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Номер 52, страница 190 - гдз по геометрии 9 класс учебник Солтан, Солтан
Авторы: Солтан Г. Н., Солтан А. Е., Жумадилова А. Ж.
Тип: Учебник
Издательство: Кокшетау
Год издания: 2019 - 2026
ISBN: 978-601-317-432-7
Рекомендовано Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 9 классе
Задания для итоговой самопроверки - номер 52, страница 190.
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" => 1136790 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "190" "field_page_end" => null "field_url" => "/9-klass/geometrija/kokshetau-soltan-uchebnik/vopr-550" "field_display_title" => "52" "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 {#1046 #items: array:1 [ 0 => App\Models\Branch {#1045 #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" => 1136233 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Задания для итоговой самопроверки" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1047 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "182" "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 {#1078 #items: array:1 [ 0 => App\Models\Book {#1048 #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" => 4312 "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 {#1050 #items: array:1 [ 0 => App\Models\Term {#1049 #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 {#1051 #items: array:1 [ 0 => App\Models\Term {#1052 #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" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ "field_accusative_case" => "девятый" "field_creative_case" => "девятым" "field_dative_case" => "девятому" "field_genitive_case" => "девятого" "field_nominative_case" => "девятый" "field_prepositional_case" => "девятом" ] "field_translit" => "devjatyj" ] #original: array:6 [ "id" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ "field_accusative_case" => "девятый" "field_creative_case" => "девятым" "field_dative_case" => "девятому" "field_genitive_case" => "девятого" "field_nominative_case" => "девятый" "field_prepositional_case" => "девятом" ] "field_translit" => "devjatyj" ] #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 {#1053 #items: array:1 [ 0 => App\Models\Term {#1054 #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" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "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" => "Kokshetau" ] #original: array:6 [ "id" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "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" => "Kokshetau" ] #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 {#1055 #items: array:3 [ 0 => App\Models\Term {#1056 #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" => 5836 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Г" "field_patronymic" => "Н" "field_surname_rp" => null ] #original: array:12 [ "id" => 5836 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Г" "field_patronymic" => "Н" "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 {#1057 #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" => 7004 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алла" "field_patronymic" => "Евгеньевна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7004 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алла" "field_patronymic" => "Евгеньевна" "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 => "*" ] } 2 => 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:12 [ "id" => 7005 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумадилова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Аманбала" "field_patronymic" => "Жумадиловна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7005 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумадилова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Аманбала" "field_patronymic" => "Жумадиловна" "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 {#1059 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1060 #items: array:1 [ 0 => App\Models\Term {#1061 #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 {#1062 #items: array:1 [ 0 => App\Models\Term {#1063 #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 {#1064 #items: array:1 [ 0 => App\Models\Term {#1065 #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 {#1066 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1067 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1068 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1069 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1070 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1071 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "600" "field_priority" => "4" "field_default_folder" => "/geometrija_09/Kokshetau-u19/" "field_isbn" => "978-601-317-432-7" "field_cover" => array:1 [ 0 => "/media/geometrija_09/Kokshetau-u19/covers/cover1.webp?ts=1753337603" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_09/Kokshetau-u19/covers/cover1.webp?ts=1753337603" "alt" => "" "width" => "1632" "height" => "2472" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1072 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1073 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1074 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/geometrija/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1075 #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" => 380 "subject" => 543 "class_subject" => 391 ] ] #original: array:50 [ "id" => 4312 "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 {#1050} "field_class" => Illuminate\Database\Eloquent\Collection {#1051} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1053} "field_author" => Illuminate\Database\Eloquent\Collection {#1055} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1059} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1060} "field_country" => Illuminate\Database\Eloquent\Collection {#1062} "field_city" => Illuminate\Database\Eloquent\Collection {#1064} "field_series" => Illuminate\Database\Eloquent\Collection {#1066} "field_umk" => Illuminate\Database\Eloquent\Collection {#1067} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1068} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1069} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1070} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1071} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "600" "field_priority" => "4" "field_default_folder" => "/geometrija_09/Kokshetau-u19/" "field_isbn" => "978-601-317-432-7" "field_cover" => array:1 [ 0 => "/media/geometrija_09/Kokshetau-u19/covers/cover1.webp?ts=1753337603" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_09/Kokshetau-u19/covers/cover1.webp?ts=1753337603" "alt" => "" "width" => "1632" "height" => "2472" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1072} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1073} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1074} "field_url" => "/9-klass/geometrija/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1075} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 543 "class_subject" => 391 ] ] #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 {#1076 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1136233 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Задания для итоговой самопроверки" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1047} "field_page_start" => "182" "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 {#1078} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1076} ] #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 {#1038 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1042 #items: array:3 [ 0 => App\Models\Element {#1083 #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" => 1357568 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1093 #items: array:1 [ 0 => App\Models\Edition {#1092 #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" => 5037 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1097 #items: array:1 [ 0 => App\Models\Term {#1094 #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" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "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" => "Kokshetau" ] #original: array:6 [ "id" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "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" => "Kokshetau" ] #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 {#1101 #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 {#1098 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5037 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1097} "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 {#1101} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1095} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1098} "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" => "1136790" "type" => "task" ] "text" => "<p><strong>52.</strong> В треугольнике $ABC$ $AC = 2$ дм, $\angle A = 30^\circ$, $\angle C = 45^\circ$. Найдите высоту треугольника, проведенную к стороне $AC$.</p><p>$((\sqrt{3} - 1)$ дм)</p>" "img" => array:1 [ 0 => array:5 [ "name" => "52-1.jpg" "alt" => null "width" => "1503" "height" => 301 "path" => "/media/geometrija_09/Kokshetau-u19/0-samoprov/52-1.webp?ts=1754475131" ] ] ] #original: array:7 [ "id" => 1357568 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1093} "task" => array:2 [ "refs" => "1136790" "type" => "task" ] "text" => "<p><strong>52.</strong> В треугольнике $ABC$ $AC = 2$ дм, $\angle A = 30^\circ$, $\angle C = 45^\circ$. Найдите высоту треугольника, проведенную к стороне $AC$.</p><p>$((\sqrt{3} - 1)$ дм)</p>" "img" => array:1 [ 0 => array:5 [ "name" => "52-1.jpg" "alt" => null "width" => "1503" "height" => 301 "path" => "/media/geometrija_09/Kokshetau-u19/0-samoprov/52-1.webp?ts=1754475131" ] ] ] #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 {#1085 #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" => 1358168 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1102 #items: array:1 [ 0 => App\Models\Edition {#1086 #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" => 5038 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1089 #items: array:1 [ 0 => App\Models\Term {#1084 #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 {#1087 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1090 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1088 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5038 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1089} "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 {#1087} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1090} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1088} "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" => "1136790" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "52-1.jpg" "alt" => null "width" => "1884" "height" => 2030 "path" => "/media/geometrija_09/Kokshetau-u19/1-samoprov/52-1.webp?ts=1754475352" ] ] ] #original: array:6 [ "id" => 1358168 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1102} "task" => array:2 [ "refs" => "1136790" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "52-1.jpg" "alt" => null "width" => "1884" "height" => 2030 "path" => "/media/geometrija_09/Kokshetau-u19/1-samoprov/52-1.webp?ts=1754475352" ] ] ] #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 {#1096 #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" => 1561766 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1110 #items: array:1 [ 0 => App\Models\Edition {#1099 #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" => 5682 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1103 #items: array:1 [ 0 => App\Models\Term {#1100 #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 …2 ] "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 [ …6] "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 {#1104 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1106 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1107 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5682 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1103} "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 {#1104} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1106} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1107} "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" => "1136790" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>треугольник $ABC$;</p><p>$AC = 2$ дм;</p><p>$\angle A = 30^\circ$;</p><p>$\angle C = 45^\circ$.