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Номер 2.2, страница 57 - 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 2.2, страница 57.
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" => 1122670 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "57" "field_page_end" => null "field_url" => "/8-klass/geometrija/shynybekov-uchebnik/2-2" "field_display_title" => "2.2" "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" => 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 {#1050 #items: array:1 [ 0 => App\Models\Term {#1047 #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 [ "field_accusative_case" => "Раздел" "field_creative_case" => "Разделом" "field_dative_case" => "Разделу" "field_genitive_case" => "Раздела" "field_nominative_case" => "Раздел" "field_prepositional_case" => "Разделе" ] "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 [ "field_accusative_case" => "Раздел" "field_creative_case" => "Разделом" "field_dative_case" => "Разделу" "field_genitive_case" => "Раздела" "field_nominative_case" => "Раздел" "field_prepositional_case" => "Разделе" ] "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 {#1058 #items: array:1 [ 0 => App\Models\Book {#1057 #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 {#1062 #items: array:1 [ 0 => App\Models\Term {#1059 #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 {#1070 #items: array:1 [ 0 => App\Models\Term {#1069 #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 [ "field_accusative_case" => "восьмой" "field_creative_case" => "восьмым" "field_dative_case" => "восьмому" "field_genitive_case" => "восьмого" "field_nominative_case" => "восьмой" "field_prepositional_case" => "восьмом" ] "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 [ "field_accusative_case" => "восьмой" "field_creative_case" => "восьмым" "field_dative_case" => "восьмому" "field_genitive_case" => "восьмого" "field_nominative_case" => "восьмой" "field_prepositional_case" => "восьмом" ] "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 {#1073 #items: array:1 [ 0 => App\Models\Term {#1072 #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 [ "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" => "atamu'ra" ] #original: array:6 [ "id" => 6991 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Атамұра" "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" => "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 {#1060 #items: array:3 [ 0 => App\Models\Term {#1078 #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 {#1079 #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 {#1080 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array: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 {#1075 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1076 #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 {#1084 #items: array:1 [ 0 => App\Models\Term {#1083 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 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 {#1087 #items: array:1 [ 0 => App\Models\Term {#1086 #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 {#1065 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1074 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1063 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1081 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1066 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1064 #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 {#1067 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1088 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1089 #items: [] #escapeWhenCastingToString: false } "field_url" => "/8-klass/geometrija/shynybekov-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1090 #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 {#1062} "field_class" => Illuminate\Database\Eloquent\Collection {#1070} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1073} "field_author" => Illuminate\Database\Eloquent\Collection {#1060} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1075} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1076} "field_country" => Illuminate\Database\Eloquent\Collection {#1084} "field_city" => Illuminate\Database\Eloquent\Collection {#1087} "field_series" => Illuminate\Database\Eloquent\Collection {#1065} "field_umk" => Illuminate\Database\Eloquent\Collection {#1074} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1063} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1081} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1066} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1064} "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 {#1067} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1088} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1089} "field_url" => "/8-klass/geometrija/shynybekov-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1090} "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 {#1048 #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 {#1050} "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 {#1058} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1048} ] #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 {#1113 #items: array:2 [ 0 => App\Models\Branch {#1121 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 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 {#1122 #items: array:1 [ 0 => App\Models\Term {#1047} ] #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 {#1123 #items: array:1 [ 0 => App\Models\Book {#1057} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1124 #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 {#1122} "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 {#1123} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1124} ] #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 {#1125 #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 {#1126 #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 {#1127 #items: array:1 [ 0 => App\Models\Book {#1057} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1128 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #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 {#1126} "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 {#1127} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1128} ] #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 {#1038 #items: array:4 [ 0 => App\Models\Element {#1102 #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" => 1305033 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1055 #items: array:1 [ 0 => App\Models\Edition {#1093 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5006 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Учебник rus" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1092 #items: array:1 [ …1] #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" => "0rus-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1049 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1053 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1054 #items: [] #escapeWhenCastingToString: false } "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 {#1092} "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 {#1049} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1053} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1054} "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" => "1122670" "type" => "task" ] "text" => "<p>2.2. Постройте угол, косинус которого равен:</p><p><strong>1)</strong> $ \frac{3}{5} $;</p><p><strong>2)</strong> $ \frac{4}{9} $;</p><p><strong>3)</strong> 0,5;</p><p><strong>4)</strong> 0,8.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "2-1.jpg" "alt" => null "width" => "1629" "height" => 177 "path" => "/media/geometrija_08/Shynybekov-u17/0rus-2/2-1.webp?ts=1753776351" ] ] ] #original: array:7 [ "id" => 1305033 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1055} "task" => array:2 [ "refs" => "1122670" "type" => "task" ] "text" => "<p>2.2. Постройте угол, косинус которого равен:</p><p><strong>1)</strong> $ \frac{3}{5} $;</p><p><strong>2)</strong> $ \frac{4}{9} $;</p><p><strong>3)</strong> 0,5;</p><p><strong>4)</strong> 0,8.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "2-1.jpg" "alt" => null "width" => "1629" "height" => 177 "path" => "/media/geometrija_08/Shynybekov-u17/0rus-2/2-1.webp?ts=1753776351" ] ] ] #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 {#1100 #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" => 1305382 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1051 #items: array:1 [ 0 => App\Models\Edition {#1043 #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 {#1041 #items: array:1 [ …1] #escapeWhenCastingToString: false } "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 {#1114 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1112 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1111 #items: [] #escapeWhenCastingToString: false } "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 {#1041} "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 {#1114} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1112} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1111} "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" => "1122670" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "2-1.jpg" "alt" => null "width" => "1767" "height" => 220 "path" => "/media/geometrija_08/Shynybekov-u17/0kz-2/2-1.webp?ts=1753776398" ] ] ] #original: array:6 [ "id" => 1305382 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1051} "task" => array:2 [ "refs" => "1122670" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "2-1.jpg" "alt" => null "width" => "1767" "height" => 220 "path" => "/media/geometrija_08/Shynybekov-u17/0kz-2/2-1.webp?ts=1753776398" ] ] ] #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 {#1101 #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" => 1306051 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116 #items: array:1 [ 0 => App\Models\Edition {#1117 #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 {#1138 #items: array:1 [ …1] #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 {#1135 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1131 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1115 #items: [] #escapeWhenCastingToString: false } "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 {#1138} "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 {#1135} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1131} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1115} "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" => "1122670" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "2-1.jpg" "alt" => null "width" => "1407" "height" => 603 "path" => "/media/geometrija_08/Shynybekov-u17/1-2/2-1.webp?ts=1753776461" ] ] ] #original: array:6 [ "id" => 1306051 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116} "task" => array:2 [ "refs" => "1122670" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "2-1.jpg" "alt" => null "width" => "1407" "height" => 603 "path" => "/media/geometrija_08/Shynybekov-u17/1-2/2-1.webp?ts=1753776461" ] ] ] #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 {#1098 #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" => 1480905 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1132 #items: array:1 [ 0 => App\Models\Edition {#1133 #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 {#1149 #items: array:1 [ …1] #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 {#1146 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1142 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1139 #items: [] #escapeWhenCastingToString: false } "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 {#1149} "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 {#1146} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1142} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1139} "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" => "1122670" "type" => "task" ] "text" => "<p>Для построения угла по заданному значению его косинуса мы будем использовать определение косинуса острого угла в прямоугольном треугольнике. Косинус угла — это отношение прилежащего к этому углу катета к гипотенузе. Следовательно, для построения угла $ \alpha $, косинус которого равен $ \frac{a}{c} $, необходимо построить прямоугольный треугольник с прилежащим катетом $ a $ и гипотенузой $ c $. Угол между этим катетом и гипотенузой и будет искомым углом $ \alpha $.</p><p><strong>1</strong>) $ \frac{3}{5} $</p><p>Чтобы построить угол, косинус которого равен $ \frac{3}{5} $, нужно построить прямоугольный треугольник, у которого прилежащий катет равен 3 условным единицам, а гипотенуза — 5 условным единицам.<br><strong>Порядок построения:</strong><br>1. Начертим луч с началом в точке A. На этом луче отложим отрезок AB, равный 3 условным единицам. Этот отрезок будет прилежащим катетом.<br>2. В точке B восстановим перпендикуляр к лучу AB.<br>3. Из точки A как из центра проведем дугу окружности радиусом 5 условных единиц (длина гипотенузы).<br>4. Точку пересечения дуги с перпендикуляром обозначим C.<br>5. Соединим точки A и C.<br>В результате построен прямоугольный треугольник ABC с прямым углом B. Угол $ \angle BAC $ является искомым, так как по построению $ \cos(\angle BAC) = \frac{AB}{AC} = \frac{3}{5} $.<br><strong>Ответ:</strong> Угол $ \angle BAC $ в построенном треугольнике является искомым.</p><p><strong>2</strong>) $ \frac{4}{9} $</p><p>Построим прямоугольный треугольник, у которого прилежащий катет равен 4 условным единицам, а гипотенуза — 9 условным единицам.<br><strong>Порядок построения:</strong><br>1. Начертим луч с началом в точке A и отложим на нем отрезок AB длиной 4 единицы.<br>2. В точке B построим перпендикуляр к AB.<br>3. Из точки A проведем дугу окружности радиусом 9 единиц.<br>4. Точку пересечения дуги с перпендикуляром обозначим C.<br>5. Соединим точки A и C.<br>Полученный угол $ \angle BAC $ в прямоугольном треугольнике ABC является искомым, поскольку $ \cos(\angle BAC) = \frac{AB}{AC} = \frac{4}{9} $.<br><strong>Ответ:</strong> Угол $ \angle BAC $ в построенном треугольнике является искомым.</p><p><strong>3</strong>) 0,5</p><p>Сначала представим десятичную дробь в виде обыкновенной: $ 0,5 = \frac{5}{10} = \frac{1}{2} $.<br>Теперь задача сводится к построению угла, косинус которого равен $ \frac{1}{2} $. Построим прямоугольный треугольник с прилежащим катетом равным 1 условной единице и гипотенузой равной 2 условным единицам.<br><strong>Порядок построения:</strong><br>1. Начертим луч с началом в точке A и отложим на нем отрезок AB длиной 1 единицу.<br>2. В точке B построим перпендикуляр к AB.<br>3. Из точки A проведем дугу окружности радиусом 2 единицы.<br>4. Точку пересечения дуги с перпендикуляром обозначим C.<br>5. Соединим точки A и C.<br>Полученный угол $ \angle BAC $ в прямоугольном треугольнике ABC является искомым, так как $ \cos(\angle BAC) = \frac{AB}{AC} = \frac{1}{2} $. (Этот угол равен $ 60^\circ $).<br><strong>Ответ:</strong> Угол $ \angle BAC $ в построенном треугольнике является искомым.</p><p><strong>4</strong>) 0,8</p><p>Представим десятичную дробь в виде обыкновенной и сократим ее: $ 0,8 = \frac{8}{10} = \frac{4}{5} $.<br>Задача сводится к построению угла, косинус которого равен $ \frac{4}{5} $. Построим прямоугольный треугольник с прилежащим катетом равным 4 условным единицам и гипотенузой равной 5 условным единицам.<br><strong>Порядок построения:</strong><br>1. Начертим луч с началом в точке A и отложим на нем отрезок AB длиной 4 единицы.<br>2. В точке B построим перпендикуляр к AB.<br>3. Из точки A проведем дугу окружности радиусом 5 единиц.<br>4. Точку пересечения дуги с перпендикуляром обозначим C.<br>5. Соединим точки A и C.<br>Полученный угол $ \angle BAC $ в прямоугольном треугольнике ABC является искомым, поскольку $ \cos(\angle BAC) = \frac{AB}{AC} = \frac{4}{5} $.<br><strong>Ответ:</strong> Угол $ \angle BAC $ в построенном треугольнике является искомым.</p>" ] #original: array:6 [ "id" => 1480905 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1132} "task" => array:2 [ "refs" => "1122670" "type" => "task" ] "text" => "<p>Для построения угла по заданному значению его косинуса мы будем использовать определение косинуса острого угла в прямоугольном треугольнике. Косинус угла — это отношение прилежащего к этому углу катета к гипотенузе. Следовательно, для построения угла $ \alpha $, косинус которого равен $ \frac{a}{c} $, необходимо построить прямоугольный треугольник с прилежащим катетом $ a $ и гипотенузой $ c $. Угол между этим катетом и гипотенузой и будет искомым углом $ \alpha $.