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Номер 32, страница 238 - гдз по геометрии 10-11 класс учебник Атанасян, Бутузов
Авторы: Атанасян Л. С., Бутузов В. Ф., Кадомцев С. Б., Позняк Э. Г., Киселёва Л. С.
Тип: Учебник
Издательство: Просвещение
Год издания: 2019 - 2026
Уровень обучения: базовый и углублённый
Цвет обложки: коричневый с ромбами
ISBN: 978-5-09-103606-0 (2023)
Допущено Министерством просвещения Российской Федерации
Математика: алгебра и начала математического анализа, геометрия
Популярные ГДЗ в 10 классе
Задачи для подготовки к ЕГЭ. 16 - номер 32, страница 238.
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" => 745431 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "238" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/eg-5-32" "field_display_title" => "32" "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" => 745563 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Задачи для подготовки к ЕГЭ" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1047 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "229" "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 {#1085 #items: array:1 [ 0 => App\Models\Book {#1048 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 36 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1050 #items: array:1 [ 0 => App\Models\Term {#1049 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6477 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Геометрия" "field_abbreviated_name" => null "field_cases" => array:6 [ …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 {#1051 #items: array:2 [ 0 => App\Models\Term {#1052 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ …6] "field_translit" => "desjatyj" ] #original: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ …6] "field_translit" => "desjatyj" ] #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 {#1053 #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" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ …6] "field_translit" => "odinadcatyj" ] #original: array:6 [ "id" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ …6] "field_translit" => "odinadcatyj" ] #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 {#1054 #items: array:1 [ 0 => App\Models\Term {#1055 #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" => 5153 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #original: array:6 [ "id" => 5153 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #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 {#1056 #items: array:5 [ 0 => App\Models\Term {#1057 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 3753 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Атанасян" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Левон" "field_patronymic" => "Сергеевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 3753 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Атанасян" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Левон" "field_patronymic" => "Сергеевич" "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 {#1058 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 3949 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Бутузов" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Валентин" "field_patronymic" => "Фёдорович" "field_surname_rp" => null ] #original: array:12 [ "id" => 3949 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Бутузов" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Валентин" "field_patronymic" => "Фёдорович" "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 {#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:12 [ "id" => 4518 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кадомцев" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Сергей" "field_patronymic" => "Борисович" "field_surname_rp" => null ] #original: array:12 [ "id" => 4518 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кадомцев" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Сергей" "field_patronymic" => "Борисович" "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 => "*" ] } 3 => App\Models\Term {#1060 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 5494 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Позняк" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Эдуард" "field_patronymic" => "Генрихович" "field_surname_rp" => null ] #original: array:12 [ "id" => 5494 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Позняк" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Эдуард" "field_patronymic" => "Генрихович" "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 => "*" ] } 4 => 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:12 [ "id" => 6905 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Киселёва" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Людмила" "field_patronymic" => "Сергеевна" "field_surname_rp" => null ] #original: array:12 [ "id" => 6905 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Киселёва" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Людмила" "field_patronymic" => "Сергеевна" "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 {#1062 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1063 #items: array:1 [ 0 => App\Models\Term {#1064 #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 {#1065 #items: array:1 [ 0 => App\Models\Term {#1066 #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" => 9 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Россия" "field_cases" => array:6 [ …6] "field_translit" => "rossiya" ] #original: array:6 [ "id" => 9 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Россия" "field_cases" => array:6 [ …6] "field_translit" => "rossiya" ] #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 {#1067 #items: array:1 [ 0 => App\Models\Term {#1068 #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" => 15 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Москва" "field_cases" => array:6 [ …6] "field_translit" => "moskva" ] #original: array:6 [ "id" => 15 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Москва" "field_cases" => array:6 [ …6] "field_translit" => "moskva" ] #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 {#1069 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1070 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1071 #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" => 41 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "базовый и углублённый" "field_cases" => array:6 [ …6] "field_translit" => "bazovyy i uglublennyy" ] #original: array:6 [ "id" => 41 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "базовый и углублённый" "field_cases" => array:6 [ …6] "field_translit" => "bazovyy i uglublennyy" ] #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_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1073 #items: array:1 [ 0 => App\Models\Term {#1074 #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" => 6706 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "ФГОС (старый)" "field_cases" => array:6 [ …6] "field_color" => null "field_end_year" => "2022" "field_start_year" => "2019" "field_translit" => "fgos-old" "field_type" => null ] #original: array:10 [ "id" => 6706 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "ФГОС (старый)" "field_cases" => array:6 [ …6] "field_color" => null "field_end_year" => "2022" "field_start_year" => "2019" "field_translit" => "fgos-old" "field_type" => 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_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1075 #items: array:1 [ 0 => App\Models\Term {#1076 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 33 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "стереотипное" "field_cases" => array:6 [ …6] "field_translit" => "stereotipnoe" ] #original: array:6 [ "id" => 33 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "стереотипное" "field_cases" => array:6 [ …6] "field_translit" => "stereotipnoe" ] #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_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1077 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => "с ромбами" "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для общеобразовательных организаций" "field_allowed" => "Допущено Министерством просвещения Российской Федерации" "field_reserve_field" => "Математика: алгебра и начала математического анализа, геометрия" "field_link_to_source" => null "field_tasks_count" => "1147" "field_priority" => "12" "field_default_folder" => "/geometrija_10/atanasjan-u23/" "field_isbn" => "978-5-09-103606-0 (2023)" "field_cover" => array:1 [ 0 => "/media/geometrija_10/atanasjan-u23/covers/cover1.webp?ts=1739363242" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_10/atanasjan-u23/covers/cover1.webp?ts=1739363242" "alt" => "" "width" => "1200" "height" => "1661" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1078 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1079 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1080 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1081 #items: array:1 [ 0 => App\Models\Term {#1082 #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" => 51 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "коричневый" "field_cases" => array:6 [ …6] "field_translit" => "korichnevyy" ] #original: array:6 [ "id" => 51 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "коричневый" "field_cases" => array:6 [ …6] "field_translit" => "korichnevyy" ] #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 "breadcrumbs" => array:3 [ "class" => 381 "subject" => 543 "class_subject" => 392 ] ] #original: array:50 [ "id" => 36 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1050} "field_class" => Illuminate\Database\Eloquent\Collection {#1051} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1054} "field_author" => Illuminate\Database\Eloquent\Collection {#1056} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1062} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1063} "field_country" => Illuminate\Database\Eloquent\Collection {#1065} "field_city" => Illuminate\Database\Eloquent\Collection {#1067} "field_series" => Illuminate\Database\Eloquent\Collection {#1069} "field_umk" => Illuminate\Database\Eloquent\Collection {#1070} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1071} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1073} "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1075} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1077} "field_under_the_edition_degree" => null "field_cover_description" => "с ромбами" "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для общеобразовательных организаций" "field_allowed" => "Допущено Министерством просвещения Российской Федерации" "field_reserve_field" => "Математика: алгебра и начала математического анализа, геометрия" "field_link_to_source" => null "field_tasks_count" => "1147" "field_priority" => "12" "field_default_folder" => "/geometrija_10/atanasjan-u23/" "field_isbn" => "978-5-09-103606-0 (2023)" "field_cover" => array:1 [ 0 => "/media/geometrija_10/atanasjan-u23/covers/cover1.webp?ts=1739363242" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_10/atanasjan-u23/covers/cover1.webp?ts=1739363242" "alt" => "" "width" => "1200" "height" => "1661" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1078} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1079} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1080} "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1081} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 543 "class_subject" => 392 ] ] #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 {#1083 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 745563 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Задачи для подготовки к ЕГЭ" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1047} "field_page_start" => "229" "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 {#1085} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1083} ] #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 {#1119 #items: array:2 [ 0 => App\Models\Branch {#1127 #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" => 745563 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Задачи для подготовки к ЕГЭ" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1128 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "229" "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 {#1129 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1130 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 745563 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Задачи для подготовки к ЕГЭ" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1128} "field_page_start" => "229" "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 {#1129} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1130} ] #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 {#1131 #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" => 745568 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "16" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1132 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "236" "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 {#1133 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1134 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 745568 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "16" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1132} "field_page_start" => "236" "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 {#1133} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1134} ] #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 {#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:7 [ "id" => 940978 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1092 #items: array:1 [ 0 => App\Models\Edition {#1101 #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" => 2315 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1096 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_process_formula" => "wiris" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_content_source" => null ] #original: array:21 [ "id" => 2315 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1096 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_process_formula" => "wiris" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1089 …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" => "745431" "type" => "task" ] "text" => "<p><strong>32.