</p><p>Перевод в СИ:</p><p>$AC = 2$ дм $ = 0.2$ м.</p><p><strong>Найти:</strong></p><p>высоту $BH$, проведенную к стороне $AC$.</p><p><strong>Решение:</strong></p><p>Пусть $BH$ — высота треугольника $ABC$, проведенная из вершины $B$ к стороне $AC$. Точка $H$ лежит на отрезке $AC$, так как углы $A$ и $C$ острые (меньше $90^\circ$).</p><p>Высота $BH$ перпендикулярна стороне $AC$, поэтому треугольники $ABH$ и $CBH$ являются прямоугольными.</p><p>Рассмотрим прямоугольный треугольник $ABH$:</p><p>$\angle A = 30^\circ$.</p><p>Тангенс угла $A$ определяется как отношение противолежащего катета $BH$ к прилежащему катету $AH$:</p><p>$\tan A = \frac{BH}{AH}$</p><p>Отсюда, $AH = \frac{BH}{\tan A} = \frac{BH}{\tan 30^\circ}$.</p><p>Известно, что $\tan 30^\circ = \frac{1}{\sqrt{3}}$.</p><p>Следовательно, $AH = \frac{BH}{1/\sqrt{3}} = BH\sqrt{3}$.</p><p>Рассмотрим прямоугольный треугольник $CBH$:</p><p>$\angle C = 45^\circ$.</p><p>Тангенс угла $C$ определяется как отношение противолежащего катета $BH$ к прилежащему катету $HC$:</p><p>$\tan C = \frac{BH}{HC}$</p><p>Отсюда, $HC = \frac{BH}{\tan C} = \frac{BH}{\tan 45^\circ}$.</p><p>Известно, что $\tan 45^\circ = 1$.</p><p>Следовательно, $HC = \frac{BH}{1} = BH$.</p><p>Длина стороны $AC$ равна сумме длин отрезков $AH$ и $HC$:</p><p>$AC = AH + HC$</p><p>Подставим найденные выражения для $AH$ и $HC$:</p><p>$AC = BH\sqrt{3} + BH$</p><p>$AC = BH(\sqrt{3} + 1)$</p><p>Выразим высоту $BH$:</p><p>$BH = \frac{AC}{\sqrt{3} + 1}$</p><p>Подставим значение $AC = 2$ дм:</p><p>$BH = \frac{2}{\sqrt{3} + 1}$</p><p>Чтобы избавиться от иррациональности в знаменателе, умножим числитель и знаменатель на сопряженное выражение $(\sqrt{3} - 1)$:</p><p>$BH = \frac{2}{(\sqrt{3} + 1)} \cdot \frac{(\sqrt{3} - 1)}{(\sqrt{3} - 1)}$</p><p>Используем формулу разности квадратов $(a+b)(a-b) = a^2 - b^2$:</p><p>$BH = \frac{2(\sqrt{3} - 1)}{(\sqrt{3})^2 - (1)^2}$</p><p>$BH = \frac{2(\sqrt{3} - 1)}{3 - 1}$</p><p>$BH = \frac{2(\sqrt{3} - 1)}{2}$</p><p>$BH = \sqrt{3} - 1$</p><p>Единица измерения — дециметры (дм).</p><p><strong>Ответ:</strong></p><p>Высота треугольника, проведенная к стороне $AC$, равна $(\sqrt{3} - 1)$ дм.</p>" ] #original: array:6 [ "id" => 1561766 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1110} "task" => array:2 [ "refs" => "1136790" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>треугольник $ABC$;</p><p>$AC = 2$ дм;</p><p>$\angle A = 30^\circ$;</p><p>$\angle C = 45^\circ$.</p><p>Перевод в СИ:</p><p>$AC = 2$ дм $ = 0.2$ м.</p><p><strong>Найти:</strong></p><p>высоту $BH$, проведенную к стороне $AC$.</p><p><strong>Решение:</strong></p><p>Пусть $BH$ — высота треугольника $ABC$, проведенная из вершины $B$ к стороне $AC$. Точка $H$ лежит на отрезке $AC$, так как углы $A$ и $C$ острые (меньше $90^\circ$).</p><p>Высота $BH$ перпендикулярна стороне $AC$, поэтому треугольники $ABH$ и $CBH$ являются прямоугольными.</p><p>Рассмотрим прямоугольный треугольник $ABH$:</p><p>$\angle A = 30^\circ$.</p><p>Тангенс угла $A$ определяется как отношение противолежащего катета $BH$ к прилежащему катету $AH$:</p><p>$\tan A = \frac{BH}{AH}$</p><p>Отсюда, $AH = \frac{BH}{\tan A} = \frac{BH}{\tan 30^\circ}$.</p><p>Известно, что $\tan 30^\circ = \frac{1}{\sqrt{3}}$.</p><p>Следовательно, $AH = \frac{BH}{1/\sqrt{3}} = BH\sqrt{3}$.</p><p>Рассмотрим прямоугольный треугольник $CBH$:</p><p>$\angle C = 45^\circ$.</p><p>Тангенс угла $C$ определяется как отношение противолежащего катета $BH$ к прилежащему катету $HC$:</p><p>$\tan C = \frac{BH}{HC}$</p><p>Отсюда, $HC = \frac{BH}{\tan C} = \frac{BH}{\tan 45^\circ}$.</p><p>Известно, что $\tan 45^\circ = 1$.</p><p>Следовательно, $HC = \frac{BH}{1} = BH$.</p><p>Длина стороны $AC$ равна сумме длин отрезков $AH$ и $HC$:</p><p>$AC = AH + HC$</p><p>Подставим найденные выражения для $AH$ и $HC$:</p><p>$AC = BH\sqrt{3} + BH$</p><p>$AC = BH(\sqrt{3} + 1)$</p><p>Выразим высоту $BH$:</p><p>$BH = \frac{AC}{\sqrt{3} + 1}$</p><p>Подставим значение $AC = 2$ дм:</p><p>$BH = \frac{2}{\sqrt{3} + 1}$</p><p>Чтобы избавиться от иррациональности в знаменателе, умножим числитель и знаменатель на сопряженное выражение $(\sqrt{3} - 1)$:</p><p>$BH = \frac{2}{(\sqrt{3} + 1)} \cdot \frac{(\sqrt{3} - 1)}{(\sqrt{3} - 1)}$</p><p>Используем формулу разности квадратов $(a+b)(a-b) = a^2 - b^2$:</p><p>$BH = \frac{2(\sqrt{3} - 1)}{(\sqrt{3})^2 - (1)^2}$</p><p>$BH = \frac{2(\sqrt{3} - 1)}{3 - 1}$</p><p>$BH = \frac{2(\sqrt{3} - 1)}{2}$</p><p>$BH = \sqrt{3} - 1$</p><p>Единица измерения — дециметры (дм).</p><p><strong>Ответ:</strong></p><p>Высота треугольника, проведенная к стороне $AC$, равна $(\sqrt{3} - 1)$ дм.</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" => "1136791" "type" => "task" ] "previous" => array:2 [ "refs" => "1136789" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1080 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1077 #items: array:1 [ 0 => App\Models\BookPage {#1040 #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" => 1116581 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "190" "field_url" => "/9-klass/geometrija/kokshetau-soltan-uchebnik/page-190" "field_display_title" => "190" "field_folder" => "1" "field_image_name" => "190" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1043 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1108 #items: array:1 [ 0 => App\Models\Book {#1048} ] #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 {#1109 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1120 #items: array:1 [ 0 => App\Models\Element {#1119 #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" => 1294846 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1121 #items: array:1 [ 0 => App\Models\Edition {#1092} ] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1116581" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "190-1.jpg" "alt" => null "width" => "2009" "height" => 2972 "path" => "/media/geometrija_09/Kokshetau-u19/0-1/190-1.webp?ts=1753338192" ] ] ] #original: array:6 [ "id" => 1294846 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1121} "book_page" => array:2 [ "refs" => "1116581" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "190-1.jpg" "alt" => null "width" => "2009" "height" => 2972 "path" => "/media/geometrija_09/Kokshetau-u19/0-1/190-1.webp?ts=1753338192" ] ] ] #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" => "1116582" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1116580" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1181 #items: array:4 [ 0 => App\Models\Task {#1191 #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" => 1136788 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "190" "field_page_end" => null "field_url" => "/9-klass/geometrija/kokshetau-soltan-uchebnik/vopr-548" "field_display_title" => "50" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1192 #items: array:1 [ 0 => App\Models\Term {#1036} ] #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 {#1193 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1194 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1195 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1198 #items: array:3 [ 0 => App\Models\Element {#1207 #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" => 1357566 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1208 …2} "task" => array:2 [ …2] "text" => "<strong>50. Длина гипотенузы прямоугольного треугольника равна 12 см.</strong><p>Найдите длины его катетов, если известно, что длина большего катета равна среднему арифметическому длин меньшего катета и гипотенузы.</p><p>(9,6 см, 7,2 см)</p>" "img" => array:1 [ …1] ] #original: array:7 [ "id" => 1357566 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1208 …2} "task" => array:2 [ …2] "text" => "<strong>50. Длина гипотенузы прямоугольного треугольника равна 12 см.</strong><p>Найдите длины его катетов, если известно, что длина большего катета равна среднему арифметическому длин меньшего катета и гипотенузы.</p><p>(9,6 см, 7,2 см)</p>" "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 => "*" ] } 1 => App\Models\Element {#1209 #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" => 1358166 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1210 …2} "task" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1358166 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1210 …2} "task" => 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 => "*" ] } 2 => App\Models\Element {#1211 #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" => 1561764 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1212 …2} "task" => array:2 [ …2] "text" => "<p><strong>Дано:</strong></p><p>Длина гипотенузы прямоугольного треугольника: $c = 12$ см</p><p>Длина большего катета ($b$) равна среднему арифметическому длин меньшего катета ($a$) и гипотенузы ($c$): $b = \frac{a + c}{2}$</p><p><strong>Перевод в СИ:</strong></p><p>$c = 12$ см $= 0.12$ м</p><p>Длины катетов $a$ и $b$ будут получены в метрах при пересчете.</p><p><strong>Найти:</strong></p><p>Длины катетов $a$ и $b$.</p><p><strong>Решение:</strong></p><p>Обозначим длины катетов как $a$ (меньший катет) и $b$ (больший катет). Длина гипотенузы $c = 12$ см.</p><p>Согласно условию задачи, длина большего катета $b$ равна среднему арифметическому длин меньшего катета $a$ и гипотенузы $c$:</p><p>$b = \frac{a + c}{2}$</p><p>Подставим известное значение гипотенузы $c = 12$ см:</p><p>$b = \frac{a + 12}{2} \quad (1)$</p><p>Для прямоугольного треугольника справедлива теорема Пифагора:</p><p>$a^2 + b^2 = c^2$</p><p>Подставим значение $c = 12$ см:</p><p>$a^2 + b^2 = 12^2$</p><p>$a^2 + b^2 = 144 \quad (2)$</p><p>Теперь подставим выражение для $b$ из уравнения $(1)$ в уравнение $(2)$:</p><p>$a^2 + \left(\frac{a + 12}{2}\right)^2 = 144$</p><p>$a^2 + \frac{(a + 12)^2}{4} = 144$</p><p>Умножим обе части уравнения на 4, чтобы избавиться от знаменателя:</p><p>$4a^2 + (a + 12)^2 = 144 \times 4$</p><p>Раскроем скобки $(a + 12)^2 = a^2 + 2 \times a \times 12 + 12^2 = a^2 + 24a + 144$:</p><p>$4a^2 + a^2 + 24a + 144 = 576$</p><p>Приведем подобные члены и перенесем все слагаемые в левую часть уравнения:</p><p>$5a^2 + 24a + 144 - 576 = 0$</p><p>$5a^2 + 24a - 432 = 0$</p><p>Это квадратное уравнение вида $Ax^2 + Bx + C = 0$, где $A = 5$, $B = 24$, $C = -432$. Найдем дискриминант $D = B^2 - 4AC$:</p><p>$D = 24^2 - 4 \times 5 \times (-432)$</p><p>$D = 576 - 20 \times (-432)$</p><p>$D = 576 + 8640$</p><p>$D = 9216$</p><p>Найдем квадратный корень из дискриминанта:</p><p>$\sqrt{D} = \sqrt{9216} = 96$</p><p>Теперь найдем значения $a$ по формуле $a = \frac{-B \pm \sqrt{D}}{2A}$:</p><p>$a_1 = \frac{-24 + 96}{2 \times 5} = \frac{72}{10} = 7.2$</p><p>$a_2 = \frac{-24 - 96}{2 \times 5} = \frac{-120}{10} = -12$</p><p>Длина отрезка не может быть отрицательной, поэтому принимаем $a = 7.2$ см.</p><p>Теперь найдем длину большего катета $b$, используя уравнение $(1)$:</p><p>$b = \frac{a + 12}{2} = \frac{7.2 + 12}{2} = \frac{19.2}{2} = 9.6$</p><p>Таким образом, длины катетов равны 7.2 см и 9.6 см. Проверим, что $a$ является меньшим катетом, а $b$ большим: $7.2 < 9.6$, что соответствует условию задачи.</p><p><strong>Ответ:</strong></p><p>Длины катетов составляют 9.6 см и 7.2 см.</p>" ] #original: array:6 [ "id" => 1561764 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1212 …2} "task" => array:2 [ …2] "text" => "<p><strong>Дано:</strong></p><p>Длина гипотенузы прямоугольного треугольника: $c = 12$ см</p><p>Длина большего катета ($b$) равна среднему арифметическому длин меньшего катета ($a$) и гипотенузы ($c$): $b = \frac{a + c}{2}$</p><p><strong>Перевод в СИ:</strong></p><p>$c = 12$ см $= 0.12$ м</p><p>Длины катетов $a$ и $b$ будут получены в метрах при пересчете.</p><p><strong>Найти:</strong></p><p>Длины катетов $a$ и $b$.</p><p><strong>Решение:</strong></p><p>Обозначим длины катетов как $a$ (меньший катет) и $b$ (больший катет). Длина гипотенузы $c = 12$ см.