</p><p><strong>1</strong>) $ \frac{3}{5} $</p><p>Чтобы построить угол, косинус которого равен $ \frac{3}{5} $, нужно построить прямоугольный треугольник, у которого прилежащий катет равен 3 условным единицам, а гипотенуза — 5 условным единицам.<br><strong>Порядок построения:</strong><br>1. Начертим луч с началом в точке A. На этом луче отложим отрезок AB, равный 3 условным единицам. Этот отрезок будет прилежащим катетом.<br>2. В точке B восстановим перпендикуляр к лучу AB.<br>3. Из точки A как из центра проведем дугу окружности радиусом 5 условных единиц (длина гипотенузы).<br>4. Точку пересечения дуги с перпендикуляром обозначим C.<br>5. Соединим точки A и C.<br>В результате построен прямоугольный треугольник ABC с прямым углом B. Угол $ \angle BAC $ является искомым, так как по построению $ \cos(\angle BAC) = \frac{AB}{AC} = \frac{3}{5} $.<br><strong>Ответ:</strong> Угол $ \angle BAC $ в построенном треугольнике является искомым.</p><p><strong>2</strong>) $ \frac{4}{9} $</p><p>Построим прямоугольный треугольник, у которого прилежащий катет равен 4 условным единицам, а гипотенуза — 9 условным единицам.<br><strong>Порядок построения:</strong><br>1. Начертим луч с началом в точке A и отложим на нем отрезок AB длиной 4 единицы.<br>2. В точке B построим перпендикуляр к AB.<br>3. Из точки A проведем дугу окружности радиусом 9 единиц.<br>4. Точку пересечения дуги с перпендикуляром обозначим C.<br>5. Соединим точки A и C.<br>Полученный угол $ \angle BAC $ в прямоугольном треугольнике ABC является искомым, поскольку $ \cos(\angle BAC) = \frac{AB}{AC} = \frac{4}{9} $.<br><strong>Ответ:</strong> Угол $ \angle BAC $ в построенном треугольнике является искомым.</p><p><strong>3</strong>) 0,5</p><p>Сначала представим десятичную дробь в виде обыкновенной: $ 0,5 = \frac{5}{10} = \frac{1}{2} $.<br>Теперь задача сводится к построению угла, косинус которого равен $ \frac{1}{2} $. Построим прямоугольный треугольник с прилежащим катетом равным 1 условной единице и гипотенузой равной 2 условным единицам.<br><strong>Порядок построения:</strong><br>1. Начертим луч с началом в точке A и отложим на нем отрезок AB длиной 1 единицу.<br>2. В точке B построим перпендикуляр к AB.<br>3. Из точки A проведем дугу окружности радиусом 2 единицы.<br>4. Точку пересечения дуги с перпендикуляром обозначим C.<br>5. Соединим точки A и C.<br>Полученный угол $ \angle BAC $ в прямоугольном треугольнике ABC является искомым, так как $ \cos(\angle BAC) = \frac{AB}{AC} = \frac{1}{2} $. (Этот угол равен $ 60^\circ $).<br><strong>Ответ:</strong> Угол $ \angle BAC $ в построенном треугольнике является искомым.</p><p><strong>4</strong>) 0,8</p><p>Представим десятичную дробь в виде обыкновенной и сократим ее: $ 0,8 = \frac{8}{10} = \frac{4}{5} $.<br>Задача сводится к построению угла, косинус которого равен $ \frac{4}{5} $. Построим прямоугольный треугольник с прилежащим катетом равным 4 условным единицам и гипотенузой равной 5 условным единицам.<br><strong>Порядок построения:</strong><br>1. Начертим луч с началом в точке A и отложим на нем отрезок AB длиной 4 единицы.<br>2. В точке B построим перпендикуляр к AB.<br>3. Из точки A проведем дугу окружности радиусом 5 единиц.<br>4. Точку пересечения дуги с перпендикуляром обозначим C.<br>5. Соединим точки A и C.<br>Полученный угол $ \angle BAC $ в прямоугольном треугольнике ABC является искомым, поскольку $ \cos(\angle BAC) = \frac{AB}{AC} = \frac{4}{5} $.<br><strong>Ответ:</strong> Угол $ \angle BAC $ в построенном треугольнике является искомым.</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" => "1122671" "type" => "task" ] "previous" => array:2 [ "refs" => "1122669" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1104 #items: array:1 [ 0 => App\Models\Book {#1057} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1042 #items: array:1 [ 0 => App\Models\BookPage {#1109 #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" => 1114259 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "57" "field_url" => "/8-klass/geometrija/shynybekov-uchebnik/page-57" "field_display_title" => "57" "field_folder" => "1" "field_image_name" => "57" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1108 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1040 #items: array:1 [ 0 => App\Models\Book {#1057} ] #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 {#1097 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1096 #items: array:2 [ 0 => App\Models\Element {#1150 #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" => 1290378 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1151 #items: array:1 [ …1] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1114259" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ …5] ] ] #original: array:6 [ "id" => 1290378 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1151} "book_page" => array:2 [ "refs" => "1114259" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ …5] ] ] #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 {#1153 #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" => 1290402 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1154 #items: array:1 [ …1] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1114259" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ …5] ] ] #original: array:6 [ "id" => 1290402 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1154} "book_page" => array:2 [ "refs" => "1114259" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ …5] ] ] #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" => "1114260" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1114258" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1228 #items: array:6 [ 0 => App\Models\Task {#1240 #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" => 1122668 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "57" "field_page_end" => null "field_url" => "/8-klass/geometrija/shynybekov-uchebnik/tvorrab-57" "field_display_title" => "Творческая работа" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1241 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1242 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1243 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1244 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1247 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1122669" "type" => "task" ] "previous" => array:2 [ "refs" => "1122667" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1253 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1114259" "type" => "book_page" ] ] #original: array:24 [ "id" => 1122668 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "57" "field_page_end" => null "field_url" => "/8-klass/geometrija/shynybekov-uchebnik/tvorrab-57" "field_display_title" => "Творческая работа" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1241} "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 {#1242} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1243} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1244} "content" => Illuminate\Database\Eloquent\Collection {#1247} "next" => array:2 [ "refs" => "1122669" "type" => "task" ] "previous" => array:2 [ "refs" => "1122667" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1253} "page" => array:2 [ "refs" => "1114259" "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 {#1246 #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" => 1122669 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "57" "field_page_end" => null "field_url" => "/8-klass/geometrija/shynybekov-uchebnik/2-1" "field_display_title" => "2.1" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1245 #items: array:1 [ …1] #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 {#1252 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1254 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1250 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1251 #items: array:4 [ …4] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1122670" "type" => "task" ] "previous" => array:2 [ "refs" => "1122668" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1269 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1114259" "type" => "book_page" ] ] #original: array:24 [ "id" => 1122669 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "57" "field_page_end" => null "field_url" => "/8-klass/geometrija/shynybekov-uchebnik/2-1" "field_display_title" => "2.1" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1245} "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 {#1252} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1254} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1250} "content" => Illuminate\Database\Eloquent\Collection {#1251} "next" => array:2 [ "refs" => "1122670" "type" => "task" ] "previous" => array:2 [ "refs" => "1122668" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1269} "page" => array:2 [ "refs" => "1114259" "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 {#1249 #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" => 1122670 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "57" "field_page_end" => null "field_url" => "/8-klass/geometrija/shynybekov-uchebnik/2-2" "field_display_title" => "2.2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1248 #items: array:1 [ …1] #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 {#1267 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1266 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1268 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1264 #items: array:4 [ …4] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1122671" "type" => "task" ] "previous" => array:2 [ "refs" => "1122669" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1262 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1114259" "type" => "book_page" ] ] #original: array:24 [ "id" => 1122670 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "57" "field_page_end" => null "field_url" => "/8-klass/geometrija/shynybekov-uchebnik/2-2" "field_display_title" => "2.2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1248} "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 {#1267} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1266} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1268} "content" => Illuminate\Database\Eloquent\Collection {#1264} "next" => array:2 [ "refs" => "1122671" "type" => "task" ] "previous" => array:2 [ "refs" => "1122669" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1262} "page" => array:2 [ "refs" => "1114259" "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 {#1265 #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" => 1122671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "57" "field_page_end" => null "field_url" => "/8-klass/geometrija/shynybekov-uchebnik/2-3" "field_display_title" => "2.3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1263 #items: array:1 [ …1] #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 {#1279 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1280 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1281 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1283 #items: array:4 [ …4] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1122672" "type" => "task" ] "previous" => array:2 [ "refs" => "1122670" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1292 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1114259" "type" => "book_page" ] ] #original: array:24 [ "id" => 1122671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "57" "field_page_end" => null "field_url" => "/8-klass/geometrija/shynybekov-uchebnik/2-3" "field_display_title" => "2.3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1263} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1279} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1280} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1281} "content" => Illuminate\Database\Eloquent\Collection {#1283} "next" => array:2 [ "refs" => "1122672" "type" => "task" ] "previous" => array:2 [ "refs" => "1122670" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1292} "page" => array:2 [ "refs" => "1114259" "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 => "*" ] } 4 => App\Models\Task {#1284 #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" => 1122672 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "57" "field_page_end" => null "field_url" => "/8-klass/geometrija/shynybekov-uchebnik/2-4" "field_display_title" => "2.4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1282 #items: array:1 [ …1] #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 {#1290 #items: [] …1 } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1289 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1291 …2} "content" => Illuminate\Database\Eloquent\Collection {#1285 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1308 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1122672 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "57" "field_page_end" => null "field_url" => "/8-klass/geometrija/shynybekov-uchebnik/2-4" "field_display_title" => "2.4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1282} "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 {#1290 …1} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1289 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1291 …2} "content" => Illuminate\Database\Eloquent\Collection {#1285 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1308 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 5 => App\Models\Task {#1288 #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" => 1122673 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "57" "field_page_end" => null "field_url" => "/8-klass/geometrija/shynybekov-uchebnik/2-5" "field_display_title" => "2.5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1287 …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 {#1306 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1305 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1307 …2} "content" => Illuminate\Database\Eloquent\Collection {#1286 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1324 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1122673 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "57" "field_page_end" => null "field_url" => "/8-klass/geometrija/shynybekov-uchebnik/2-5" "field_display_title" => "2.5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1287 …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 {#1306 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1305 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1307 …2} "content" => Illuminate\Database\Eloquent\Collection {#1286 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1324 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 1114259 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "57" "field_url" => "/8-klass/geometrija/shynybekov-uchebnik/page-57" "field_display_title" => "57" "field_folder" => "1" "field_image_name" => "57" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1108} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1040} "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 {#1097} "content" => Illuminate\Database\Eloquent\Collection {#1096} "next" => array:2 [ "refs" => "1114260" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1114258" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1228} ] #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" => 1122670 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "57" "field_page_end" => null "field_url" => "/8-klass/geometrija/shynybekov-uchebnik/2-2" "field_display_title" => "2.2" "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 {#1113} "content" => Illuminate\Database\Eloquent\Collection {#1038} "next" => array:2 [ "refs" => "1122671" "type" => "task" ] "previous" => array:2 [ "refs" => "1122669" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1104} "page" => array:2 [ "refs" => "1114259" "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 => "*" ] }
№2.2 (с. 57)
Учебник rus. №2.2 (с. 57)
скриншот условия
2.2. Постройте угол, косинус которого равен:
1) $ \frac{3}{5} $;
2) $ \frac{4}{9} $;
3) 0,5;
4) 0,8.
Учебник kz. №2.2 (с. 57)
Решение. №2.2 (с. 57)
Решение 2 rus. №2.2 (с. 57)
Для построения угла по заданному значению его косинуса мы будем использовать определение косинуса острого угла в прямоугольном треугольнике. Косинус угла — это отношение прилежащего к этому углу катета к гипотенузе. Следовательно, для построения угла $ \alpha $, косинус которого равен $ \frac{a}{c} $, необходимо построить прямоугольный треугольник с прилежащим катетом $ a $ и гипотенузой $ c $. Угол между этим катетом и гипотенузой и будет искомым углом $ \alpha $.
