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Номер 2.28, страница 59 - гдз по геометрии 8 класс учебник Шыныбеков, Шыныбеков
Авторы: Шыныбеков А. Н., Шыныбеков Д. А., Жумабаев Р. Н.
Тип: Учебник
Издательство: Атамұра
Год издания: 2018 - 2026
ISBN: 978-601-331-161-6
Рекомендовано Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 8 классе
Раздел 2. Соотношения между сторонами и углами прямоугольного треугольника. 2.1. Теорема о пропорциональных отрезках. Теорема Пифагора - номер 2.28, страница 59.
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" => 1122696 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "59" "field_page_end" => null "field_url" => "/8-klass/geometrija/shynybekov-uchebnik/2-28" "field_display_title" => "2.28" "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 {#1072 #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" => 1122381 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Соотношения между сторонами и углами прямоугольного треугольника" "field_branch_order" => "2" "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" => 30 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Раздел" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 30 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Раздел" "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" => "52" "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" => 2592 "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 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2018" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 8 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "538" "field_priority" => "6" "field_default_folder" => "/geometrija_08/Shynybekov-u17/" "field_isbn" => "978-601-331-161-6" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1067 …2} "field_url" => "/8-klass/geometrija/shynybekov-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 2592 "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 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2018" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 8 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "538" "field_priority" => "6" "field_default_folder" => "/geometrija_08/Shynybekov-u17/" "field_isbn" => "978-601-331-161-6" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1067 …2} "field_url" => "/8-klass/geometrija/shynybekov-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 {#1069 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1122381 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Соотношения между сторонами и углами прямоугольного треугольника" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039} "field_page_start" => "52" "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 {#1069} ] #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 {#1106 #items: array:2 [ 0 => App\Models\Branch {#1114 #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" => 1122381 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Соотношения между сторонами и углами прямоугольного треугольника" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1118 #items: array:1 [ 0 => App\Models\Term {#1115 #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" => 30 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Раздел" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 30 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Раздел" "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" => "52" "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 {#1117 #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" => 2592 "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 {#1119 #items: array:1 [ 0 => App\Models\Term {#1116 #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 {#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" => 5457 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "8" "field_cases" => array:6 [ …6] "field_translit" => "vosmoj" ] #original: array:6 [ "id" => 5457 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "8" "field_cases" => array:6 [ …6] "field_translit" => "vosmoj" ] #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 {#1122 #items: array:1 [ 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:6 [ "id" => 6991 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Атамұра" "field_cases" => array:6 [ …6] "field_translit" => "atamu'ra" ] #original: array:6 [ "id" => 6991 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Атамұра" "field_cases" => array:6 [ …6] "field_translit" => "atamu'ra" ] #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 {#1124 #items: array:3 [ 0 => App\Models\Term {#1125 #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" => 6992 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Шыныбеков" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Абдухали" "field_patronymic" => "Насырович" "field_surname_rp" => null ] #original: array:12 [ "id" => 6992 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Шыныбеков" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Абдухали" "field_patronymic" => "Насырович" "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 {#1126 #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" => 6993 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Шыныбеков" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Данияр" "field_patronymic" => "Абдухалиевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 6993 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Шыныбеков" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Данияр" "field_patronymic" => "Абдухалиевич" "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 {#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:12 [ "id" => 7001 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумабаев" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Ринат" "field_patronymic" => "Нурланович" "field_surname_rp" => null ] #original: array:12 [ "id" => 7001 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумабаев" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Ринат" "field_patronymic" => "Нурланович" "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 {#1128 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1129 #items: array:1 [ 0 => App\Models\Term {#1130 #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 {#1131 #items: array:1 [ 0 => App\Models\Term {#1132 #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 {#1133 #items: array:1 [ 0 => App\Models\Term {#1134 #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 {#1135 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1136 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1137 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1138 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1139 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1140 