</strong> Окружности с центрами O₁ и O₂ касаются друг друга извне в точке C. Прямая касается этих окружностей в точках A и B, отличных от точки C. Найдите угол AO₂B, если <span class="long">tg ∠ABC = <span class="fraction"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math></span>. </span></p>" "img" => array:1 [ 0 => array:5 [ "name" => "32-1.jpg" "alt" => null "width" => "1534" "height" => 243 "path" => "/media/geometrija_10/atanasjan-u23/0-eg-5/32-1.webp?ts=1739437047" ] ] ] #original: array:7 [ "id" => 940978 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1092} "task" => array:2 [ "refs" => "745431" "type" => "task" ] "text" => "<p><strong>32.</strong> Окружности с центрами O₁ и O₂ касаются друг друга извне в точке C. Прямая касается этих окружностей в точках A и B, отличных от точки C. Найдите угол AO₂B, если <span class="long">tg ∠ABC = <span class="fraction"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math></span>. </span></p>" "img" => array:1 [ 0 => array:5 [ "name" => "32-1.jpg" "alt" => null "width" => "1534" "height" => 243 "path" => "/media/geometrija_10/atanasjan-u23/0-eg-5/32-1.webp?ts=1739437047" ] ] ] #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 {#1109 #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" => 939795 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1098 #items: array:1 [ 0 => App\Models\Edition {#1106 #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" => 2316 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 1" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1104 …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" => "Пелеванюк" "field_moderator" => "filin" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1103 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1102 …2} "field_content_source" => null ] #original: array:21 [ "id" => 2316 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 1" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1104 …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" => "Пелеванюк" "field_moderator" => "filin" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1103 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1102 …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" => "745431" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "32-1.jpg" "alt" => null "width" => "2047" "height" => 2198 "path" => "/media/geometrija_10/atanasjan-u23/1-eg-5/32-1.webp?ts=1739436246" ] 1 => array:5 [ "name" => "32-2.jpg" "alt" => null "width" => "2047" "height" => 933 "path" => "/media/geometrija_10/atanasjan-u23/1-eg-5/32-2.webp?ts=1739436246" ] ] ] #original: array:6 [ "id" => 939795 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1098} "task" => array:2 [ "refs" => "745431" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "32-1.jpg" "alt" => null "width" => "2047" "height" => 2198 "path" => "/media/geometrija_10/atanasjan-u23/1-eg-5/32-1.webp?ts=1739436246" ] 1 => array:5 [ "name" => "32-2.jpg" "alt" => null "width" => "2047" "height" => 933 "path" => "/media/geometrija_10/atanasjan-u23/1-eg-5/32-2.webp?ts=1739436246" ] ] ] #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 {#1094 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 942977 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1040 #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" => 2317 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1090 …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" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_comment" => null "field_similar_book" => 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#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 {#1086 #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" => 1352744 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1123 #items: array:1 [ 0 => App\Models\Edition {#1084 #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" => 5120 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 6" "field_order" => "7" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1034 …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" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "6-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1118 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1117 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5120 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 6" "field_order" => "7" "field_publisher" => 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array:2 [ "refs" => "745431" "type" => "task" ] "text" => "<p>Пусть $r_1$ и $r_2$ — радиусы окружностей с центрами в точках $O_1$ и $O_2$ соответственно, где $A$ — точка касания на первой окружности, а $B$ — на второй. Таким образом, $O_1A = r_1$ и $O_2B = r_2$.