1) $ \frac{3}{5} $
Чтобы построить угол, косинус которого равен $ \frac{3}{5} $, нужно построить прямоугольный треугольник, у которого прилежащий катет равен 3 условным единицам, а гипотенуза — 5 условным единицам.
Порядок построения:
1. Начертим луч с началом в точке A. На этом луче отложим отрезок AB, равный 3 условным единицам. Этот отрезок будет прилежащим катетом.
2. В точке B восстановим перпендикуляр к лучу AB.
3. Из точки A как из центра проведем дугу окружности радиусом 5 условных единиц (длина гипотенузы).
4. Точку пересечения дуги с перпендикуляром обозначим C.
5. Соединим точки A и C.
В результате построен прямоугольный треугольник ABC с прямым углом B. Угол $ \angle BAC $ является искомым, так как по построению $ \cos(\angle BAC) = \frac{AB}{AC} = \frac{3}{5} $.
Ответ: Угол $ \angle BAC $ в построенном треугольнике является искомым.
2) $ \frac{4}{9} $
Построим прямоугольный треугольник, у которого прилежащий катет равен 4 условным единицам, а гипотенуза — 9 условным единицам.
Порядок построения:
1. Начертим луч с началом в точке A и отложим на нем отрезок AB длиной 4 единицы.
2. В точке B построим перпендикуляр к AB.
3. Из точки A проведем дугу окружности радиусом 9 единиц.
4. Точку пересечения дуги с перпендикуляром обозначим C.
5. Соединим точки A и C.
Полученный угол $ \angle BAC $ в прямоугольном треугольнике ABC является искомым, поскольку $ \cos(\angle BAC) = \frac{AB}{AC} = \frac{4}{9} $.
Ответ: Угол $ \angle BAC $ в построенном треугольнике является искомым.
3) 0,5
Сначала представим десятичную дробь в виде обыкновенной: $ 0,5 = \frac{5}{10} = \frac{1}{2} $.
Теперь задача сводится к построению угла, косинус которого равен $ \frac{1}{2} $. Построим прямоугольный треугольник с прилежащим катетом равным 1 условной единице и гипотенузой равной 2 условным единицам.
Порядок построения:
1. Начертим луч с началом в точке A и отложим на нем отрезок AB длиной 1 единицу.
2. В точке B построим перпендикуляр к AB.
3. Из точки A проведем дугу окружности радиусом 2 единицы.
4. Точку пересечения дуги с перпендикуляром обозначим C.
5. Соединим точки A и C.
Полученный угол $ \angle BAC $ в прямоугольном треугольнике ABC является искомым, так как $ \cos(\angle BAC) = \frac{AB}{AC} = \frac{1}{2} $. (Этот угол равен $ 60^\circ $).
Ответ: Угол $ \angle BAC $ в построенном треугольнике является искомым.
4) 0,8
Представим десятичную дробь в виде обыкновенной и сократим ее: $ 0,8 = \frac{8}{10} = \frac{4}{5} $.
Задача сводится к построению угла, косинус которого равен $ \frac{4}{5} $. Построим прямоугольный треугольник с прилежащим катетом равным 4 условным единицам и гипотенузой равной 5 условным единицам.
Порядок построения:
1. Начертим луч с началом в точке A и отложим на нем отрезок AB длиной 4 единицы.
2. В точке B построим перпендикуляр к AB.
3. Из точки A проведем дугу окружности радиусом 5 единиц.
4. Точку пересечения дуги с перпендикуляром обозначим C.
5. Соединим точки A и C.
Полученный угол $ \angle BAC $ в прямоугольном треугольнике ABC является искомым, поскольку $ \cos(\angle BAC) = \frac{AB}{AC} = \frac{4}{5} $.
Ответ: Угол $ \angle BAC $ в построенном треугольнике является искомым.
Другие задания:
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