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Номер 23.4, страница 132 - 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 4. Окружность. Геометрические построения. Параграф 23. Задачи на построение - номер 23.4, страница 132.
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" => 1129902 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "132" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/23-4" "field_display_title" => "23.4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #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 {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1071 #items: array:1 [ 0 => App\Models\Branch {#1034 #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" => 1129379 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Окружность. Геометрические построения" "field_branch_order" => "4" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039 #items: array:1 [ 0 => App\Models\Term {#1038 #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" => "101" "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 {#1040 #items: array:1 [ 0 => App\Models\Book {#1041 #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 {#1042 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1044 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1048 #items: array:2 [ …2] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1051 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1052 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1054 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1056 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1058 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1059 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1060 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1061 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1062 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1063 #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 [ …4] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1064 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1065 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1066 #items: [] #escapeWhenCastingToString: false } "field_url" => "/7-klass/geometrija/smirnova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1067 #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 {#1042} "field_class" => Illuminate\Database\Eloquent\Collection {#1044} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046} "field_author" => Illuminate\Database\Eloquent\Collection {#1048} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1051} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1052} "field_country" => Illuminate\Database\Eloquent\Collection {#1054} "field_city" => Illuminate\Database\Eloquent\Collection {#1056} "field_series" => Illuminate\Database\Eloquent\Collection {#1058} "field_umk" => Illuminate\Database\Eloquent\Collection {#1059} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1060} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1061} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1062} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1063} "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 [ …4] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1064} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1065} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1066} "field_url" => "/7-klass/geometrija/smirnova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1067} "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 {#1068 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1129379 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Окружность. Геометрические построения" "field_branch_order" => "4" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039} "field_page_start" => "101" "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 {#1040} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1068} ] #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 {#1104 #items: array:2 [ 0 => App\Models\Branch {#1112 #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" => 1129379 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Окружность. Геометрические построения" "field_branch_order" => "4" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1116 #items: array:1 [ 0 => App\Models\Term {#1113 #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" …5 ] "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 [ …6] "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" => "101" "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 {#1144 #items: array:1 [ 0 => App\Models\Book {#1115 #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 {#1117 #items: array:1 [ 0 => App\Models\Term {#1114 #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 [ …6] "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 [ …6] "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 {#1118 #items: array:1 [ 0 => App\Models\Term {#1119 #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 [ …6] "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 [ …6] "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 {#1120 #items: array:1 [ 0 => App\Models\Term {#1121 #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 [ …6] "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 [ …6] "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 {#1122 #items: array:2 [ 0 => App\Models\Term {#1123 #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 {#1124 #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 {#1125 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1126 #items: array:1 [ 0 => App\Models\Term {#1127 #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 [ …6] "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 [ …6] "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 {#1128 #items: array:1 [ 0 => App\Models\Term {#1129 #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 [ …6] "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 [ …6] "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 {#1130 #items: array:1 [ 0 => App\Models\Term {#1131 #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 [ …6] "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 [ …6] "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 {#1132 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1133 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1134 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1135 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1136 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1137 #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 {#1138 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1139 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1140 #items: [] #escapeWhenCastingToString: false } "field_url" => "/7-klass/geometrija/smirnova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1141 #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 {#1117} "field_class" => Illuminate\Database\Eloquent\Collection {#1118} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1120} "field_author" => Illuminate\Database\Eloquent\Collection {#1122} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1125} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1126} "field_country" => Illuminate\Database\Eloquent\Collection {#1128} "field_city" => Illuminate\Database\Eloquent\Collection {#1130} "field_series" => Illuminate\Database\Eloquent\Collection {#1132} "field_umk" => Illuminate\Database\Eloquent\Collection {#1133} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1134} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1135} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1136} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1137} "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 {#1138} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1139} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1140} "field_url" => "/7-klass/geometrija/smirnova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1141} "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 {#1142 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1129379 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Окружность. Геометрические построения" "field_branch_order" => "4" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1116} "field_page_start" => "101" "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 {#1144} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1142} ] #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 {#1143 #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" => 1129385 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Задачи на построение" "field_branch_order" => "23" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1148 #items: array:1 [ 0 => App\Models\Term {#1145 #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 [ …6] "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 [ …6] "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" => "128" "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 {#1147 #items: array:1 [ 0 => App\Models\Book {#1115} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1146 #items: array:1 [ 0 => App\Models\Branch {#1034} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1129385 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Задачи на построение" "field_branch_order" => "23" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1148} "field_page_start" => "128" "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 {#1147} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1146} ] #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 {#1101 #items: array:3 [ 0 => App\Models\Element {#1094 #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" => 1320052 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1086 #items: array:1 [ 0 => App\Models\Edition {#1093 #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 {#1091 …2} "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 {#1089 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1090 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1087 …2} "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 {#1091 …2} "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 {#1089 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1090 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1087 …2} "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" => "1129902" "type" => "task" ] "text" => "<p><strong>23.4.</strong> Постройте треугольник $ABC$ по двум данным сторонам и углу между ними.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "4-1.jpg" "alt" => null "width" => "1741" "height" => 205 "path" => "/media/geometrija_07/Smirnova-u17/0-23/4-1.webp?ts=1753982488" ] ] ] #original: array:7 [ "id" => 1320052 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1086} "task" => array:2 [ "refs" => "1129902" "type" => "task" ] "text" => "<p><strong>23.4.</strong> Постройте треугольник $ABC$ по двум данным сторонам и углу между ними.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "4-1.jpg" "alt" => null "width" => "1741" "height" => 205 "path" => "/media/geometrija_07/Smirnova-u17/0-23/4-1.webp?ts=1753982488" ] ] ] #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 {#1088 #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" => 1320463 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1078 #items: array:1 [ 0 => App\Models\Edition {#1085 #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 {#1081 …2} "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 {#1083 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1082 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1079 …2} "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 {#1081 …2} "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 {#1083 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1082 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1079 …2} "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" => "1129902" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "4-1.jpg" "alt" => null "width" => "1525" "height" => 741 "path" => "/media/geometrija_07/Smirnova-u17/1-23/4-1.webp?ts=1753982604" ] 1 => array:5 [ "name" => "4-2.jpg" "alt" => null "width" => "1525" "height" => 2353 "path" => "/media/geometrija_07/Smirnova-u17/1-23/4-2.webp?ts=1753982604" ] 2 => array:5 [ "name" => "4-3.jpg" "alt" => null "width" => "1525" "height" => 593 "path" => "/media/geometrija_07/Smirnova-u17/1-23/4-3.webp?ts=1753982604" ] ] ] #original: array:6 [ "id" => 1320463 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1078} "task" => array:2 [ "refs" => "1129902" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "4-1.jpg" "alt" => null "width" => "1525" "height" => 741 "path" => "/media/geometrija_07/Smirnova-u17/1-23/4-1.webp?ts=1753982604" ] 1 => array:5 [ "name" => "4-2.jpg" "alt" => null "width" => "1525" "height" => 2353 "path" => "/media/geometrija_07/Smirnova-u17/1-23/4-2.webp?ts=1753982604" ] 2 => array:5 [ "name" => "4-3.jpg" "alt" => null "width" => "1525" "height" => 593 "path" => "/media/geometrija_07/Smirnova-u17/1-23/4-3.webp?ts=1753982604" ] ] ] #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 {#1080 #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" => 1541725 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1107 #items: array:1 [ 0 => App\Models\Edition {#1077 #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 {#1076 …2} "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 {#1072 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1070 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1103 …2} "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 {#1076 …2} "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 {#1072 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1070 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1103 …2} "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" => "1129902" "type" => "task" ] "text" => "<p>Задача о построении треугольника по двум сторонам и углу между ними является одной из основных задач на построение в курсе геометрии. Для её решения используются циркуль и линейка.