</p><p><strong>1. Доказательство того, что $\angle ACB = 90^\circ$</strong></p><p>Проведем общую касательную к обеим окружностям через точку их касания $C$. Пусть эта касательная пересекает прямую $AB$ в точке $M$. По свойству касательных, проведенных из одной точки к окружности, отрезки касательных равны. Для окружности с центром $O_1$ имеем $MA = MC$. Для окружности с центром $O_2$ имеем $MB = MC$. Следовательно, $MA = MB = MC$.</p><p>Это означает, что точка $M$ является серединой отрезка $AB$, а медиана $CM$ треугольника $ABC$ равна половине стороны $AB$, к которой она проведена. По признаку прямоугольного треугольника, треугольник $ABC$ является прямоугольным с прямым углом при вершине $C$. Таким образом, $\angle ACB = 90^\circ$.</p><p><strong>2. Нахождение отношения радиусов $r_1$ и $r_2$</strong></p><p>В прямоугольном треугольнике $ABC$ тангенс угла $ABC$ равен отношению противолежащего катета $AC$ к прилежащему катету $BC$: $\text{tg}(\angle ABC) = \frac{AC}{BC}$.По условию задачи $\text{tg}(\angle ABC) = \frac{1}{2}$, следовательно, $\frac{AC}{BC} = \frac{1}{2}$.</p><p>Теперь свяжем длины катетов $AC$ и $BC$ с радиусами окружностей.По теореме об угле между касательной и хордой, угол, образованный касательной $AB$ и хордой $BC$, равен вписанному углу, опирающемуся на дугу $BC$. Центральный угол $\angle BO_2C$, опирающийся на ту же дугу, в два раза больше этого вписанного угла.Значит, $\angle BO_2C = 2 \angle ABC$.</p><p>Аналогично для первой окружности, $\angle AO_1C = 2 \angle BAC$. Поскольку $\triangle ABC$ прямоугольный, $\angle BAC = 90^\circ - \angle ABC$.Следовательно, $\angle AO_1C = 2(90^\circ - \angle ABC) = 180^\circ - 2\angle ABC$.</p><p>Рассмотрим равнобедренный треугольник $AO_1C$ ($O_1A=O_1C=r_1$). Угол при основании $\angle O_1CA = \frac{180^\circ - \angle AO_1C}{2} = \frac{180^\circ - (180^\circ - 2\angle ABC)}{2} = \angle ABC$.По теореме синусов в $\triangle AO_1C$:$\frac{AC}{\sin(\angle AO_1C)} = \frac{r_1}{\sin(\angle O_1CA)} \implies AC = \frac{r_1 \sin(180^\circ - 2\angle ABC)}{\sin(\angle ABC)} = \frac{r_1 \sin(2\angle ABC)}{\sin(\angle ABC)} = 2r_1\cos(\angle ABC)$.</p><p>Рассмотрим равнобедренный треугольник $BO_2C$ ($O_2B=O_2C=r_2$). Угол при основании $\angle O_2CB = \frac{180^\circ - \angle BO_2C}{2} = \frac{180^\circ - 2\angle ABC}{2} = 90^\circ - \angle ABC$.По теореме синусов в $\triangle BO_2C$:$\frac{BC}{\sin(\angle BO_2C)} = \frac{r_2}{\sin(\angle O_2CB)} \implies BC = \frac{r_2 \sin(2\angle ABC)}{\sin(90^\circ - \angle ABC)} = \frac{r_2 \cdot 2\sin(\angle ABC)\cos(\angle ABC)}{\cos(\angle ABC)} = 2r_2\sin(\angle ABC)$.</p><p>Теперь подставим полученные выражения для $AC$ и $BC$ в соотношение $\frac{AC}{BC} = \frac{1}{2}$:$\frac{2r_1\cos(\angle ABC)}{2r_2\sin(\angle ABC)} = \frac{1}{2}$$\frac{r_1}{r_2} \text{ctg}(\angle ABC) = \frac{1}{2}$</p><p>Так как $\text{tg}(\angle ABC) = \frac{1}{2}$, то $\text{ctg}(\angle ABC) = 2$.$\frac{r_1}{r_2} \cdot 2 = \frac{1}{2} \implies \frac{r_1}{r_2} = \frac{1}{4}$.</p><p><strong>3. Нахождение угла $AO_2B$</strong></p><p>Рассмотрим треугольник $ABO_2$. Радиус $O_2B$ проведен в точку касания $B$, поэтому он перпендикулярен касательной $AB$. Следовательно, $\angle ABO_2 = 90^\circ$, и треугольник $ABO_2$ является прямоугольным.</p><p>В прямоугольном треугольнике $ABO_2$ тангенс угла $AO_2B$ равен отношению противолежащего катета $AB$ к прилежащему катету $O_2B$:$\text{tg}(\angle AO_2B) = \frac{AB}{O_2B}$</p><p>Длина отрезка общей внешней касательной $AB$ для двух окружностей, касающихся внешне с радиусами $r_1$ и $r_2$, вычисляется по формуле $AB = 2\sqrt{r_1 r_2}$.Подставляем это выражение и $O_2B = r_2$:$\text{tg}(\angle AO_2B) = \frac{2\sqrt{r_1 r_2}}{r_2} = 2\sqrt{\frac{r_1 r_2}{r_2^2}} = 2\sqrt{\frac{r_1}{r_2}}$.</p><p>Мы уже нашли, что $\frac{r_1}{r_2} = \frac{1}{4}$. Подставляем это значение в формулу для тангенса:$\text{tg}(\angle AO_2B) = 2\sqrt{\frac{1}{4}} = 2 \cdot \frac{1}{2} = 1$.</p><p>Угол в прямоугольном треугольнике, тангенс которого равен 1, составляет $45^\circ$.Следовательно, $\angle AO_2B = 45^\circ$.</p><p><strong>Ответ:</strong> $45^\circ$.</p>" ] #original: array:6 [ "id" => 1352744 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1123} "task" => array:2 [ "refs" => "745431" "type" => "task" ] "text" => "<p>Пусть $r_1$ и $r_2$ — радиусы окружностей с центрами в точках $O_1$ и $O_2$ соответственно, где $A$ — точка касания на первой окружности, а $B$ — на второй. Таким образом, $O_1A = r_1$ и $O_2B = r_2$.