому" "field_genitive_case" => "седьмого" "field_nominative_case" => "седьмой" "field_prepositional_case" => "седьмом" ] "field_translit" => "sedmoj" ] #original: array:6 [ "id" => 5456 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "7" "field_cases" => array:6 [ "field_accusative_case" => "седьмой" "field_creative_case" => "седьмым" "field_dative_case" => "седьмому" "field_genitive_case" => "седьмого" "field_nominative_case" => "седьмой" "field_prepositional_case" => "седьмом" ] "field_translit" => "sedmoj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1071 #items: array:1 [ 0 => App\Models\Term {#1070 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #original: array:6 [ "id" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1064 #items: array:2 [ 0 => App\Models\Term {#1075 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 5801 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Смирнов" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Владимир" "field_patronymic" => "Алексеевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 5801 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Смирнов" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Владимир" "field_patronymic" => "Алексеевич" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1076 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7017 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Туяков" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Есенкельды" "field_patronymic" => "Алыбаевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 7017 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Туяков" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Есенкельды" "field_patronymic" => "Алыбаевич" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1072 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1068 #items: array:1 [ 0 => App\Models\Term {#1062 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1081 #items: array:1 [ 0 => App\Models\Term {#1080 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "kazakhstan" ] #original: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "kazakhstan" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1084 #items: array:1 [ 0 => App\Models\Term {#1083 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almaty" ] #original: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almaty" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1067 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1074 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1059 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1078 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1077 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1065 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2017" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 7 класса общеобразовательной школы" "field_allowed" => "Утверждено Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "528" "field_priority" => "2" "field_default_folder" => "/geometrija_07/Smirnova-u17/" "field_isbn" => "978-601-07-0873-0" "field_cover" => array:1 [ 0 => "/media/geometrija_07/Smirnova-u17/covers/cover1.webp?ts=1753259592" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_07/Smirnova-u17/covers/cover1.webp?ts=1753259592" "alt" => "" "width" => "1632" "height" => "2472" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1066 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1085 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1086 #items: [] #escapeWhenCastingToString: false } "field_url" => "/7-klass/geometrija/smirnova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1087 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 378 "subject" => 543 "class_subject" => 389 ] ] #original: array:50 [ "id" => 2907 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1061} "field_class" => Illuminate\Database\Eloquent\Collection {#1063} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1071} "field_author" => Illuminate\Database\Eloquent\Collection {#1064} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1072} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1068} "field_country" => Illuminate\Database\Eloquent\Collection {#1081} "field_city" => Illuminate\Database\Eloquent\Collection {#1084} "field_series" => Illuminate\Database\Eloquent\Collection {#1067} "field_umk" => Illuminate\Database\Eloquent\Collection {#1074} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1059} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1078} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1077} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1065} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2017" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 7 класса общеобразовательной школы" "field_allowed" => "Утверждено Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "528" "field_priority" => "2" "field_default_folder" => "/geometrija_07/Smirnova-u17/" "field_isbn" => "978-601-07-0873-0" "field_cover" => array:1 [ 0 => "/media/geometrija_07/Smirnova-u17/covers/cover1.webp?ts=1753259592" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_07/Smirnova-u17/covers/cover1.webp?ts=1753259592" "alt" => "" "width" => "1632" "height" => "2472" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1066} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1085} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1086} "field_url" => "/7-klass/geometrija/smirnova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1087} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 378 "subject" => 543 "class_subject" => 389 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1047 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1129366 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Треугольники" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1049} "field_page_start" => "39" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1057} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1047} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1109 #items: array:2 [ 0 => App\Models\Branch {#1117 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1129366 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Треугольники" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1118 #items: array:1 [ 0 => App\Models\Term {#1046} ] #escapeWhenCastingToString: false } "field_page_start" => "39" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1119 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1120 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1129366 