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2018" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 8 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "538" "field_priority" => "6" "field_default_folder" => "/geometrija_08/Shynybekov-u17/" "field_isbn" => "978-601-331-161-6" "field_cover" => array:1 [ 0 => "/media/geometrija_08/Shynybekov-u17/covers/cover1.webp?ts=1753258124" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_08/Shynybekov-u17/covers/cover1.webp?ts=1753258124" "alt" => "" "width" => "1632" "height" => "2472" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1141 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1142 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1143 #items: [] #escapeWhenCastingToString: false } "field_url" => "/8-klass/geometrija/shynybekov-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1144 #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" => 379 "subject" => 543 "class_subject" => 390 ] ] #original: array:50 [ "id" => 2592 "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 {#1119} "field_class" => Illuminate\Database\Eloquent\Collection {#1120} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1122} "field_author" => Illuminate\Database\Eloquent\Collection {#1124} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1128} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1129} "field_country" => Illuminate\Database\Eloquent\Collection {#1131} "field_city" => Illuminate\Database\Eloquent\Collection {#1133} "field_series" => Illuminate\Database\Eloquent\Collection {#1135} "field_umk" => Illuminate\Database\Eloquent\Collection {#1136} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1137} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1138} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1139} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1140} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2018" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 8 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "538" "field_priority" => "6" "field_default_folder" => "/geometrija_08/Shynybekov-u17/" "field_isbn" => "978-601-331-161-6" "field_cover" => array:1 [ 0 => "/media/geometrija_08/Shynybekov-u17/covers/cover1.webp?ts=1753258124" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_08/Shynybekov-u17/covers/cover1.webp?ts=1753258124" "alt" => "" "width" => "1632" "height" => "2472" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1141} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1142} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1143} "field_url" => "/8-klass/geometrija/shynybekov-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1144} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 379 "subject" => 543 "class_subject" => 390 ] ] #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 {#1145 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1122381 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Соотношения между сторонами и углами прямоугольного треугольника" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1118} "field_page_start" => "52" "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 {#1145} ] #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 {#1146 #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" => 1122382 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "2.1. Теорема о пропорциональных отрезках. Теорема Пифагора" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1148 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "52" "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 {#1149 #items: array:1 [ 0 => App\Models\Book {#1117} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1150 #items: array:1 [ 0 => App\Models\Branch {#1034} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1122382 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "2.1. Теорема о пропорциональных отрезках. Теорема Пифагора" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1148} "field_page_start" => "52" "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 {#1149} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1150} ] #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 {#1070 #items: array:4 [ 0 => App\Models\Element {#1095 #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" => 1305083 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1086 #items: array:1 [ 0 => App\Models\Edition {#1094 #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" => 5006 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Учебник rus" "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" => "0rus-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1090 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5006 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Учебник rus" "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" => "0rus-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1090 …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" => "1122696" "type" => "task" ] "text" => "<p><strong>2.28.</strong> Сформулируйте теорему, обратную теореме Пифагора.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "28-1.jpg" "alt" => null "width" => "1310" "height" => 179 "path" => "/media/geometrija_08/Shynybekov-u17/0rus-2/28-1.webp?ts=1753776374" ] ] ] #original: array:7 [ "id" => 1305083 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1086} "task" => array:2 [ "refs" => "1122696" "type" => "task" ] "text" => "<p><strong>2.28.</strong> Сформулируйте теорему, обратную теореме Пифагора.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "28-1.jpg" "alt" => null "width" => "1310" "height" => 179 "path" => "/media/geometrija_08/Shynybekov-u17/0rus-2/28-1.webp?ts=1753776374" ] ] ] #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 {#1087 #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" => 1305433 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1079 #items: array:1 [ 0 => App\Models\Edition {#1088 #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" => 5007 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Учебник kz" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "image" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0kz-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1083 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1080 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5007 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Учебник kz" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "image" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0kz-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1083 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1080 …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" => "1122696" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "28-1.jpg" "alt" => null "width" => "1304" "height" => 171 "path" => "/media/geometrija_08/Shynybekov-u17/0kz-2/28-1.webp?ts=1753776421" ] ] ] #original: array:6 [ "id" => 1305433 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1079} "task" => array:2 [ "refs" => "1122696" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "28-1.jpg" "alt" => null "width" => "1304" "height" => 171 "path" => "/media/geometrija_08/Shynybekov-u17/0kz-2/28-1.webp?ts=1753776421" ] ] ] #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 {#1081 #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" => 1306093 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1109 #items: array:1 [ 0 => App\Models\Edition {#1078 #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" => 5008 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1077 …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 {#1073 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1071 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5008 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1077 …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 {#1073 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1071 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1105 …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" => "1122696" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "28-1.jpg" "alt" => null "width" => "1622" "height" => 199 "path" => "/media/geometrija_08/Shynybekov-u17/1-2/28-1.webp?ts=1753776482" ] ] ] #original: array:6 [ "id" => 1306093 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1109} "task" => array:2 [ "refs" => "1122696" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "28-1.jpg" "alt" => null "width" => "1622" "height" => 199 "path" => "/media/geometrija_08/Shynybekov-u17/1-2/28-1.webp?ts=1753776482" ] ] ] #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\Element {#1104 #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" => 1480931 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1155 #items: array:1 [ 0 => App\Models\Edition {#1113 #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" => 5435 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 rus" "field_order" => "4" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1110 …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 {#1108 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1151 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1152 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5435 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 rus" "field_order" => "4" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1110 …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 {#1108 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1151 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1152 …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" => "1122696" "type" => "task" ] "text" => "<p>Для того чтобы сформулировать теорему, обратную теореме Пифагора, сначала следует вспомнить прямую теорему Пифагора. Она гласит, что <em>в прямоугольном треугольнике квадрат длины гипотенузы равен сумме квадратов длин катетов</em>. Если $a$ и $b$ — длины катетов, а $c$ — длина гипотенузы, то выполняется равенство: $c^2 = a^2 + b^2$.</p><p>Теорема, обратная теореме Пифагора, формулируется путем замены местами условия и заключения прямой теоремы.<br><strong>Формулировка теоремы, обратной теореме Пифагора:</strong><br><em>Если в треугольнике квадрат длины одной стороны равен сумме квадратов длин двух других сторон, то этот треугольник является прямоугольным.</em><br>При этом прямой угол — это угол, лежащий напротив той стороны, квадрат которой равен сумме квадратов двух других (то есть, наибольшей стороны).</p><p><strong>Доказательство:</strong><br>Пусть нам дан треугольник $\triangle ABC$ со сторонами $a$, $b$ и $c$, для которого выполняется равенство $c^2 = a^2 + b^2$. Необходимо доказать, что угол $\angle C$, лежащий напротив стороны $c$, является прямым ($90^\circ$).<br>Для доказательства рассмотрим второй треугольник $\triangle A'B'C'$, у которого угол $\angle C'$ — прямой, а катеты $A'C'$ и $B'C'$ равны сторонам $b$ и $a$ соответственно: $A'C' = b$, $B'C' = a$.<br>Для этого построенного прямоугольного треугольника $\triangle A'B'C'$ применим прямую теорему Пифагора, чтобы найти квадрат его гипотенузы $c'$:<br>$(c')^2 = (A'C')^2 + (B'C')^2 = b^2 + a^2$.<br>По условию задачи для исходного треугольника $\triangle ABC$ нам дано, что $c^2 = a^2 + b^2$.<br>Сравнивая два полученных выражения, видим, что $(c')^2 = c^2$. Так как длины сторон являются положительными величинами, то отсюда следует, что $c' = c$.<br>Таким образом, треугольники $\triangle ABC$ и $\triangle A'B'C'$ имеют соответственно равные стороны: $a=a$, $b=b$ и $c=c'$.<br>Следовательно, по третьему признаку равенства треугольников (по трем сторонам), треугольник $\triangle ABC$ равен треугольнику $\triangle A'B'C'$ ($\triangle ABC \cong \triangle A'B'C'$).<br>Из равенства треугольников следует равенство их соответствующих углов. В частности, угол $\angle C$ треугольника $\triangle ABC$ равен углу $\angle C'$ треугольника $\triangle A'B'C'$.<br>Поскольку по построению угол $\angle C'$ является прямым ($\angle C' = 90^\circ$), то и угол $\angle C$ также равен $90^\circ$.<br>Это доказывает, что треугольник $\triangle ABC$ является прямоугольным.</p><p>Ответ: Если квадрат одной стороны треугольника равен сумме квадратов двух других его сторон (например, $c^2 = a^2 + b^2$), то такой треугольник является прямоугольным, и прямой угол в нем находится напротив стороны $c$.</p>" ] #original: array:6 [ "id" => 1480931 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1155} "task" => array:2 [ "refs" => "1122696" "type" => "task" ] "text" => "<p>Для того чтобы сформулировать теорему, обратную теореме Пифагора, сначала следует вспомнить прямую теорему Пифагора. Она гласит, что <em>в прямоугольном треугольнике квадрат длины гипотенузы равен сумме квадратов длин катетов</em>. Если $a$ и $b$ — длины катетов, а $c$ — длина гипотенузы, то выполняется равенство: $c^2 = a^2 + b^2$.</p><p>Теорема, обратная теореме Пифагора, формулируется путем замены местами условия и заключения прямой теоремы.<br><strong>Формулировка теоремы, обратной теореме Пифагора:</strong><br><em>Если в треугольнике квадрат длины одной стороны равен сумме квадратов длин двух других сторон, то этот треугольник является прямоугольным.</em><br>При этом прямой угол — это угол, лежащий напротив той стороны, квадрат которой равен сумме квадратов двух других (то есть, наибольшей стороны).</p><p><strong>Доказательство:</strong><br>Пусть нам дан треугольник $\triangle ABC$ со сторонами $a$, $b$ и $c$, для которого выполняется равенство $c^2 = a^2 + b^2$. Необходимо доказать, что угол $\angle C$, лежащий напротив стороны $c$, является прямым ($90^\circ$).