</p><p>Согласно условию задачи, длина большего катета $b$ равна среднему арифметическому длин меньшего катета $a$ и гипотенузы $c$:</p><p>$b = \frac{a + c}{2}$</p><p>Подставим известное значение гипотенузы $c = 12$ см:</p><p>$b = \frac{a + 12}{2} \quad (1)$</p><p>Для прямоугольного треугольника справедлива теорема Пифагора:</p><p>$a^2 + b^2 = c^2$</p><p>Подставим значение $c = 12$ см:</p><p>$a^2 + b^2 = 12^2$</p><p>$a^2 + b^2 = 144 \quad (2)$</p><p>Теперь подставим выражение для $b$ из уравнения $(1)$ в уравнение $(2)$:</p><p>$a^2 + \left(\frac{a + 12}{2}\right)^2 = 144$</p><p>$a^2 + \frac{(a + 12)^2}{4} = 144$</p><p>Умножим обе части уравнения на 4, чтобы избавиться от знаменателя:</p><p>$4a^2 + (a + 12)^2 = 144 \times 4$</p><p>Раскроем скобки $(a + 12)^2 = a^2 + 2 \times a \times 12 + 12^2 = a^2 + 24a + 144$:</p><p>$4a^2 + a^2 + 24a + 144 = 576$</p><p>Приведем подобные члены и перенесем все слагаемые в левую часть уравнения:</p><p>$5a^2 + 24a + 144 - 576 = 0$</p><p>$5a^2 + 24a - 432 = 0$</p><p>Это квадратное уравнение вида $Ax^2 + Bx + C = 0$, где $A = 5$, $B = 24$, $C = -432$. Найдем дискриминант $D = B^2 - 4AC$:</p><p>$D = 24^2 - 4 \times 5 \times (-432)$</p><p>$D = 576 - 20 \times (-432)$</p><p>$D = 576 + 8640$</p><p>$D = 9216$</p><p>Найдем квадратный корень из дискриминанта:</p><p>$\sqrt{D} = \sqrt{9216} = 96$</p><p>Теперь найдем значения $a$ по формуле $a = \frac{-B \pm \sqrt{D}}{2A}$:</p><p>$a_1 = \frac{-24 + 96}{2 \times 5} = \frac{72}{10} = 7.2$</p><p>$a_2 = \frac{-24 - 96}{2 \times 5} = \frac{-120}{10} = -12$</p><p>Длина отрезка не может быть отрицательной, поэтому принимаем $a = 7.2$ см.</p><p>Теперь найдем длину большего катета $b$, используя уравнение $(1)$:</p><p>$b = \frac{a + 12}{2} = \frac{7.2 + 12}{2} = \frac{19.2}{2} = 9.6$</p><p>Таким образом, длины катетов равны 7.2 см и 9.6 см. Проверим, что $a$ является меньшим катетом, а $b$ большим: $7.2 < 9.6$, что соответствует условию задачи.</p><p><strong>Ответ:</strong></p><p>Длины катетов составляют 9.6 см и 7.2 см.</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" => "1136789" "type" => "task" ] "previous" => array:2 [ "refs" => "1136787" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1204 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1116581" "type" => "book_page" ] ] #original: array:24 [ "id" => 1136788 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "190" "field_page_end" => null "field_url" => "/9-klass/geometrija/kokshetau-soltan-uchebnik/vopr-548" "field_display_title" => "50" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1192} "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 {#1193} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1194} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1195} "content" => Illuminate\Database\Eloquent\Collection {#1198} "next" => array:2 [ "refs" => "1136789" "type" => "task" ] "previous" => array:2 [ "refs" => "1136787" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1204} "page" => array:2 [ "refs" => "1116581" "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 {#1197 #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" => 1136789 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "190" "field_page_end" => null "field_url" => "/9-klass/geometrija/kokshetau-soltan-uchebnik/vopr-549" "field_display_title" => "51" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1196 #items: array:1 [ 0 => App\Models\Term {#1036} ] #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 {#1203 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1205 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1201 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1200 #items: array:3 [ 0 => App\Models\Element {#1221 #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" => 1357567 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1222 …2} "task" => array:2 [ …2] "text" => "<p><strong>51. a)</strong> Найдите наибольшую площадь четырехугольника, вписанного в окружность, радиус которой 9 см.</p><p><strong>б)</strong> На расстоянии 8 м от центра $O$ окружности радиуса 6 м взята точка $M$. Найдите наибольшую площадь треугольника $OMX$, где $X$ – произвольная точка окружности.</p><p>(a) 162 $см^2$; б) 24 $м^2$)</p>" "img" => array:1 [ …1] ] #original: array:7 [ "id" => 1357567 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1222 …2} "task" => array:2 [ …2] "text" => "<p><strong>51. a)</strong> Найдите наибольшую площадь четырехугольника, вписанного в окружность, радиус которой 9 см.</p><p><strong>б)</strong> На расстоянии 8 м от центра $O$ окружности радиуса 6 м взята точка $M$. Найдите наибольшую площадь треугольника $OMX$, где $X$ – произвольная точка окружности.</p><p>(a) 162 $см^2$; б) 24 $м^2$)</p>" "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 => "*" ] } 1 => App\Models\Element {#1223 #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" => 1358167 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1224 …2} "task" => array:2 [ …2] "img" => array:2 [ …2] ] #original: array:6 [ "id" => 1358167 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1224 …2} "task" => array:2 [ …2] "img" => 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\Element {#1225 #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" => 1561765 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1226 …2} "task" => array:2 [ …2] "text" => "<p><strong>a)</strong></p><p><strong>Дано:</strong></p><p>Радиус окружности $R = 9$ см</p><p><strong>Перевод в СИ:</strong></p><p>$R = 9 \text{ см} = 0.09 \text{ м}$</p><p><strong>Найти:</strong></p><p>Наибольшую площадь четырехугольника $S_{max}$</p><p><strong>Решение:</strong></p><p>Среди всех четырехугольников, вписанных в окружность, наибольшую площадь имеет квадрат. Диагональ такого квадрата равна диаметру окружности.</p><p>Пусть сторона квадрата равна $a$. Тогда его диагональ $d$ связана со стороной формулой $d = a\sqrt{2}$.</p><p>Диаметр окружности $D$ равен двум радиусам, то есть $D = 2R$.</p><p>Поскольку диагональ квадрата является диаметром окружности, имеем $a\sqrt{2} = 2R$.</p><p>Выразим сторону квадрата $a$ через радиус $R$: $a = \frac{2R}{\sqrt{2}} = R\sqrt{2}$.</p><p>Площадь квадрата $S$ вычисляется как $S = a^2$. Подставим выражение для $a$:</p><p>$S_{max} = (R\sqrt{2})^2 = R^2 \cdot (\sqrt{2})^2 = 2R^2$.</p><p>Подставим заданное значение радиуса $R = 9$ см:</p><p>$S_{max} = 2 \cdot (9 \text{ см})^2 = 2 \cdot 81 \text{ см}^2 = 162 \text{ см}^2$.</p><p><strong>Ответ:</strong> 162 см$^2$</p><p><strong>б)</strong></p><p><strong>Дано:</strong></p><p>Расстояние от центра $O$ окружности до точки $M$: $OM = 8$ м</p><p>Радиус окружности: $R = 6$ м</p><p>Точка $X$ – произвольная точка окружности.</p><p><strong>Перевод в СИ:</strong></p><p>Все величины даны в системе СИ.