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Треугольники" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1118} "field_page_start" => "39" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1119} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1120} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Branch {#1121 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1129373 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Прямоугольные треугольники" "field_branch_order" => "13" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1125 #items: array:1 [ 0 => App\Models\Term {#1122 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ "field_accusative_case" => "Параграф" "field_creative_case" => "Параграфом" "field_dative_case" => "Параграфу" "field_genitive_case" => "Параграфа" "field_nominative_case" => "Параграф" "field_prepositional_case" => "Параграфе" ] "field_page_check_not_needed" => null "field_short_name" => "§" ] #original: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ "field_accusative_case" => "Параграф" "field_creative_case" => "Параграфом" "field_dative_case" => "Параграфу" "field_genitive_case" => "Параграфа" "field_nominative_case" => "Параграф" "field_prepositional_case" => "Параграфе" ] "field_page_check_not_needed" => null "field_short_name" => "§" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "72" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1124 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1123 #items: array:1 [ 0 => App\Models\Branch {#1044} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1129373 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Прямоугольные треугольники" "field_branch_order" => "13" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1125} "field_page_start" => "72" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1124} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1123} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1106 #items: array:3 [ 0 => App\Models\Element {#1099 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 1320163 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1091 #items: array:1 [ 0 => App\Models\Edition {#1098 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5009 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1096 #items: array:1 [ 0 => App\Models\Term {#1097 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #original: array:6 [ "id" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1094 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1095 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1092 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5009 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1096} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1094} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1095} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1092} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1129679" "type" => "task" ] "text" => "<p><strong>1.</strong> Какой треугольник называется прямоугольным?</p><p><strong>2.</strong> Что называется <em>гипотенузой</em> и <em>катетами</em> прямоугольного треугольника?</p><p><strong>3.</strong> Сформулируйте признаки равенства прямоугольных треугольников.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "13-1.jpg" "alt" => null "width" => "1741" "height" => 308 "path" => "/media/geometrija_07/Smirnova-u17/0-vopr/13-1.webp?ts=1753982528" ] ] ] #original: array:7 [ "id" => 1320163 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1091} "task" => array:2 [ "refs" => "1129679" "type" => "task" ] "text" => "<p><strong>1.</strong> Какой треугольник называется прямоугольным?</p><p><strong>2.</strong> Что называется <em>гипотенузой</em> и <em>катетами</em> прямоугольного треугольника?</p><p><strong>3.</strong> Сформулируйте признаки равенства прямоугольных треугольников.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "13-1.jpg" "alt" => null "width" => "1741" "height" => 308 "path" => "/media/geometrija_07/Smirnova-u17/0-vopr/13-1.webp?ts=1753982528" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1093 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1320513 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1048 #items: array:1 [ 0 => App\Models\Edition {#1090 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5010 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1054 #items: array:1 [ 0 => App\Models\Term {#1089 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6714 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Gdz" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "gdz" ] #original: array:6 [ "id" => 6714 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Gdz" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "gdz" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1088 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1052 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1053 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5010 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1054} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1088} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1052} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1053} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1129679" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "13-1.jpg" "alt" => null "width" => "1484" "height" => 2429 "path" => "/media/geometrija_07/Smirnova-u17/1-vopr/13-1.webp?ts=1753982643" ] 1 => array:5 [ "name" => "13-2.jpg" "alt" => null "width" => "1484" "height" => 1005 "path" => "/media/geometrija_07/Smirnova-u17/1-vopr/13-2.webp?ts=1753982643" ] ] ] #original: array:6 [ "id" => 1320513 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1048} "task" => array:2 [ "refs" => "1129679" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "13-1.jpg" "alt" => null "width" => "1484" "height" => 2429 "path" => "/media/geometrija_07/Smirnova-u17/1-vopr/13-1.webp?ts=1753982643" ] 1 => array:5 [ "name" => "13-2.jpg" "alt" => null "width" => "1484" "height" => 1005 "path" => "/media/geometrija_07/Smirnova-u17/1-vopr/13-2.webp?ts=1753982643" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Element {#1051 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1541775 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1112 #items: array:1 [ 0 => App\Models\Edition {#1050 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5631 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1040 #items: array:1 [ 0 => App\Models\Term {#1042 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 7012 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Gemini 2.5 Pro" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "gemini 2.5 pro" ] #original: array:6 [ "id" => 7012 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Gemini 2.5 Pro" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "gemini 2.5 pro" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1038 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1108 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5631 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1040} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1035} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1038} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1108} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1129679" "type" => "task" ] "text" => "<p><strong>1.