<br>Для доказательства рассмотрим второй треугольник $\triangle A'B'C'$, у которого угол $\angle C'$ — прямой, а катеты $A'C'$ и $B'C'$ равны сторонам $b$ и $a$ соответственно: $A'C' = b$, $B'C' = a$.<br>Для этого построенного прямоугольного треугольника $\triangle A'B'C'$ применим прямую теорему Пифагора, чтобы найти квадрат его гипотенузы $c'$:<br>$(c')^2 = (A'C')^2 + (B'C')^2 = b^2 + a^2$.<br>По условию задачи для исходного треугольника $\triangle ABC$ нам дано, что $c^2 = a^2 + b^2$.<br>Сравнивая два полученных выражения, видим, что $(c')^2 = c^2$. Так как длины сторон являются положительными величинами, то отсюда следует, что $c' = c$.<br>Таким образом, треугольники $\triangle ABC$ и $\triangle A'B'C'$ имеют соответственно равные стороны: $a=a$, $b=b$ и $c=c'$.<br>Следовательно, по третьему признаку равенства треугольников (по трем сторонам), треугольник $\triangle ABC$ равен треугольнику $\triangle A'B'C'$ ($\triangle ABC \cong \triangle A'B'C'$).<br>Из равенства треугольников следует равенство их соответствующих углов. В частности, угол $\angle C$ треугольника $\triangle ABC$ равен углу $\angle C'$ треугольника $\triangle A'B'C'$.<br>Поскольку по построению угол $\angle C'$ является прямым ($\angle C' = 90^\circ$), то и угол $\angle C$ также равен $90^\circ$.<br>Это доказывает, что треугольник $\triangle ABC$ является прямоугольным.</p><p>Ответ: Если квадрат одной стороны треугольника равен сумме квадратов двух других его сторон (например, $c^2 = a^2 + b^2$), то такой треугольник является прямоугольным, и прямой угол в нем находится напротив стороны $c$.</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" => "1122697" "type" => "task" ] "previous" => array:2 [ "refs" => "1122695" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1097 #items: array:1 [ 0 => App\Models\Book {#1117} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1074 #items: array:1 [ 0 => App\Models\BookPage {#1102 #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" => 1114261 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "59" "field_url" => "/8-klass/geometrija/shynybekov-uchebnik/page-59" "field_display_title" => "59" "field_folder" => "1" "field_image_name" => "59" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1101 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1075 #items: array:1 [ 0 => App\Models\Book {#1117} ] #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 {#1153 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1156 #items: array:2 [ 0 => App\Models\Element {#1165 #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" => 1290382 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1166 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1290382 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1166 …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 => "*" ] } 1 => App\Models\Element {#1167 #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" => 1290407 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1168 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1290407 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1168 …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" => "1114262" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1114260" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1272 #items: array:13 [ 0 => 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" => 1122685 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "59" "field_page_end" => null "field_url" => "/8-klass/geometrija/shynybekov-uchebnik/2-17" "field_display_title" => "2.17" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1293 …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 {#1294 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1295 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1296 …2} "content" => Illuminate\Database\Eloquent\Collection {#1298 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1307 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1122685 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "59" "field_page_end" => null "field_url" => "/8-klass/geometrija/shynybekov-uchebnik/2-17" "field_display_title" => "2.17" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1293 …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 {#1294 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1295 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1296 …2} "content" => Illuminate\Database\Eloquent\Collection {#1298 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1307 …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 {#1299 #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" => 1122686 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "59" "field_page_end" => null "field_url" => "/8-klass/geometrija/shynybekov-uchebnik/2-18" "field_display_title" => "2.18" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1297 …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 {#1305 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1304 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1306 …2} "content" => Illuminate\Database\Eloquent\Collection {#1300 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1323 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1122686 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "59" "field_page_end" => null "field_url" => "/8-klass/geometrija/shynybekov-uchebnik/2-18" "field_display_title" => "2.18" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1297 …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 {#1305 …2} "top_parent_branch" => 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"/8-klass/geometrija/shynybekov-uchebnik/2-19" "field_display_title" => "2.19" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1302 …2} "field_metatags_title" => null …14 ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 3 => App\Models\Task {#1319 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true 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#dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 11 => App\Models\Task {#1447 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 12 => App\Models\Task {#1463 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 1114261 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "59" "field_url" => "/8-klass/geometrija/shynybekov-uchebnik/page-59" "field_display_title" => "59" "field_folder" => "1" "field_image_name" => "59" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1101} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1075} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null 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[] #guarded: array:1 [ 0 => "*" ] }
№2.28 (с. 59)
Учебник rus. №2.28 (с. 59)
скриншот условия
2.28. Сформулируйте теорему, обратную теореме Пифагора.