</p><p><strong>Найти:</strong></p><p>Наибольшую площадь треугольника $OMX$, $S_{OMX,max}$</p><p><strong>Решение:</strong></p><p>Площадь треугольника $OMX$ определяется по формуле $S_{OMX} = \frac{1}{2} \cdot \text{основание} \cdot \text{высота}$.</p><p>В качестве основания треугольника $OMX$ возьмем отрезок $OM$. Его длина фиксирована и равна $OM = 8$ м.</p><p>Тогда площадь треугольника $S_{OMX} = \frac{1}{2} \cdot OM \cdot h_X$, где $h_X$ – высота треугольника, опущенная из вершины $X$ на прямую, содержащую основание $OM$.</p><p>Для того чтобы площадь треугольника была наибольшей, необходимо максимизировать высоту $h_X$.</p><p>Высота $h_X$ – это перпендикулярное расстояние от точки $X$ (лежащей на окружности) до прямой $OM$.</p><p>Максимальное расстояние от точки на окружности до прямой, проходящей через центр этой окружности $O$ (прямая $OM$), достигается, когда радиус $OX$ перпендикулярен прямой $OM$.</p><p>В этом случае высота $h_X$ будет равна радиусу окружности $R$.</p><p>Таким образом, максимальная высота $h_{X,max} = R = 6$ м.</p><p>Подставим значения $OM$ и $R$ в формулу площади:</p><p>$S_{OMX,max} = \frac{1}{2} \cdot OM \cdot R$</p><p>$S_{OMX,max} = \frac{1}{2} \cdot 8 \text{ м} \cdot 6 \text{ м}$</p><p>$S_{OMX,max} = \frac{1}{2} \cdot 48 \text{ м}^2$</p><p>$S_{OMX,max} = 24 \text{ м}^2$</p><p><strong>Ответ:</strong> 24 м$^2$</p>" ] #original: array:6 [ "id" => 1561765 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1226 …2} "task" => array:2 [ …2] "text" => "<p><strong>a)</strong></p><p><strong>Дано:</strong></p><p>Радиус окружности $R = 9$ см</p><p><strong>Перевод в СИ:</strong></p><p>$R = 9 \text{ см} = 0.09 \text{ м}$</p><p><strong>Найти:</strong></p><p>Наибольшую площадь четырехугольника $S_{max}$</p><p><strong>Решение:</strong></p><p>Среди всех четырехугольников, вписанных в окружность, наибольшую площадь имеет квадрат. Диагональ такого квадрата равна диаметру окружности.</p><p>Пусть сторона квадрата равна $a$. Тогда его диагональ $d$ связана со стороной формулой $d = a\sqrt{2}$.</p><p>Диаметр окружности $D$ равен двум радиусам, то есть $D = 2R$.</p><p>Поскольку диагональ квадрата является диаметром окружности, имеем $a\sqrt{2} = 2R$.</p><p>Выразим сторону квадрата $a$ через радиус $R$: $a = \frac{2R}{\sqrt{2}} = R\sqrt{2}$.</p><p>Площадь квадрата $S$ вычисляется как $S = a^2$. Подставим выражение для $a$:</p><p>$S_{max} = (R\sqrt{2})^2 = R^2 \cdot (\sqrt{2})^2 = 2R^2$.</p><p>Подставим заданное значение радиуса $R = 9$ см:</p><p>$S_{max} = 2 \cdot (9 \text{ см})^2 = 2 \cdot 81 \text{ см}^2 = 162 \text{ см}^2$.</p><p><strong>Ответ:</strong> 162 см$^2$</p><p><strong>б)</strong></p><p><strong>Дано:</strong></p><p>Расстояние от центра $O$ окружности до точки $M$: $OM = 8$ м</p><p>Радиус окружности: $R = 6$ м</p><p>Точка $X$ – произвольная точка окружности.</p><p><strong>Перевод в СИ:</strong></p><p>Все величины даны в системе СИ.</p><p><strong>Найти:</strong></p><p>Наибольшую площадь треугольника $OMX$, $S_{OMX,max}$</p><p><strong>Решение:</strong></p><p>Площадь треугольника $OMX$ определяется по формуле $S_{OMX} = \frac{1}{2} \cdot \text{основание} \cdot \text{высота}$.</p><p>В качестве основания треугольника $OMX$ возьмем отрезок $OM$. Его длина фиксирована и равна $OM = 8$ м.</p><p>Тогда площадь треугольника $S_{OMX} = \frac{1}{2} \cdot OM \cdot h_X$, где $h_X$ – высота треугольника, опущенная из вершины $X$ на прямую, содержащую основание $OM$.</p><p>Для того чтобы площадь треугольника была наибольшей, необходимо максимизировать высоту $h_X$.</p><p>Высота $h_X$ – это перпендикулярное расстояние от точки $X$ (лежащей на окружности) до прямой $OM$.</p><p>Максимальное расстояние от точки на окружности до прямой, проходящей через центр этой окружности $O$ (прямая $OM$), достигается, когда радиус $OX$ перпендикулярен прямой $OM$.</p><p>В этом случае высота $h_X$ будет равна радиусу окружности $R$.</p><p>Таким образом, максимальная высота $h_{X,max} = R = 6$ м.</p><p>Подставим значения $OM$ и $R$ в формулу площади:</p><p>$S_{OMX,max} = \frac{1}{2} \cdot OM \cdot R$</p><p>$S_{OMX,max} = \frac{1}{2} \cdot 8 \text{ м} \cdot 6 \text{ м}$</p><p>$S_{OMX,max} = \frac{1}{2} \cdot 48 \text{ м}^2$</p><p>$S_{OMX,max} = 24 \text{ м}^2$</p><p><strong>Ответ:</strong> 24 м$^2$</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" => "1136790" "type" => "task" ] "previous" => array:2 [ "refs" => "1136788" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1218 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1116581" "type" => "book_page" ] ] #original: array:24 [ "id" => 1136789 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "190" "field_page_end" => null "field_url" => "/9-klass/geometrija/kokshetau-soltan-uchebnik/vopr-549" "field_display_title" => "51" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1196} "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 {#1203} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1205} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1201} "content" => Illuminate\Database\Eloquent\Collection {#1200} "next" => array:2 [ "refs" => "1136790" "type" => "task" ] "previous" => array:2 [ "refs" => "1136788" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1218} "page" => array:2 [ "refs" => "1116581" "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 => "*" ] } 2 => App\Models\Task {#1202 #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" => 1136790 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "190" "field_page_end" => null "field_url" => "/9-klass/geometrija/kokshetau-soltan-uchebnik/vopr-550" "field_display_title" => "52" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1199 #items: array:1 [ 0 => App\Models\Term {#1036} ] #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 {#1217 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1219 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1215 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1213 #items: array:3 [ 0 => App\Models\Element {#1083} 1 => App\Models\Element {#1085} 2 => App\Models\Element {#1096} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1136791" "type" => "task" ] "previous" => array:2 [ "refs" => "1136789" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1216 