</strong> Прямоугольным называется треугольник, у которого один из углов является прямым, то есть его градусная мера составляет $90^\circ$. Прямой угол на чертежах принято обозначать небольшим квадратом.</p><svg width="220" height="150" xmlns="http://www.w3.org/2000/svg"><polygon points="20,130 180,130 20,20" style="fill:rgb(240, 248, 255);stroke:black;stroke-width:2"></polygon><rect x="20" y="110" width="20" height="20" style="fill:none;stroke:black;stroke-width:1.5"></rect><text x="10" y="25" font-family="sans-serif" font-size="16">A</text><text x="10" y="145" font-family="sans-serif" font-size="16">C</text><text x="185" y="145" font-family="sans-serif" font-size="16">B</text><text x="45" y="105" font-family="sans-serif" font-size="14">Угол $\angle C = 90^\circ$</text></svg><p>Ответ: Прямоугольный треугольник — это треугольник, в котором один угол прямой ($90^\circ$).</p><p><strong>2.</strong> Стороны в прямоугольном треугольнике носят особые названия:</p><p>• <strong>Катеты</strong> — это две стороны, образующие прямой угол. На рисунке ниже стороны $AC$ и $BC$ являются катетами.</p><p>• <strong>Гипотенуза</strong> — это сторона, которая лежит напротив прямого угла. Гипотенуза всегда является самой длинной стороной прямоугольного треугольника. На рисунке ниже сторона $AB$ является гипотенузой.</p><svg width="280" height="180" xmlns="http://www.w3.org/2000/svg"><polygon points="30,150 220,150 30,30" style="fill:rgb(240, 248, 255);stroke:black;stroke-width:2"></polygon><rect x="30" y="130" width="20" height="20" style="fill:none;stroke:black;stroke-width:1.5"></rect><text x="15" y="35" font-family="sans-serif" font-size="16">A</text><text x="15" y="165" font-family="sans-serif" font-size="16">C</text><text x="225" y="165" font-family="sans-serif" font-size="16">B</text><text x="45" y="90" font-family="sans-serif" font-size="16" fill="blue" transform="rotate(-90 45 90)">Катет</text><text x="110" y="165" font-family="sans-serif" font-size="16" fill="blue">Катет</text><text x="110" y="80" font-family="sans-serif" font-size="16" fill="red" transform="rotate(-35 125 80)">Гипотенуза</text></svg><p>Ответ: Катеты — стороны, образующие прямой угол; гипотенуза — сторона, лежащая напротив прямого угла.</p><p><strong>3.</strong> Признаки равенства прямоугольных треугольников — это специальные теоремы, позволяющие установить равенство двух прямоугольных треугольников по меньшему количеству элементов, чем в общих признаках.</p><p><strong>По двум катетам:</strong> Если катеты одного прямоугольного треугольника соответственно равны катетам другого, то такие треугольники равны.</p><svg width="260" height="110" xmlns="http://www.w3.org/2000/svg"><polygon points="10,100 110,100 10,10" style="fill:none;stroke:black;stroke-width:2"></polygon><rect x="10" y="80" width="20" height="20" style="fill:none;stroke:black;stroke-width:1"></rect><line x1="55" y1="105" x2="65" y2="105" stroke="blue" stroke-width="2"></line><line x1="5" y1="50" x2="5" y2="60" stroke="red" stroke-width="2"></line><line x1="5" y1="65" x2="5" y2="75" stroke="red" stroke-width="2"></line><polygon points="150,100 250,100 150,10" style="fill:none;stroke:black;stroke-width:2"></polygon><rect x="150" y="80" width="20" height="20" style="fill:none;stroke:black;stroke-width:1"></rect><line x1="195" y1="105" x2="205" y2="105" stroke="blue" stroke-width="2"></line><line x1="145" y1="50" x2="145" y2="60" stroke="red" stroke-width="2"></line><line x1="145" y1="65" x2="145" y2="75" stroke="red" stroke-width="2"></line></svg><p><strong>По катету и прилежащему острому углу:</strong> Если катет и прилежащий к нему острый угол одного прямоугольного треугольника соответственно равны катету и прилежащему острому углу другого, то такие треугольники равны.</p><svg width="260" height="110" xmlns="http://www.w3.org/2000/svg"><polygon points="10,100 110,100 10,10" style="fill:none;stroke:black;stroke-width:2"></polygon><rect x="10" y="80" width="20" height="20" style="fill:none;stroke:black;stroke-width:1"></rect><line x1="55" y1="105" x2="65" y2="105" stroke="blue" stroke-width="2"></line><path d="M 90,100 A 15 15, 0, 0, 0, 100 88" fill="none" stroke="red" stroke-width="1.5"></path><polygon points="150,100 250,100 150,10" style="fill:none;stroke:black;stroke-width:2"></polygon><rect x="150" y="80" width="20" height="20" style="fill:none;stroke:black;stroke-width:1"></rect><line x1="195" y1="105" x2="205" y2="105" stroke="blue" stroke-width="2"></line><path d="M 230,100 A 15 15, 0, 0, 0, 240 88" fill="none" stroke="red" stroke-width="1.5"></path></svg><p><strong>По гипотенузе и острому углу:</strong> Если гипотенуза и острый угол одного прямоугольного треугольника соответственно равны гипотенузе и острому углу другого, то такие треугольники равны.</p><svg width="260" height="110" xmlns="http://www.w3.org/2000/svg"><polygon points="10,100 110,100 10,10" style="fill:none;stroke:black;stroke-width:2"></polygon><rect x="10" y="80" width="20" height="20" style="fill:none;stroke:black;stroke-width:1"></rect><line x1="50" y1="45" x2="70" y2="65" stroke="blue" stroke-width="2" transform="rotate(-45 60 55)"></line><path d="M 10,25 A 15 15, 0, 0, 1, 23 15" fill="none" stroke="red" stroke-width="1.5"></path><polygon points="150,100 250,100 150,10" style="fill:none;stroke:black;stroke-width:2"></polygon><rect x="150" y="80" width="20" height="20" style="fill:none;stroke:black;stroke-width:1"></rect><line x1="190" y1="45" x2="210" y2="65" stroke="blue" stroke-width="2" transform="rotate(-45 200 55)"></line><path d="M 150,25 A 15 15, 0, 0, 1, 163 15" fill="none" stroke="red" stroke-width="1.5"></path></svg><p><strong>По катету и противолежащему острому углу:</strong> Если катет и противолежащий ему острый угол одного прямоугольного треугольника соответственно равны катету и противолежащему ему острому углу другого, то такие треугольники равны.