</p><p><strong>Построение</strong></p><p>Пусть даны два отрезка с длинами $p_1$ и $p_2$, и угол $\alpha$. Требуется построить треугольник $ABC$ так, чтобы $AB=p_1$, $AC=p_2$ и $\angle BAC = \alpha$.</p><p>Алгоритм построения следующий:<br>1. Проведем произвольную прямую и отметим на ней точку $A$, которая будет одной из вершин треугольника.<br>2. С помощью циркуля отложим на этой прямой от точки $A$ отрезок $AC$, равный по длине отрезку $p_2$.<br>3. От луча $AC$ в заданной полуплоскости отложим угол, равный данному углу $\alpha$. Для этого построим луч $AM$ так, чтобы $\angle CAM = \alpha$.<br>4. На луче $AM$ отложим отрезок $AB$, равный по длине отрезку $p_1$.<br>5. Соединим точки $B$ и $C$ отрезком.<br>Треугольник $ABC$, полученный в результате этих действий, является искомым.</p><svg width="400" height="250" xmlns="http://www.w3.org/2000/svg"><line x1="100" y1="200" x2="350" y2="200" stroke="gray" stroke-width="1" stroke-dasharray="5 3"/><line x1="100" y1="200" x2="29.3" y2="129.3" stroke="gray" stroke-width="1" stroke-dasharray="5 3"/><polygon points="100,200 250,200 29.3,129.3" stroke="black" stroke-width="2.5" fill="rgba(173, 216, 230, 0.4)"/><circle cx="100" cy="200" r="3" fill="black"/><text x="95" y="220" font-family="serif" font-size="20">A</text><circle cx="250" cy="200" r="3" fill="black"/><text x="255" y="220" font-family="serif" font-size="20">C</text><circle cx="29.3" cy="129.3" r="3" fill="black"/><text x="15" y="125" font-family="serif" font-size="20">B</text><text x="170" y="195" font-family="serif" font-size="16" font-style="italic">p₂</text><text x="50" y="155" font-family="serif" font-size="16" font-style="italic">p₁</text><path d="M 130 200 A 30 30 0 0 1 79.3 179.3" stroke="darkred" stroke-width="1.5" fill="none"/><text x="105" y="180" font-family="serif" font-size="16" font-style="italic" fill="darkred">α</text></svg><p><strong>Доказательство</strong></p><p>В построенном треугольнике $ABC$ сторона $AC$ по построению равна $p_2$, сторона $AB$ по построению равна $p_1$, а угол $\angle BAC$ между сторонами $AB$ и $AC$ по построению равен $\alpha$. Таким образом, построенный треугольник полностью удовлетворяет условиям задачи.</p><p><strong>Исследование</strong></p><p>Задача имеет решение, если длины заданных отрезков $p_1$ и $p_2$ являются положительными числами, а заданный угол $\alpha$ больше $0^\circ$ и меньше $180^\circ$ (то есть, неразвернутый угол). При соблюдении этих условий построение всегда возможно. Согласно первому признаку равенства треугольников (по двум сторонам и углу между ними), все треугольники, построенные по заданным элементам, будут равны. Следовательно, задача имеет единственное решение (с точностью до конгруэнтности).</p><p>Ответ: Алгоритм построения искомого треугольника $ABC$ описан выше. Треугольник существует и единственен, если данные стороны имеют положительную длину, а угол больше $0^\circ$ и меньше $180^\circ$.</p>" ] #original: array:6 [ "id" => 1541725 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1107} "task" => array:2 [ "refs" => "1129902" "type" => "task" ] "text" => "<p>Задача о построении треугольника по двум сторонам и углу между ними является одной из основных задач на построение в курсе геометрии. Для её решения используются циркуль и линейка.</p><p><strong>Построение</strong></p><p>Пусть даны два отрезка с длинами $p_1$ и $p_2$, и угол $\alpha$. Требуется построить треугольник $ABC$ так, чтобы $AB=p_1$, $AC=p_2$ и $\angle BAC = \alpha$.</p><p>Алгоритм построения следующий:<br>1. Проведем произвольную прямую и отметим на ней точку $A$, которая будет одной из вершин треугольника.<br>2. С помощью циркуля отложим на этой прямой от точки $A$ отрезок $AC$, равный по длине отрезку $p_2$.<br>3. От луча $AC$ в заданной полуплоскости отложим угол, равный данному углу $\alpha$. Для этого построим луч $AM$ так, чтобы $\angle CAM = \alpha$.<br>4. На луче $AM$ отложим отрезок $AB$, равный по длине отрезку $p_1$.<br>5. Соединим точки $B$ и $C$ отрезком.<br>Треугольник $ABC$, полученный в результате этих действий, является искомым.</p><svg width="400" height="250" xmlns="http://www.w3.org/2000/svg"><line x1="100" y1="200" x2="350" y2="200" stroke="gray" stroke-width="1" stroke-dasharray="5 3"/><line x1="100" y1="200" x2="29.3" y2="129.3" stroke="gray" stroke-width="1" stroke-dasharray="5 3"/><polygon points="100,200 250,200 29.3,129.3" stroke="black" stroke-width="2.5" fill="rgba(173, 216, 230, 0.4)"/><circle cx="100" cy="200" r="3" fill="black"/><text x="95" y="220" font-family="serif" font-size="20">A</text><circle cx="250" cy="200" r="3" fill="black"/><text x="255" y="220" font-family="serif" font-size="20">C</text><circle cx="29.3" cy="129.3" r="3" fill="black"/><text x="15" y="125" font-family="serif" font-size="20">B</text><text x="170" y="195" font-family="serif" font-size="16" font-style="italic">p₂</text><text x="50" y="155" font-family="serif" font-size="16" font-style="italic">p₁</text><path d="M 130 200 A 30 30 0 0 1 79.3 179.3" stroke="darkred" stroke-width="1.5" fill="none"/><text x="105" y="180" font-family="serif" font-size="16" font-style="italic" fill="darkred">α</text></svg><p><strong>Доказательство</strong></p><p>В построенном треугольнике $ABC$ сторона $AC$ по построению равна $p_2$, сторона $AB$ по построению равна $p_1$, а угол $\angle BAC$ между сторонами $AB$ и $AC$ по построению равен $\alpha$. Таким образом, построенный треугольник полностью удовлетворяет условиям задачи.</p><p><strong>Исследование</strong></p><p>Задача имеет решение, если длины заданных отрезков $p_1$ и $p_2$ являются положительными числами, а заданный угол $\alpha$ больше $0^\circ$ и меньше $180^\circ$ (то есть, неразвернутый угол). При соблюдении этих условий построение всегда возможно. Согласно первому признаку равенства треугольников (по двум сторонам и углу между ними), все треугольники, построенные по заданным элементам, будут равны. Следовательно, задача имеет единственное решение (с точностью до конгруэнтности).</p><p>Ответ: Алгоритм построения искомого треугольника $ABC$ описан выше. Треугольник существует и единственен, если данные стороны имеют положительную длину, а угол больше $0^\circ$ и меньше $180^\circ$.