</p><p><strong>1. Доказательство того, что $\angle ACB = 90^\circ$</strong></p><p>Проведем общую касательную к обеим окружностям через точку их касания $C$. Пусть эта касательная пересекает прямую $AB$ в точке $M$. По свойству касательных, проведенных из одной точки к окружности, отрезки касательных равны. Для окружности с центром $O_1$ имеем $MA = MC$. Для окружности с центром $O_2$ имеем $MB = MC$. Следовательно, $MA = MB = MC$.</p><p>Это означает, что точка $M$ является серединой отрезка $AB$, а медиана $CM$ треугольника $ABC$ равна половине стороны $AB$, к которой она проведена. По признаку прямоугольного треугольника, треугольник $ABC$ является прямоугольным с прямым углом при вершине $C$. Таким образом, $\angle ACB = 90^\circ$.</p><p><strong>2. Нахождение отношения радиусов $r_1$ и $r_2$</strong></p><p>В прямоугольном треугольнике $ABC$ тангенс угла $ABC$ равен отношению противолежащего катета $AC$ к прилежащему катету $BC$: $\text{tg}(\angle ABC) = \frac{AC}{BC}$.По условию задачи $\text{tg}(\angle ABC) = \frac{1}{2}$, следовательно, $\frac{AC}{BC} = \frac{1}{2}$.</p><p>Теперь свяжем длины катетов $AC$ и $BC$ с радиусами окружностей.По теореме об угле между касательной и хордой, угол, образованный касательной $AB$ и хордой $BC$, равен вписанному углу, опирающемуся на дугу $BC$. Центральный угол $\angle BO_2C$, опирающийся на ту же дугу, в два раза больше этого вписанного угла.Значит, $\angle BO_2C = 2 \angle ABC$.</p><p>Аналогично для первой окружности, $\angle AO_1C = 2 \angle BAC$. Поскольку $\triangle ABC$ прямоугольный, $\angle BAC = 90^\circ - \angle ABC$.Следовательно, $\angle AO_1C = 2(90^\circ - \angle ABC) = 180^\circ - 2\angle ABC$.</p><p>Рассмотрим равнобедренный треугольник $AO_1C$ ($O_1A=O_1C=r_1$). Угол при основании $\angle O_1CA = \frac{180^\circ - \angle AO_1C}{2} = \frac{180^\circ - (180^\circ - 2\angle ABC)}{2} = \angle ABC$.По теореме синусов в $\triangle AO_1C$:$\frac{AC}{\sin(\angle AO_1C)} = \frac{r_1}{\sin(\angle O_1CA)} \implies AC = \frac{r_1 \sin(180^\circ - 2\angle ABC)}{\sin(\angle ABC)} = \frac{r_1 \sin(2\angle ABC)}{\sin(\angle ABC)} = 2r_1\cos(\angle ABC)$.</p><p>Рассмотрим равнобедренный треугольник $BO_2C$ ($O_2B=O_2C=r_2$). Угол при основании $\angle O_2CB = \frac{180^\circ - \angle BO_2C}{2} = \frac{180^\circ - 2\angle ABC}{2} = 90^\circ - \angle ABC$.По теореме синусов в $\triangle BO_2C$:$\frac{BC}{\sin(\angle BO_2C)} = \frac{r_2}{\sin(\angle O_2CB)} \implies BC = \frac{r_2 \sin(2\angle ABC)}{\sin(90^\circ - \angle ABC)} = \frac{r_2 \cdot 2\sin(\angle ABC)\cos(\angle ABC)}{\cos(\angle ABC)} = 2r_2\sin(\angle ABC)$.</p><p>Теперь подставим полученные выражения для $AC$ и $BC$ в соотношение $\frac{AC}{BC} = \frac{1}{2}$:$\frac{2r_1\cos(\angle ABC)}{2r_2\sin(\angle ABC)} = \frac{1}{2}$$\frac{r_1}{r_2} \text{ctg}(\angle ABC) = \frac{1}{2}$</p><p>Так как $\text{tg}(\angle ABC) = \frac{1}{2}$, то $\text{ctg}(\angle ABC) = 2$.$\frac{r_1}{r_2} \cdot 2 = \frac{1}{2} \implies \frac{r_1}{r_2} = \frac{1}{4}$.</p><p><strong>3. Нахождение угла $AO_2B$</strong></p><p>Рассмотрим треугольник $ABO_2$. Радиус $O_2B$ проведен в точку касания $B$, поэтому он перпендикулярен касательной $AB$. Следовательно, $\angle ABO_2 = 90^\circ$, и треугольник $ABO_2$ является прямоугольным.</p><p>В прямоугольном треугольнике $ABO_2$ тангенс угла $AO_2B$ равен отношению противолежащего катета $AB$ к прилежащему катету $O_2B$:$\text{tg}(\angle AO_2B) = \frac{AB}{O_2B}$</p><p>Длина отрезка общей внешней касательной $AB$ для двух окружностей, касающихся внешне с радиусами $r_1$ и $r_2$, вычисляется по формуле $AB = 2\sqrt{r_1 r_2}$.Подставляем это выражение и $O_2B = r_2$:$\text{tg}(\angle AO_2B) = \frac{2\sqrt{r_1 r_2}}{r_2} = 2\sqrt{\frac{r_1 r_2}{r_2^2}} = 2\sqrt{\frac{r_1}{r_2}}$.</p><p>Мы уже нашли, что $\frac{r_1}{r_2} = \frac{1}{4}$. Подставляем это значение в формулу для тангенса:$\text{tg}(\angle AO_2B) = 2\sqrt{\frac{1}{4}} = 2 \cdot \frac{1}{2} = 1$.</p><p>Угол в прямоугольном треугольнике, тангенс которого равен 1, составляет $45^\circ$.Следовательно, $\angle AO_2B = 45^\circ$.</p><p><strong>Ответ:</strong> $45^\circ$.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "745432" "type" => "task" ] "previous" => array:2 [ "refs" => "745430" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1110 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1039 #items: array:1 [ 0 => App\Models\BookPage {#1114 #connection: 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№32 (с. 238)
Условие. №32 (с. 238)
скриншот условия
32. Окружности с центрами O₁ и O₂ касаются друг друга извне в точке C. Прямая касается этих окружностей в точках A и B, отличных от точки C. Найдите угол AO₂B, если tg ∠ABC = .
Решение 1. №32 (с. 238)
Решение 2. №32 (с. 238)
Решение 6. №32 (с. 238)
Пусть $r_1$ и $r_2$ — радиусы окружностей с центрами в точках $O_1$ и $O_2$ соответственно, где $A$ — точка касания на первой окружности, а $B$ — на второй. Таким образом, $O_1A = r_1$ и $O_2B = r_2$.