</p><svg width="260" height="110" xmlns="http://www.w3.org/2000/svg"><polygon points="10,100 110,100 10,10" style="fill:none;stroke:black;stroke-width:2"></polygon><rect x="10" y="80" width="20" height="20" style="fill:none;stroke:black;stroke-width:1"></rect><line x1="55" y1="105" x2="65" y2="105" stroke="blue" stroke-width="2"></line><path d="M 10,25 A 15 15, 0, 0, 1, 23 15" fill="none" stroke="red" stroke-width="1.5"></path><polygon points="150,100 250,100 150,10" style="fill:none;stroke:black;stroke-width:2"></polygon><rect x="150" y="80" width="20" height="20" style="fill:none;stroke:black;stroke-width:1"></rect><line x1="195" y1="105" x2="205" y2="105" stroke="blue" stroke-width="2"></line><path d="M 150,25 A 15 15, 0, 0, 1, 163 15" fill="none" stroke="red" stroke-width="1.5"></path></svg><p><strong>По гипотенузе и катету:</strong> Если гипотенуза и катет одного прямоугольного треугольника соответственно равны гипотенузе и катету другого, то такие треугольники равны.</p><svg width="260" height="110" xmlns="http://www.w3.org/2000/svg"><polygon points="10,100 110,100 10,10" style="fill:none;stroke:black;stroke-width:2"></polygon><rect x="10" y="80" width="20" height="20" style="fill:none;stroke:black;stroke-width:1"></rect><line x1="50" y1="45" x2="70" y2="65" stroke="blue" stroke-width="2" transform="rotate(-45 60 55)"></line><line x1="55" y1="105" x2="65" y2="105" stroke="red" stroke-width="2"></line><polygon points="150,100 250,100 150,10" style="fill:none;stroke:black;stroke-width:2"></polygon><rect x="150" y="80" width="20" height="20" style="fill:none;stroke:black;stroke-width:1"></rect><line x1="190" y1="45" x2="210" y2="65" stroke="blue" stroke-width="2" transform="rotate(-45 200 55)"></line><line x1="195" y1="105" x2="205" y2="105" stroke="red" stroke-width="2"></line></svg><p>Ответ: Прямоугольные треугольники равны, если выполняется одно из условий: 1) равны их два катета; 2) равны катет и прилежащий острый угол; 3) равны гипотенуза и острый угол; 4) равны катет и противолежащий острый угол; 5) равны гипотенуза и катет.</p>" ] #original: array:6 [ "id" => 1541775 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1112} "task" => array:2 [ "refs" => "1129679" "type" => "task" ] "text" => "<p><strong>1.</strong> Прямоугольным называется треугольник, у которого один из углов является прямым, то есть его градусная мера составляет $90^\circ$. Прямой угол на чертежах принято обозначать небольшим квадратом.</p><svg width="220" height="150" xmlns="http://www.w3.org/2000/svg"><polygon points="20,130 180,130 20,20" style="fill:rgb(240, 248, 255);stroke:black;stroke-width:2"></polygon><rect x="20" y="110" width="20" height="20" style="fill:none;stroke:black;stroke-width:1.5"></rect><text x="10" y="25" font-family="sans-serif" font-size="16">A</text><text x="10" y="145" font-family="sans-serif" font-size="16">C</text><text x="185" y="145" font-family="sans-serif" font-size="16">B</text><text x="45" y="105" font-family="sans-serif" font-size="14">Угол $\angle C = 90^\circ$</text></svg><p>Ответ: Прямоугольный треугольник — это треугольник, в котором один угол прямой ($90^\circ$).</p><p><strong>2.</strong> Стороны в прямоугольном треугольнике носят особые названия:</p><p>• <strong>Катеты</strong> — это две стороны, образующие прямой угол. На рисунке ниже стороны $AC$ и $BC$ являются катетами.</p><p>• <strong>Гипотенуза</strong> — это сторона, которая лежит напротив прямого угла. Гипотенуза всегда является самой длинной стороной прямоугольного треугольника. На рисунке ниже сторона $AB$ является гипотенузой.</p><svg width="280" height="180" xmlns="http://www.w3.org/2000/svg"><polygon points="30,150 220,150 30,30" style="fill:rgb(240, 248, 255);stroke:black;stroke-width:2"></polygon><rect x="30" y="130" width="20" height="20" style="fill:none;stroke:black;stroke-width:1.5"></rect><text x="15" y="35" font-family="sans-serif" font-size="16">A</text><text x="15" y="165" font-family="sans-serif" font-size="16">C</text><text x="225" y="165" font-family="sans-serif" font-size="16">B</text><text x="45" y="90" font-family="sans-serif" font-size="16" fill="blue" transform="rotate(-90 45 90)">Катет</text><text x="110" y="165" font-family="sans-serif" font-size="16" fill="blue">Катет</text><text x="110" y="80" font-family="sans-serif" font-size="16" fill="red" transform="rotate(-35 125 80)">Гипотенуза</text></svg><p>Ответ: Катеты — стороны, образующие прямой угол; гипотенуза — сторона, лежащая напротив прямого угла.</p><p><strong>3.</strong> Признаки равенства прямоугольных треугольников — это специальные теоремы, позволяющие установить равенство двух прямоугольных треугольников по меньшему количеству элементов, чем в общих признаках.</p><p><strong>По двум катетам:</strong> Если катеты одного прямоугольного треугольника соответственно равны катетам другого, то такие треугольники равны.</p><svg width="260" height="110" xmlns="http://www.w3.org/2000/svg"><polygon points="10,100 110,100 10,10" style="fill:none;stroke:black;stroke-width:2"></polygon><rect x="10" y="80" width="20" height="20" style="fill:none;stroke:black;stroke-width:1"></rect><line x1="55" y1="105" x2="65" y2="105" stroke="blue" stroke-width="2"></line><line x1="5" y1="50" x2="5" y2="60" stroke="red" stroke-width="2"></line><line x1="5" y1="65" x2="5" y2="75" stroke="red" stroke-width="2"></line><polygon points="150,100 250,100 150,10" style="fill:none;stroke:black;stroke-width:2"></polygon><rect x="150" y="80" width="20" height="20" style="fill:none;stroke:black;stroke-width:1"></rect><line x1="195" y1="105" x2="205" y2="105" stroke="blue" stroke-width="2"></line><line x1="145" y1="50" x2="145" y2="60" stroke="red" stroke-width="2"></line><line x1="145" y1="65" x2="145" y2="75" stroke="red" stroke-width="2"></line></svg><p><strong>По катету и прилежащему острому углу:</strong> Если катет и прилежащий к нему острый угол одного прямоугольного треугольника соответственно равны катету и прилежащему острому углу другого, то такие треугольники равны.</p><svg width="260" height="110" xmlns="http://www.w3.org/2000/svg"><polygon points="10,100 110,100 10,10" style="fill:none;stroke:black;stroke-width:2"></polygon><rect x="10" y="80" width="20" height="20" style="fill:none;stroke:black;stroke-width:1"></rect><line x1="55" y1="105" x2="65" y2="105" stroke="blue" stroke-width="2"></line><path d="M 90,100 A 15 15, 0, 0, 0, 100 88" fill="none" stroke="red" stroke-width="1.5"></path><polygon points="150,100 250,100 150,10" style="fill:none;stroke:black;stroke-width:2"></polygon><rect x="150" y="80" width="20" height="20" style="fill:none;stroke:black;stroke-width:1"></rect><line x1="195" y1="105" x2="205" y2="105" stroke="blue" stroke-width="2"></line><path d="M 230,100 A 15 15, 0, 0, 0, 240 88" fill="none" stroke="red" stroke-width="1.5"></path></svg><p><strong>По гипотенузе и острому углу:</strong> Если гипотенуза и острый угол одного прямоугольного треугольника соответственно равны гипотенузе и острому углу другого, то такие треугольники равны.