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1116581" "type" => "book_page" ] ] #original: array:24 [ "id" => 1136790 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "190" "field_page_end" => null "field_url" => "/9-klass/geometrija/kokshetau-soltan-uchebnik/vopr-550" "field_display_title" => "52" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1199} "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 {#1217} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1219} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1215} "content" => Illuminate\Database\Eloquent\Collection {#1213} "next" => array:2 [ "refs" => "1136791" "type" => "task" ] "previous" => array:2 [ "refs" => "1136789" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1216} "page" => array:2 [ "refs" => "1116581" "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 => "*" ] } 3 => App\Models\Task {#1214 #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" => 1136791 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "190" "field_page_end" => null "field_url" => "/9-klass/geometrija/kokshetau-soltan-uchebnik/vopr-551" "field_display_title" => "53" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1227 #items: array:1 [ 0 => App\Models\Term {#1036} ] #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 {#1229 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1230 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #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" => 1357569 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1243 …2} "task" => array:2 [ …2] "text" => "<p><strong>53.</strong> Катеты прямоугольного треугольника равны 9 см и 40 см. Найдите расстояние от центра вписанной в треугольник окружности до центра описанной около него окружности. $(2,5\sqrt{41} \text{ см})$</p>" "img" => array:1 [ …1] ] #original: array:7 [ "id" => 1357569 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1243 …2} "task" => array:2 [ …2] "text" => "<p><strong>53.</strong> Катеты прямоугольного треугольника равны 9 см и 40 см. Найдите расстояние от центра вписанной в треугольник окружности до центра описанной около него окружности. $(2,5\sqrt{41} \text{ см})$</p>" "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 => "*" ] } 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" => 1358169 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1245 …2} "task" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1358169 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1245 …2} "task" => 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 => "*" ] } 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" => 1561767 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1247 …2} "task" => array:2 [ …2] "text" => "<p><strong>Дано</strong></p><p>Прямоугольный треугольник со следующими длинами катетов:</p><p>$a = 9 \, \text{см}$<br>$b = 40 \, \text{см}$</p><p><strong>Найти</strong></p><p>Расстояние $d$ от центра вписанной окружности до центра описанной окружности.</p><p><strong>Решение</strong></p><p><strong>1.</strong> Найдем длину гипотенузы $c$ прямоугольного треугольника, используя теорему Пифагора: $c^2 = a^2 + b^2$ $c^2 = 9^2 + 40^2$ $c^2 = 81 + 1600$ $c^2 = 1681$ $c = \sqrt{1681}$ $c = 41 \, \text{см}$</p><p><strong>2.</strong> Найдем радиус $r$ вписанной окружности для прямоугольного треугольника. Формула для радиуса вписанной окружности в прямоугольный треугольник: $r = \frac{a + b - c}{2}$ $r = \frac{9 + 40 - 41}{2}$ $r = \frac{49 - 41}{2}$ $r = \frac{8}{2}$ $r = 4 \, \text{см}$</p><p><strong>3.</strong> Найдем радиус $R$ описанной окружности для прямоугольного треугольника. Центр описанной окружности прямоугольного треугольника лежит на середине гипотенузы, а ее радиус равен половине длины гипотенузы: $R = \frac{c}{2}$ $R = \frac{41}{2}$ $R = 20.5 \, \text{см}$</p><p><strong>4.</strong> Определим координаты центров окружностей. Пусть вершины прямоугольного треугольника находятся в точках $A(0, a)$, $B(b, 0)$ и $C(0, 0)$ (с прямым углом в точке $C$). В нашем случае, $A(0, 9)$, $B(40, 0)$, $C(0, 0)$ (или $A(0, 40)$, $B(9, 0)$, $C(0, 0)$ - это не повлияет на результат, так как оси симметричны). Центр вписанной окружности $O_i$ имеет координаты $(r, r)$, так как он равноудален от катетов и находится внутри угла: $O_i = (4, 4)$ Центр описанной окружности $O_o$ является серединой гипотенузы. Если вершины гипотенузы $A(0, a)$ и $B(b, 0)$, то координаты $O_o$ будут: $O_o = \left( \frac{0 + b}{2}, \frac{a + 0}{2} \right)$ $O_o = \left( \frac{40}{2}, \frac{9}{2} \right)$ $O_o = (20, 4.5)$</p><p><strong>5.</strong> Найдем расстояние $d$ между центрами $O_i(x_1, y_1)$ и $O_o(x_2, y_2)$ по формуле расстояния между двумя точками: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ $d = \sqrt{(20 - 4)^2 + (4.5 - 4)^2}$ $d = \sqrt{(16)^2 + (0.5)^2}$ $d = \sqrt{256 + 0.25}$ $d = \sqrt{256.25}$</p><p>Представим $256.25$ в виде дроби: $256.25 = \frac{25625}{100} = \frac{1025}{4}$. $d = \sqrt{\frac{1025}{4}}$ $d = \frac{\sqrt{1025}}{\sqrt{4}}$ $d = \frac{\sqrt{25 \cdot 41}}{2}$ $d = \frac{5\sqrt{41}}{2}$ $d = 2.5\sqrt{41} \, \text{см}$</p><p><strong>Ответ</strong></p><p>Расстояние от центра вписанной окружности до центра описанной окружности составляет $2.5\sqrt{41} \, \text{см}$.</p>" ] #original: array:6 [ "id" => 1561767 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1247 …2} "task" => array:2 [ …2] "text" => "<p><strong>Дано</strong></p><p>Прямоугольный треугольник со следующими длинами катетов:</p><p>$a = 9 \, \text{см}$<br>$b = 40 \, \text{см}$</p><p><strong>Найти</strong></p><p>Расстояние $d$ от центра вписанной окружности до центра описанной окружности.</p><p><strong>Решение</strong></p><p><strong>1.</strong> Найдем длину гипотенузы $c$ прямоугольного треугольника, используя теорему Пифагора: $c^2 = a^2 + b^2$ $c^2 = 9^2 + 40^2$ $c^2 = 81 + 1600$ $c^2 = 1681$ $c = \sqrt{1681}$ $c = 41 \, \text{см}$</p><p><strong>2.</strong> Найдем радиус $r$ вписанной окружности для прямоугольного треугольника. Формула для радиуса вписанной окружности в прямоугольный треугольник: $r = \frac{a + b - c}{2}$ $r = \frac{9 + 40 - 41}{2}$ $r = \frac{49 - 41}{2}$ $r = \frac{8}{2}$ $r = 4 \, \text{см}$</p><p><strong>3.