Учебник kz. №2.28 (с. 59)
Решение. №2.28 (с. 59)
Решение 2 rus. №2.28 (с. 59)
Для того чтобы сформулировать теорему, обратную теореме Пифагора, сначала следует вспомнить прямую теорему Пифагора. Она гласит, что в прямоугольном треугольнике квадрат длины гипотенузы равен сумме квадратов длин катетов. Если $a$ и $b$ — длины катетов, а $c$ — длина гипотенузы, то выполняется равенство: $c^2 = a^2 + b^2$.
Теорема, обратная теореме Пифагора, формулируется путем замены местами условия и заключения прямой теоремы.
Формулировка теоремы, обратной теореме Пифагора:
Если в треугольнике квадрат длины одной стороны равен сумме квадратов длин двух других сторон, то этот треугольник является прямоугольным.
При этом прямой угол — это угол, лежащий напротив той стороны, квадрат которой равен сумме квадратов двух других (то есть, наибольшей стороны).
Доказательство:
Пусть нам дан треугольник $\triangle ABC$ со сторонами $a$, $b$ и $c$, для которого выполняется равенство $c^2 = a^2 + b^2$. Необходимо доказать, что угол $\angle C$, лежащий напротив стороны $c$, является прямым ($90^\circ$).
Для доказательства рассмотрим второй треугольник $\triangle A'B'C'$, у которого угол $\angle C'$ — прямой, а катеты $A'C'$ и $B'C'$ равны сторонам $b$ и $a$ соответственно: $A'C' = b$, $B'C' = a$.
Для этого построенного прямоугольного треугольника $\triangle A'B'C'$ применим прямую теорему Пифагора, чтобы найти квадрат его гипотенузы $c'$:
$(c')^2 = (A'C')^2 + (B'C')^2 = b^2 + a^2$.
По условию задачи для исходного треугольника $\triangle ABC$ нам дано, что $c^2 = a^2 + b^2$.
Сравнивая два полученных выражения, видим, что $(c')^2 = c^2$. Так как длины сторон являются положительными величинами, то отсюда следует, что $c' = c$.
Таким образом, треугольники $\triangle ABC$ и $\triangle A'B'C'$ имеют соответственно равные стороны: $a=a$, $b=b$ и $c=c'$.
Следовательно, по третьему признаку равенства треугольников (по трем сторонам), треугольник $\triangle ABC$ равен треугольнику $\triangle A'B'C'$ ($\triangle ABC \cong \triangle A'B'C'$).
Из равенства треугольников следует равенство их соответствующих углов. В частности, угол $\angle C$ треугольника $\triangle ABC$ равен углу $\angle C'$ треугольника $\triangle A'B'C'$.
Поскольку по построению угол $\angle C'$ является прямым ($\angle C' = 90^\circ$), то и угол $\angle C$ также равен $90^\circ$.
Это доказывает, что треугольник $\triangle ABC$ является прямоугольным.
Ответ: Если квадрат одной стороны треугольника равен сумме квадратов двух других его сторон (например, $c^2 = a^2 + b^2$), то такой треугольник является прямоугольным, и прямой угол в нем находится напротив стороны $c$.
Другие задания:
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