</p><svg width="260" height="110" xmlns="http://www.w3.org/2000/svg"><polygon points="10,100 110,100 10,10" style="fill:none;stroke:black;stroke-width:2"></polygon><rect x="10" y="80" width="20" height="20" style="fill:none;stroke:black;stroke-width:1"></rect><line x1="50" y1="45" x2="70" y2="65" stroke="blue" stroke-width="2" transform="rotate(-45 60 55)"></line><path d="M 10,25 A 15 15, 0, 0, 1, 23 15" fill="none" stroke="red" stroke-width="1.5"></path><polygon points="150,100 250,100 150,10" style="fill:none;stroke:black;stroke-width:2"></polygon><rect x="150" y="80" width="20" height="20" style="fill:none;stroke:black;stroke-width:1"></rect><line x1="190" y1="45" x2="210" y2="65" stroke="blue" stroke-width="2" transform="rotate(-45 200 55)"></line><path d="M 150,25 A 15 15, 0, 0, 1, 163 15" fill="none" stroke="red" stroke-width="1.5"></path></svg><p><strong>По катету и противолежащему острому углу:</strong> Если катет и противолежащий ему острый угол одного прямоугольного треугольника соответственно равны катету и противолежащему ему острому углу другого, то такие треугольники равны.</p><svg width="260" height="110" xmlns="http://www.w3.org/2000/svg"><polygon points="10,100 110,100 10,10" style="fill:none;stroke:black;stroke-width:2"></polygon><rect x="10" y="80" width="20" height="20" style="fill:none;stroke:black;stroke-width:1"></rect><line x1="55" y1="105" x2="65" y2="105" stroke="blue" stroke-width="2"></line><path d="M 10,25 A 15 15, 0, 0, 1, 23 15" fill="none" stroke="red" stroke-width="1.5"></path><polygon points="150,100 250,100 150,10" style="fill:none;stroke:black;stroke-width:2"></polygon><rect x="150" y="80" width="20" height="20" style="fill:none;stroke:black;stroke-width:1"></rect><line x1="195" y1="105" x2="205" y2="105" stroke="blue" stroke-width="2"></line><path d="M 150,25 A 15 15, 0, 0, 1, 163 15" fill="none" stroke="red" stroke-width="1.5"></path></svg><p><strong>По гипотенузе и катету:</strong> Если гипотенуза и катет одного прямоугольного треугольника соответственно равны гипотенузе и катету другого, то такие треугольники равны.</p><svg width="260" height="110" xmlns="http://www.w3.org/2000/svg"><polygon points="10,100 110,100 10,10" style="fill:none;stroke:black;stroke-width:2"></polygon><rect x="10" y="80" width="20" height="20" style="fill:none;stroke:black;stroke-width:1"></rect><line x1="50" y1="45" x2="70" y2="65" stroke="blue" stroke-width="2" transform="rotate(-45 60 55)"></line><line x1="55" y1="105" x2="65" y2="105" stroke="red" stroke-width="2"></line><polygon points="150,100 250,100 150,10" style="fill:none;stroke:black;stroke-width:2"></polygon><rect x="150" y="80" width="20" height="20" style="fill:none;stroke:black;stroke-width:1"></rect><line x1="190" y1="45" x2="210" y2="65" stroke="blue" stroke-width="2" transform="rotate(-45 200 55)"></line><line x1="195" y1="105" x2="205" y2="105" stroke="red" stroke-width="2"></line></svg><p>Ответ: Прямоугольные треугольники равны, если выполняется одно из условий: 1) равны их два катета; 2) равны катет и прилежащий острый угол; 3) равны гипотенуза и острый угол; 4) равны катет и противолежащий острый угол; 5) равны гипотенуза и катет.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1129680" "type" => "task" ] "previous" => array:2 [ "refs" => "1129678" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1102 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1128 #items: array:1 [ 0 => App\Models\BookPage {#1127 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 1114391 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "74" "field_url" => "/7-klass/geometrija/smirnova-uchebnik/page-74" "field_display_title" => "74" "field_folder" => "1" "field_image_name" => "74" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1129 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1130 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1131 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1141 #items: array:1 [ 0 => App\Models\Element {#1140 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1290578 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1142 #items: array:1 [ 0 => App\Models\Edition {#1098} ] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1114391" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "74-1.jpg" "alt" => null "width" => "1889" "height" => 2818 "path" => "/media/geometrija_07/Smirnova-u17/0-1/74-1.webp?ts=1753260158" ] ] ] #original: array:6 [ "id" => 1290578 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1142} "book_page" => array:2 [ "refs" => "1114391" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "74-1.jpg" "alt" => null "width" => "1889" "height" => 2818 "path" => "/media/geometrija_07/Smirnova-u17/0-1/74-1.webp?ts=1753260158" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1114392" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1114390" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1210 #items: array:2 [ 0 => App\Models\Task {#1218 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1129679 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "74" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/vopr-13" "field_display_title" => "Вопросы" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1219 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1220 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1221 #items: array:1 [ 0 => App\Models\Branch {#1117} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1222 #items: array:2 [ 0 => App\Models\Branch {#1117} 1 => App\Models\Branch {#1121} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1223 #items: array:3 [ 0 => App\Models\Element {#1099} 1 => App\Models\Element {#1093} 2 => App\Models\Element {#1051} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1129680" "type" => "task" ] "previous" => array:2 [ "refs" => "1129678" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1224 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1114391" "type" => "book_page" ] ] #original: array:24 [ "id" => 1129679 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "74" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/vopr-13" "field_display_title" => "Вопросы" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1219} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1220} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1221} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1222} "content" => Illuminate\Database\Eloquent\Collection {#1223} "next" => array:2 [ "refs" => "1129680" "type" => "task" ] "previous" => array:2 [ "refs" => "1129678" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1224} "page" => array:2 [ "refs" => "1114391" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Task {#1225 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1129680 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "74" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/13-1" "field_display_title" => "13.1" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1229 #items: array:1 [ 0 => App\Models\Term {#1226 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1228 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1227 #items: array:1 [ 0 => App\Models\Branch {#1117} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1230 #items: array:2 [ 0 => App\Models\Branch {#1117} 1 => App\Models\Branch {#1121} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1233 #items: array:3 [ 0 => App\Models\Element {#1242 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 1319735 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1243 #items: array:1 [ 0 => App\Models\Edition {#1098} ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1129680" "type" => "task" ] "text" => "<p><strong>13.1.</strong> Изобразите какой-нибудь прямоугольный треугольник, катетом которого является отрезок $AC$, а вершина $B$ находится в одном из узлов сетки (рис. 13.5).</p><p><strong>а)</strong></p><p><strong>б)</strong></p><p>Рис. 13.5</p>" "img" => array:2 [ 0 => array:5 [ "name" => "1-1.jpg" "alt" => null "width" => "1741" "height" => 308 "path" => "/media/geometrija_07/Smirnova-u17/0-13/1-1.webp?ts=1753982356" ] 1 => array:5 [ "name" => "1-2.jpg" "alt" => null "width" => "1136" "height" => 647 "path" => "/media/geometrija_07/Smirnova-u17/0-13/1-2.webp?ts=1753982356" ] ] ] #original: array:7 [ "id" => 1319735 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1243} "task" => array:2 [ "refs" => "1129680" "type" => "task" ] "text" => "<p><strong>13.1.</strong> Изобразите какой-нибудь прямоугольный треугольник, катетом которого является отрезок $AC$, а вершина $B$ находится в одном из узлов сетки (рис. 13.5).</p><p><strong>а)</strong></p><p><strong>б)</strong></p><p>Рис. 13.5</p>" "img" => array:2 [ 0 => array:5 [ "name" => "1-1.jpg" "alt" => null "width" => "1741" "height" => 308 "path" => "/media/geometrija_07/Smirnova-u17/0-13/1-1.webp?ts=1753982356" ] 1 => array:5 [ "name" => "1-2.jpg" "alt" => null "width" => "1136" "height" => 647 "path" => "/media/geometrija_07/Smirnova-u17/0-13/1-2.webp?ts=1753982356" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1244 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1320263 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1245 #items: array:1 [ 0 => App\Models\Edition {#1090} ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1129680" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "1-1.jpg" "alt" => null "width" => "1420" "height" => 639 "path" => "/media/geometrija_07/Smirnova-u17/1-13/1-1.webp?ts=1753982471" ] ] ] #original: array:6 [ "id" => 1320263 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1245} "task" => array:2 [ "refs" => "1129680" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "1-1.jpg" "alt" => null "width" => "1420" "height" => 639 "path" => "/media/geometrija_07/Smirnova-u17/1-13/1-1.webp?ts=1753982471" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Element {#1246 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1541525 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1247 #items: array:1 [ 0 => App\Models\Edition {#1050} ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1129680" "type" => "task" ] "text" => "<p>Чтобы построить прямоугольный треугольник, в котором отрезок $AC$ является катетом, необходимо, чтобы прямой угол находился в одной из его вершин, то есть в точке $A$ или в точке $C$. Это означает, что второй катет (соответственно, $AB$ или $CB$) должен быть перпендикулярен катету $AC$.</p><p>Для построения перпендикулярного отрезка на сетке можно использовать простое правило, основанное на векторах или угловых коэффициентах. Если отрезок $AC$ строится смещением на $\Delta x$ клеток по горизонтали и на $\Delta y$ клеток по вертикали, то перпендикулярный ему отрезок такой же длины будет строиться смещением на $\Delta y$ клеток по горизонтали и $(-\Delta x)$ клеток по вертикали (или на $(-\Delta y)$ и $\Delta x$ клеток соответственно).</p><p><strong>а)</strong> Введем систему координат с началом в левом верхнем узле сетки, ось $Ox$ направим вправо, а ось $Oy$ — вниз. В этой системе вершина $A$ имеет координаты $(1, 1)$, а вершина $C$ — $(3, 3)$.</p><p>Чтобы переместиться из точки $A$ в точку $C$, нужно сместиться на $3-1=2$ клетки вправо и на $3-1=2$ клетки вниз. Таким образом, вектор $\vec{AC}$ имеет координаты $(2, 2)$.</p><p>Вектор, перпендикулярный вектору $(2, 2)$, будет иметь координаты $(2, -2)$ или $(-2, 2)$. Для удобства можно использовать и коллинеарные им векторы, например, $(1, -1)$ и $(-1, 1)$.</p><p>Рассмотрим два случая:</p><p><strong>1.</strong> <i>Прямой угол в вершине A.</i> Искомую вершину $B$ можно найти, отложив от точки $A(1, 1)$ перпендикулярный вектор. <ul><li>Отложив вектор $(-1, 1)$, получим точку $B(1-1, 1+1) = B(0, 2)$.</li><li>Отложив вектор $(1, -1)$, получим точку $B(1+1, 1-1) = B(2, 0)$.</li></ul></p><p><strong>2.</strong> <i>Прямой угол в вершине C.</i> Искомую вершину $B$ можно найти, отложив от точки $C(3, 3)$ перпендикулярный вектор. <ul><li>Отложив вектор $(-1, 1)$, получим точку $B(3-1, 3+1) = B(2, 4)$.</li><li>Отложив вектор $(1, -1)$, получим точку $B(3+1, 3-1) = B(4, 2)$.</li></ul></p><p>Любая из найденных четырех точек — $(0, 2)$, $(2, 0)$, $(2, 4)$, $(4, 2)$ — может быть вершиной $B$. Выберем, например, точку $B(2,4)$. В этом случае треугольник $ABC$ будет прямоугольным с прямым углом при вершине $C$.</p><p>Ответ: <svg width="150" height="150" viewBox="0 0 150 150" xmlns="http://www.w3.org/2000/svg" style="vertical-align: middle;"><defs><style> .grid { stroke: #b0e0e6; stroke-width: 1; } .line-initial { stroke: #ff0000; stroke-width: 2; } .line-solution { stroke: #0000ff; stroke-width: 2; } .point-label { font-family: sans-serif; font-size: 14px; fill: black; } .right-angle { stroke: #333; stroke-width: 1.5; fill: none; } </style></defs><!-- Grid --><path class="grid" d="M 0 0 H 150 M 0 30 H 150 M 0 60 H 150 M 0 90 H 150 M 0 120 H 150 M 0 150 H 150" /><path class="grid" d="M 0 0 V 150 M 30 0 V 150 M 60 0 V 150 M 90 0 V 150 M 120 0 V 150 M 150 0 V 150" /><!-- Triangle --><line x1="30" y1="30" x2="90" y2="90" class="line-initial" /><line x1="90" y1="90" x2="60" y2="120" class="line-solution" /><line x1="30" y1="30" x2="60" y2="120" class="line-solution" /><!-- Right angle symbol at C --><path d="M 80 90 L 80 100 L 90 100" class="right-angle"/><!