</strong> Найдем радиус $R$ описанной окружности для прямоугольного треугольника. Центр описанной окружности прямоугольного треугольника лежит на середине гипотенузы, а ее радиус равен половине длины гипотенузы: $R = \frac{c}{2}$ $R = \frac{41}{2}$ $R = 20.5 \, \text{см}$</p><p><strong>4.</strong> Определим координаты центров окружностей. Пусть вершины прямоугольного треугольника находятся в точках $A(0, a)$, $B(b, 0)$ и $C(0, 0)$ (с прямым углом в точке $C$). В нашем случае, $A(0, 9)$, $B(40, 0)$, $C(0, 0)$ (или $A(0, 40)$, $B(9, 0)$, $C(0, 0)$ - это не повлияет на результат, так как оси симметричны). Центр вписанной окружности $O_i$ имеет координаты $(r, r)$, так как он равноудален от катетов и находится внутри угла: $O_i = (4, 4)$ Центр описанной окружности $O_o$ является серединой гипотенузы. Если вершины гипотенузы $A(0, a)$ и $B(b, 0)$, то координаты $O_o$ будут: $O_o = \left( \frac{0 + b}{2}, \frac{a + 0}{2} \right)$ $O_o = \left( \frac{40}{2}, \frac{9}{2} \right)$ $O_o = (20, 4.5)$</p><p><strong>5.</strong> Найдем расстояние $d$ между центрами $O_i(x_1, y_1)$ и $O_o(x_2, y_2)$ по формуле расстояния между двумя точками: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ $d = \sqrt{(20 - 4)^2 + (4.5 - 4)^2}$ $d = \sqrt{(16)^2 + (0.5)^2}$ $d = \sqrt{256 + 0.25}$ $d = \sqrt{256.25}$</p><p>Представим $256.25$ в виде дроби: $256.25 = \frac{25625}{100} = \frac{1025}{4}$. $d = \sqrt{\frac{1025}{4}}$ $d = \frac{\sqrt{1025}}{\sqrt{4}}$ $d = \frac{\sqrt{25 \cdot 41}}{2}$ $d = \frac{5\sqrt{41}}{2}$ $d = 2.5\sqrt{41} \, \text{см}$</p><p><strong>Ответ</strong></p><p>Расстояние от центра вписанной окружности до центра описанной окружности составляет $2.5\sqrt{41} \, \text{см}$.</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" => "1136792" "type" => "task" ] "previous" => array:2 [ "refs" => "1136790" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1239 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1116581" "type" => "book_page" ] ] #original: array:24 [ "id" => 1136791 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "190" "field_page_end" => null "field_url" => "/9-klass/geometrija/kokshetau-soltan-uchebnik/vopr-551" "field_display_title" => "53" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1227} "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 {#1229} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1230} "content" => Illuminate\Database\Eloquent\Collection {#1233} "next" => array:2 [ "refs" => "1136792" "type" => "task" ] "previous" => array:2 [ "refs" => "1136790" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1239} "page" => array:2 [ "refs" => "1116581" "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" => 1116581 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "190" "field_url" => "/9-klass/geometrija/kokshetau-soltan-uchebnik/page-190" "field_display_title" => "190" "field_folder" => "1" "field_image_name" => "190" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1043} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1108} "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 {#1109} "content" => Illuminate\Database\Eloquent\Collection {#1120} "next" => array:2 [ "refs" => "1116582" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1116580" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1181} ] #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" => 1136790 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "190" "field_page_end" => null "field_url" => "/9-klass/geometrija/kokshetau-soltan-uchebnik/vopr-550" "field_display_title" => "52" "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 {#1046} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1038} "content" => Illuminate\Database\Eloquent\Collection {#1042} "next" => array:2 [ "refs" => "1136791" "type" => "task" ] "previous" => array:2 [ "refs" => "1136789" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1080} "page" => array:2 [ "refs" => "1116581" "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 => "*" ] }
№52 (с. 190)
Условие. №52 (с. 190)
скриншот условия
52. В треугольнике $ABC$ $AC = 2$ дм, $\angle A = 30^\circ$, $\angle C = 45^\circ$. Найдите высоту треугольника, проведенную к стороне $AC$.
$((\sqrt{3} - 1)$ дм)
Решение. №52 (с. 190)
Решение 2. №52 (с. 190)
Дано:
треугольник $ABC$;
$AC = 2$ дм;
$\angle A = 30^\circ$;
$\angle C = 45^\circ$.
Перевод в СИ:
$AC = 2$ дм $ = 0.2$ м.
Найти:
высоту $BH$, проведенную к стороне $AC$.
Решение:
Пусть $BH$ — высота треугольника $ABC$, проведенная из вершины $B$ к стороне $AC$. Точка $H$ лежит на отрезке $AC$, так как углы $A$ и $C$ острые (меньше $90^\circ$).
Высота $BH$ перпендикулярна стороне $AC$, поэтому треугольники $ABH$ и $CBH$ являются прямоугольными.
Рассмотрим прямоугольный треугольник $ABH$:
$\angle A = 30^\circ$.
Тангенс угла $A$ определяется как отношение противолежащего катета $BH$ к прилежащему катету $AH$:
$\tan A = \frac{BH}{AH}$
Отсюда, $AH = \frac{BH}{\tan A} = \frac{BH}{\tan 30^\circ}$.
Известно, что $\tan 30^\circ = \frac{1}{\sqrt{3}}$.
Следовательно, $AH = \frac{BH}{1/\sqrt{3}} = BH\sqrt{3}$.
Рассмотрим прямоугольный треугольник $CBH$:
$\angle C = 45^\circ$.
Тангенс угла $C$ определяется как отношение противолежащего катета $BH$ к прилежащему катету $HC$:
$\tan C = \frac{BH}{HC}$
Отсюда, $HC = \frac{BH}{\tan C} = \frac{BH}{\tan 45^\circ}$.
Известно, что $\tan 45^\circ = 1$.
Следовательно, $HC = \frac{BH}{1} = BH$.
Длина стороны $AC$ равна сумме длин отрезков $AH$ и $HC$:
$AC = AH + HC$
Подставим найденные выражения для $AH$ и $HC$:
$AC = BH\sqrt{3} + BH$
$AC = BH(\sqrt{3} + 1)$
Выразим высоту $BH$:
$BH = \frac{AC}{\sqrt{3} + 1}$
Подставим значение $AC = 2$ дм:
$BH = \frac{2}{\sqrt{3} + 1}$
Чтобы избавиться от иррациональности в знаменателе, умножим числитель и знаменатель на сопряженное выражение $(\sqrt{3} - 1)$:
$BH = \frac{2}{(\sqrt{3} + 1)} \cdot \frac{(\sqrt{3} - 1)}{(\sqrt{3} - 1)}$
Используем формулу разности квадратов $(a+b)(a-b) = a^2 - b^2$:
$BH = \frac{2(\sqrt{3} - 1)}{(\sqrt{3})^2 - (1)^2}$
$BH = \frac{2(\sqrt{3} - 1)}{3 - 1}$
$BH = \frac{2(\sqrt{3} - 1)}{2}$
$BH = \sqrt{3} - 1$
Единица измерения — дециметры (дм).
Ответ:
Высота треугольника, проведенная к стороне $AC$, равна $(\sqrt{3} - 1)$ дм.
Другие задания:
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