</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" => "1129903" "type" => "task" ] "previous" => array:2 [ "refs" => "1129901" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1097 #items: array:1 [ 0 => App\Models\Book {#1115} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1099 #items: array:1 [ 0 => App\Models\BookPage {#1073 #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" => 1114449 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "132" "field_url" => "/7-klass/geometrija/smirnova-uchebnik/page-132" "field_display_title" => "132" "field_folder" => "1" "field_image_name" => "132" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1074 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1102 #items: array:1 [ 0 => App\Models\Book {#1115} ] #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 {#1111 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1155 #items: array:1 [ 0 => App\Models\Element {#1154 #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" => 1290636 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1156 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1290636 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1156 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #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" => "1114450" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1114448" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1253 #items: array:8 [ 0 => App\Models\Task {#1267 #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" => 1129899 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "132" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/23-1" "field_display_title" => "23.1" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1268 …2} "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 {#1269 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1270 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1271 …2} "content" => Illuminate\Database\Eloquent\Collection {#1274 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1280 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1129899 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "132" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/23-1" "field_display_title" => "23.1" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1268 …2} "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 {#1269 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1270 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1271 …2} "content" => Illuminate\Database\Eloquent\Collection {#1274 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1280 …2} "page" => array:2 [ …2] ] #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 {#1273 #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" => 1129900 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "132" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/23-2" "field_display_title" => "23.2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1272 …2} "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 {#1279 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1281 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1277 …2} "content" => Illuminate\Database\Eloquent\Collection {#1276 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1294 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1129900 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "132" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/23-2" "field_display_title" => "23.2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1272 …2} "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 {#1279 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1281 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1277 …2} "content" => Illuminate\Database\Eloquent\Collection {#1276 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1294 …2} "page" => array:2 [ …2] ] #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\Task {#1278 #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" => 1129901 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "132" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/23-3" "field_display_title" => "23.3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1275 …2} "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 {#1293 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1295 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1291 …2} "content" => Illuminate\Database\Eloquent\Collection {#1290 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1308 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1129901 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "132" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/23-3" "field_display_title" => "23.3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1275 …2} "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 {#1293 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1295 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1291 …2} "content" => Illuminate\Database\Eloquent\Collection {#1290 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1308 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Task {#1292 #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" => 1129902 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "132" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/23-4" "field_display_title" => "23.4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1289 …2} "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 {#1307 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1309 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1305 …2} "content" => Illuminate\Database\Eloquent\Collection {#1303 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1306 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1129902 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "132" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/23-4" "field_display_title" => "23.4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1289 …2} "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 {#1307 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1309 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1305 …2} "content" => Illuminate\Database\Eloquent\Collection {#1303 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1306 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Task {#1304 #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" => 1129903 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "132" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/23-5" "field_display_title" => "23.5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1317 …2} "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 {#1318 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1319 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1320 …2} "content" => Illuminate\Database\Eloquent\Collection {#1323 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1329 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1129903 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "132" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/23-5" "field_display_title" => "23.5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1317 …2} "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 {#1318 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1319 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1320 …2} "content" => Illuminate\Database\Eloquent\Collection {#1323 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1329 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 5 => App\Models\Task {#1322 #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" => 1129904 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "132" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/23-6" "field_display_title" => "23.6" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1321 …2} "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 {#1328 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1330 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1326 …2} "content" => Illuminate\Database\Eloquent\Collection {#1325 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1343 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1129904 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "132" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/23-6" "field_display_title" => "23.6" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1321 …2} "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 {#1328 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1330 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1326 …2} "content" => Illuminate\Database\Eloquent\Collection {#1325 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1343 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 6 => App\Models\Task {#1327 #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" => 1129905 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "132" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/23-7" "field_display_title" => "23.7" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1324 …2} "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 {#1342 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1344 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1340 …2} "content" => Illuminate\Database\Eloquent\Collection {#1339 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1357 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1129905 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "132" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/23-7" "field_display_title" => "23.7" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1324 …2} "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 {#1342 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1344 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1340 …2} "content" => Illuminate\Database\Eloquent\Collection {#1339 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1357 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 7 => App\Models\Task {#1341 #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" => 1129906 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "132" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/23-8" "field_display_title" => "23.8" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1338 …2} "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 {#1356 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1358 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1354 …2} "content" => Illuminate\Database\Eloquent\Collection {#1353 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1371 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1129906 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "132" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/23-8" "field_display_title" => "23.8" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1338 …2} "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 {#1356 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1358 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1354 …2} "content" => Illuminate\Database\Eloquent\Collection {#1353 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1371 …2} "page" => array:2 [ …2] ] #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" => 1114449 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "132" "field_url" => "/7-klass/geometrija/smirnova-uchebnik/page-132" "field_display_title" => "132" "field_folder" => "1" "field_image_name" => "132" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1074} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1102} "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 {#1111} "content" => Illuminate\Database\Eloquent\Collection {#1155} "next" => array:2 [ "refs" => "1114450" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1114448" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1253} ] #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:24 [ "id" => 1129902 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "132" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/23-4" "field_display_title" => "23.4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037} "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 {#1035} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1071} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1104} "content" => Illuminate\Database\Eloquent\Collection {#1101} "next" => array:2 [ "refs" => "1129903" "type" => "task" ] "previous" => array:2 [ "refs" => "1129901" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1097} "page" => array:2 [ "refs" => "1114449" "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 => "*" ] }
№23.4 (с. 132)
Условие. №23.4 (с. 132)
скриншот условия
23.4. Постройте треугольник $ABC$ по двум данным сторонам и углу между ними.
Решение. №23.4 (с. 132)
Решение 2. №23.4 (с. 132)
Задача о построении треугольника по двум сторонам и углу между ними является одной из основных задач на построение в курсе геометрии. Для её решения используются циркуль и линейка.
Построение
Пусть даны два отрезка с длинами $p_1$ и $p_2$, и угол $\alpha$. Требуется построить треугольник $ABC$ так, чтобы $AB=p_1$, $AC=p_2$ и $\angle BAC = \alpha$.
Алгоритм построения следующий:
1. Проведем произвольную прямую и отметим на ней точку $A$, которая будет одной из вершин треугольника.
2. С помощью циркуля отложим на этой прямой от точки $A$ отрезок $AC$, равный по длине отрезку $p_2$.
3. От луча $AC$ в заданной полуплоскости отложим угол, равный данному углу $\alpha$. Для этого построим луч $AM$ так, чтобы $\angle CAM = \alpha$.
4. На луче $AM$ отложим отрезок $AB$, равный по длине отрезку $p_1$.
5. Соединим точки $B$ и $C$ отрезком.
Треугольник $ABC$, полученный в результате этих действий, является искомым.
Доказательство
В построенном треугольнике $ABC$ сторона $AC$ по построению равна $p_2$, сторона $AB$ по построению равна $p_1$, а угол $\angle BAC$ между сторонами $AB$ и $AC$ по построению равен $\alpha$. Таким образом, построенный треугольник полностью удовлетворяет условиям задачи.
Исследование
Задача имеет решение, если длины заданных отрезков $p_1$ и $p_2$ являются положительными числами, а заданный угол $\alpha$ больше $0^\circ$ и меньше $180^\circ$ (то есть, неразвернутый угол). При соблюдении этих условий построение всегда возможно. Согласно первому признаку равенства треугольников (по двум сторонам и углу между ними), все треугольники, построенные по заданным элементам, будут равны. Следовательно, задача имеет единственное решение (с точностью до конгруэнтности).
Ответ: Алгоритм построения искомого треугольника $ABC$ описан выше. Треугольник существует и единственен, если данные стороны имеют положительную длину, а угол больше $0^\circ$ и меньше $180^\circ$.
Другие задания:
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