-- Points and Labels --><circle cx="30" cy="30" r="3" fill="black" /><text x="20" y="25" class="point-label">A</text><circle cx="90" cy="90" r="3" fill="black" /><text x="95" y="85" class="point-label">C</text><circle cx="60" cy="120" r="3" fill="black" /><text x="50" y="138" class="point-label">B</text></svg></p><p><strong>б)</strong> Аналогично, введем систему координат с началом в левом верхнем узле сетки. В этой системе вершина $A$ имеет координаты $(2, 1)$, а вершина $C$ — $(3, 3)$.</p><p>Чтобы переместиться из точки $A$ в точку $C$, нужно сместиться на $3-2=1$ клетку вправо и на $3-1=2$ клетки вниз. Таким образом, вектор $\vec{AC}$ имеет координаты $(1, 2)$.</p><p>Вектор, перпендикулярный вектору $(1, 2)$, будет иметь координаты $(-2, 1)$ или $(2, -1)$.</p><p>Рассмотрим два случая:</p><p><strong>1.</strong> <i>Прямой угол в вершине A.</i> Отложим от точки $A(2, 1)$ перпендикулярный вектор. <ul><li>Отложив вектор $(-2, 1)$, получим точку $B(2-2, 1+1) = B(0, 2)$.</li><li>Отложив вектор $(2, -1)$, получим точку $B(2+2, 1-1) = B(4, 0)$.</li></ul></p><p><strong>2.</strong> <i>Прямой угол в вершине C.</i> Отложим от точки $C(3, 3)$ перпендикулярный вектор. <ul><li>Отложив вектор $(-2, 1)$, получим точку $B(3-2, 3+1) = B(1, 4)$.</li><li>Отложив вектор $(2, -1)$, получим точку $B(3+2, 3-1) = B(5, 2)$.</li></ul></p><p>Любая из найденных четырех точек — $(0, 2)$, $(4, 0)$, $(1, 4)$, $(5, 2)$ — может быть вершиной $B$. Выберем, например, точку $B(1,4)$. В этом случае треугольник $ABC$ будет прямоугольным с прямым углом при вершине $C$.</p><p>Ответ: <svg width="180" height="150" viewBox="0 0 180 150" xmlns="http://www.w3.org/2000/svg" style="vertical-align: middle;"><defs><style> .grid { stroke: #b0e0e6; stroke-width: 1; } .line-initial { stroke: #ff0000; stroke-width: 2; } .line-solution { stroke: #0000ff; stroke-width: 2; } .point-label { font-family: sans-serif; font-size: 14px; fill: black; } .right-angle { stroke: #333; stroke-width: 1.5; fill: none; } </style></defs><!-- Grid --><path class="grid" d="M 0 0 H 180 M 0 30 H 180 M 0 60 H 180 M 0 90 H 180 M 0 120 H 180 M 0 150 H 180" /><path class="grid" d="M 0 0 V 150 M 30 0 V 150 M 60 0 V 150 M 90 0 V 150 M 120 0 V 150 M 150 0 V 150" /><!-- Triangle --><line x1="60" y1="30" x2="90" y2="90" class="line-initial" /><line x1="90" y1="90" x2="30" y2="120" class="line-solution" /><line x1="60" y1="30" x2="30" y2="120" class="line-solution" /><!-- Right angle symbol at C --><path d="M 80 90 L 80 100 L 90 100" class="right-angle"/><!-- Points and Labels --><circle cx="60" cy="30" r="3" fill="black" /><text x="65" y="28" class="point-label">A</text><circle cx="90" cy="90" r="3" fill="black" /><text x="95" y="85" class="point-label">C</text><circle cx="30" cy="120" r="3" fill="black" /><text x="20" y="138" class="point-label">B</text></svg></p>" ] #original: array:6 [ "id" => 1541525 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1247} "task" => array:2 [ "refs" => "1129680" "type" => "task" ] "text" => "<p>Чтобы построить прямоугольный треугольник, в котором отрезок $AC$ является катетом, необходимо, чтобы прямой угол находился в одной из его вершин, то есть в точке $A$ или в точке $C$. Это означает, что второй катет (соответственно, $AB$ или $CB$) должен быть перпендикулярен катету $AC$.</p><p>Для построения перпендикулярного отрезка на сетке можно использовать простое правило, основанное на векторах или угловых коэффициентах. Если отрезок $AC$ строится смещением на $\Delta x$ клеток по горизонтали и на $\Delta y$ клеток по вертикали, то перпендикулярный ему отрезок такой же длины будет строиться смещением на $\Delta y$ клеток по горизонтали и $(-\Delta x)$ клеток по вертикали (или на $(-\Delta y)$ и $\Delta x$ клеток соответственно).</p><p><strong>а)</strong> Введем систему координат с началом в левом верхнем узле сетки, ось $Ox$ направим вправо, а ось $Oy$ — вниз. В этой системе вершина $A$ имеет координаты $(1, 1)$, а вершина $C$ — $(3, 3)$.</p><p>Чтобы переместиться из точки $A$ в точку $C$, нужно сместиться на $3-1=2$ клетки вправо и на $3-1=2$ клетки вниз. Таким образом, вектор $\vec{AC}$ имеет координаты $(2, 2)$.</p><p>Вектор, перпендикулярный вектору $(2, 2)$, будет иметь координаты $(2, -2)$ или $(-2, 2)$. Для удобства можно использовать и коллинеарные им векторы, например, $(1, -1)$ и $(-1, 1)$.</p><p>Рассмотрим два случая:</p><p><strong>1.</strong> <i>Прямой угол в вершине A.</i> Искомую вершину $B$ можно найти, отложив от точки $A(1, 1)$ перпендикулярный вектор. <ul><li>Отложив вектор $(-1, 1)$, получим точку $B(1-1, 1+1) = B(0, 2)$.</li><li>Отложив вектор $(1, -1)$, получим точку $B(1+1, 1-1) = B(2, 0)$.</li></ul></p><p><strong>2.</strong> <i>Прямой угол в вершине C.</i> Искомую вершину $B$ можно найти, отложив от точки $C(3, 3)$ перпендикулярный вектор. <ul><li>Отложив вектор $(-1, 1)$, получим точку $B(3-1, 3+1) = B(2, 4)$.</li><li>Отложив вектор $(1, -1)$, получим точку $B(3+1, 3-1) = B(4, 2)$.</li></ul></p><p>Любая из найденных четырех точек — $(0, 2)$, $(2, 0)$, $(2, 4)$, $(4, 2)$ — может быть вершиной $B$. Выберем, например, точку $B(2,4)$. В этом случае треугольник $ABC$ будет прямоугольным с прямым углом при вершине $C$.</p><p>Ответ: <svg width="150" height="150" viewBox="0 0 150 150" xmlns="http://www.w3.org/2000/svg" style="vertical-align: middle;"><defs><style> .grid { stroke: #b0e0e6; stroke-width: 1; } .line-initial { stroke: #ff0000; stroke-width: 2; } .line-solution { stroke: #0000ff; stroke-width: 2; } .point-label { font-family: sans-serif; font-size: 14px; fill: black; } .right-angle { stroke: #333; stroke-width: 1.5; fill: none; } </style></defs><!-- Grid --><path class="grid" d="M 0 0 H 150 M 0 30 H 150 M 0 60 H 150 M 0 90 H 150 M 0 120 H 150 M 0 150 H 150" /><path class="grid" d="M 0 0 V 150 M 30 0 V 150 M 60 0 V 150 M 90 0 V 150 M 120 0 V 150 M 150 0 V 150" /><!-- Triangle --><line x1="30" y1="30" x2="90" y2="90" class="line-initial" /><line x1="90" y1="90" x2="60" y2="120" class="line-solution" /><line x1="30" y1="30" x2="60" y2="120" class="line-solution" /><!-- Right angle symbol at C --><path d="M 80 90 L 80 100 L 90 100" class="right-angle"/><!-- Points and Labels --><circle cx="30" cy="30" r="3" fill="black" /><text x="20" y="25" class="point-label">A</text><circle cx="90" cy="90" r="3" fill="black" /><text x="95" y="85" class="point-label">C</text><circle cx="60" cy="120" r="3" fill="black" /><text x="50" y="138" class="point-label">B</text></svg></p><p><strong>б)</strong> Аналогично, введем систему координат с началом в левом верхнем узле сетки. В этой системе вершина $A$ имеет координаты $(2, 1)$, а вершина $C$ — $(3, 3)$.</p><p>Чтобы переместиться из точки $A$ в точку $C$, нужно сместиться на $3-2=1$ клетку вправо и на $3-1=2$ клетки вниз. Таким образом, вектор $\vec{AC}$ имеет координаты $(1, 2)$.</p><p>Вектор, перпендикулярный вектору $(1, 2)$, будет иметь координаты $(-2, 1)$ или $(2, -1)$.</p><p>Рассмотрим два случая:</p><p><strong>1.</strong> <i>Прямой угол в вершине A.</i> Отложим от точки $A(2, 1)$ перпендикулярный вектор. <ul><li>Отложив вектор $(-2, 1)$, получим точку $B(2-2, 1+1) = B(0, 2)$.</li><li>Отложив вектор $(2, -1)$, получим точку $B(2+2, 1-1) = B(4, 0)$.</li></ul></p><p><strong>2.</strong> <i>Прямой угол в вершине C.</i> Отложим от точки $C(3, 3)$ перпендикулярный вектор. <ul><li>Отложив вектор $(-2, 1)$, получим точку $B(3-2, 3+1) = B(1, 4)$.</li><li>Отложив вектор $(2, -1)$, получим точку $B(3+2, 3-1) = B(5, 2)$.</li></ul></p><p>Любая из найденных четырех точек — $(0, 2)$, $(4, 0)$, $(1, 4)$, $(5, 2)$ — может быть вершиной $B$. Выберем, например, точку $B(1,4)$. В этом случае треугольник $ABC$ будет прямоугольным с прямым углом при вершине $C$.</p><p>Ответ: <svg width="180" height="150" viewBox="0 0 180 150" xmlns="http://www.w3.org/2000/svg" style="vertical-align: middle;"><defs><style> .grid { stroke: #b0e0e6; stroke-width: 1; } .line-initial { stroke: #ff0000; stroke-width: 2; } .line-solution { stroke: #0000ff; stroke-width: 2; } .point-label { font-family: sans-serif; font-size: 14px; fill: black; } .right-angle { stroke: #333; stroke-width: 1.5; fill: none; } </style></defs><!-- Grid --><path class="grid" d="M 0 0 H 180 M 0 30 H 180 M 0 60 H 180 M 0 90 H 180 M 0 120 H 180 M 0 150 H 180" /><path class="grid" d="M 0 0 V 150 M 30 0 V 150 M 60 0 V 150 M 90 0 V 150 M 120 0 V 150 M 150 0 V 150" /><!-- Triangle --><line x1="60" y1="30" x2="90" y2="90" class="line-initial" /><line x1="90" y1="90" x2="30" y2="120" class="line-solution" /><line x1="60" y1="30" x2="30" y2="120" class="line-solution" /><!-- Right angle symbol at C --><path d="M 80 90 L 80 100 L 90 100" class="right-angle"/><!-- Points and Labels --><circle cx="60" cy="30" r="3" fill="black" /><text x="65" y="28" class="point-label">A</text><circle cx="90" cy="90" r="3" fill="black" /><text x="95" y="85" class="point-label">C</text><circle cx="30" cy="120" r="3" fill="black" /><text x="20" y="138" class="point-label">B</text></svg></p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1129681" "type" => "task" ] "previous" => array:2 [ "refs" => "1129679" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1239 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1114391" "type" => "book_page" ] ] #original: array:24 [ "id" => 1129680 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "74" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/13-1" "field_display_title" => "13.1" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1229} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1228} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1227} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1230} "content" => Illuminate\Database\Eloquent\Collection {#1233} "next" => array:2 [ "refs" => "1129681" "type" => "task" ] "previous" => array:2 [ "refs" => "1129679" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1239} "page" => array:2 [ "refs" => "1114391" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 1114391 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "74" "field_url" => "/7-klass/geometrija/smirnova-uchebnik/page-74" "field_display_title" => "74" "field_folder" => "1" "field_image_name" => "74" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1129} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1130} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => 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"field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1034} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1045} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1109} "content" => Illuminate\Database\Eloquent\Collection {#1106} "next" => array:2 [ "refs" => "1129680" "type" => "task" ] "previous" => array:2 [ "refs" => "1129678" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1102} "page" => array:2 [ "refs" => "1114391" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] }
Вопросы (с. 74)
Условие. Вопросы (с. 74)
скриншот условия
1. Какой треугольник называется прямоугольным?
2. Что называется гипотенузой и катетами прямоугольного треугольника?
3. Сформулируйте признаки равенства прямоугольных треугольников.
Решение. Вопросы (с. 74)
Решение 2. Вопросы (с. 74)
1. Прямоугольным называется треугольник, у которого один из углов является прямым, то есть его градусная мера составляет $90^\circ$. Прямой угол на чертежах принято обозначать небольшим квадратом.
Ответ: Прямоугольный треугольник — это треугольник, в котором один угол прямой ($90^\circ$).
2. Стороны в прямоугольном треугольнике носят особые названия:
• Катеты — это две стороны, образующие прямой угол. На рисунке ниже стороны $AC$ и $BC$ являются катетами.
• Гипотенуза — это сторона, которая лежит напротив прямого угла. Гипотенуза всегда является самой длинной стороной прямоугольного треугольника. На рисунке ниже сторона $AB$ является гипотенузой.
Ответ: Катеты — стороны, образующие прямой угол; гипотенуза — сторона, лежащая напротив прямого угла.
3. Признаки равенства прямоугольных треугольников — это специальные теоремы, позволяющие установить равенство двух прямоугольных треугольников по меньшему количеству элементов, чем в общих признаках.
По двум катетам: Если катеты одного прямоугольного треугольника соответственно равны катетам другого, то такие треугольники равны.
По катету и прилежащему острому углу: Если катет и прилежащий к нему острый угол одного прямоугольного треугольника соответственно равны катету и прилежащему острому углу другого, то такие треугольники равны.
По гипотенузе и острому углу: Если гипотенуза и острый угол одного прямоугольного треугольника соответственно равны гипотенузе и острому углу другого, то такие треугольники равны.
По катету и противолежащему острому углу: Если катет и противолежащий ему острый угол одного прямоугольного треугольника соответственно равны катету и противолежащему ему острому углу другого, то такие треугольники равны.
По гипотенузе и катету: Если гипотенуза и катет одного прямоугольного треугольника соответственно равны гипотенузе и катету другого, то такие треугольники равны.
Ответ: Прямоугольные треугольники равны, если выполняется одно из условий: 1) равны их два катета; 2) равны катет и прилежащий острый угол; 3) равны гипотенуза и острый угол; 4) равны катет и противолежащий острый угол; 5) равны гипотенуза и катет.
Другие задания:
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