1. Доказательство того, что $\angle ACB = 90^\circ$
Проведем общую касательную к обеим окружностям через точку их касания $C$. Пусть эта касательная пересекает прямую $AB$ в точке $M$. По свойству касательных, проведенных из одной точки к окружности, отрезки касательных равны. Для окружности с центром $O_1$ имеем $MA = MC$. Для окружности с центром $O_2$ имеем $MB = MC$. Следовательно, $MA = MB = MC$.
Это означает, что точка $M$ является серединой отрезка $AB$, а медиана $CM$ треугольника $ABC$ равна половине стороны $AB$, к которой она проведена. По признаку прямоугольного треугольника, треугольник $ABC$ является прямоугольным с прямым углом при вершине $C$. Таким образом, $\angle ACB = 90^\circ$.
2. Нахождение отношения радиусов $r_1$ и $r_2$
В прямоугольном треугольнике $ABC$ тангенс угла $ABC$ равен отношению противолежащего катета $AC$ к прилежащему катету $BC$: $\text{tg}(\angle ABC) = \frac{AC}{BC}$.По условию задачи $\text{tg}(\angle ABC) = \frac{1}{2}$, следовательно, $\frac{AC}{BC} = \frac{1}{2}$.
Теперь свяжем длины катетов $AC$ и $BC$ с радиусами окружностей.По теореме об угле между касательной и хордой, угол, образованный касательной $AB$ и хордой $BC$, равен вписанному углу, опирающемуся на дугу $BC$. Центральный угол $\angle BO_2C$, опирающийся на ту же дугу, в два раза больше этого вписанного угла.Значит, $\angle BO_2C = 2 \angle ABC$.
Аналогично для первой окружности, $\angle AO_1C = 2 \angle BAC$. Поскольку $\triangle ABC$ прямоугольный, $\angle BAC = 90^\circ - \angle ABC$.Следовательно, $\angle AO_1C = 2(90^\circ - \angle ABC) = 180^\circ - 2\angle ABC$.
Рассмотрим равнобедренный треугольник $AO_1C$ ($O_1A=O_1C=r_1$). Угол при основании $\angle O_1CA = \frac{180^\circ - \angle AO_1C}{2} = \frac{180^\circ - (180^\circ - 2\angle ABC)}{2} = \angle ABC$.По теореме синусов в $\triangle AO_1C$:$\frac{AC}{\sin(\angle AO_1C)} = \frac{r_1}{\sin(\angle O_1CA)} \implies AC = \frac{r_1 \sin(180^\circ - 2\angle ABC)}{\sin(\angle ABC)} = \frac{r_1 \sin(2\angle ABC)}{\sin(\angle ABC)} = 2r_1\cos(\angle ABC)$.
Рассмотрим равнобедренный треугольник $BO_2C$ ($O_2B=O_2C=r_2$). Угол при основании $\angle O_2CB = \frac{180^\circ - \angle BO_2C}{2} = \frac{180^\circ - 2\angle ABC}{2} = 90^\circ - \angle ABC$.По теореме синусов в $\triangle BO_2C$:$\frac{BC}{\sin(\angle BO_2C)} = \frac{r_2}{\sin(\angle O_2CB)} \implies BC = \frac{r_2 \sin(2\angle ABC)}{\sin(90^\circ - \angle ABC)} = \frac{r_2 \cdot 2\sin(\angle ABC)\cos(\angle ABC)}{\cos(\angle ABC)} = 2r_2\sin(\angle ABC)$.
Теперь подставим полученные выражения для $AC$ и $BC$ в соотношение $\frac{AC}{BC} = \frac{1}{2}$:$\frac{2r_1\cos(\angle ABC)}{2r_2\sin(\angle ABC)} = \frac{1}{2}$$\frac{r_1}{r_2} \text{ctg}(\angle ABC) = \frac{1}{2}$
Так как $\text{tg}(\angle ABC) = \frac{1}{2}$, то $\text{ctg}(\angle ABC) = 2$.$\frac{r_1}{r_2} \cdot 2 = \frac{1}{2} \implies \frac{r_1}{r_2} = \frac{1}{4}$.
3. Нахождение угла $AO_2B$
Рассмотрим треугольник $ABO_2$. Радиус $O_2B$ проведен в точку касания $B$, поэтому он перпендикулярен касательной $AB$. Следовательно, $\angle ABO_2 = 90^\circ$, и треугольник $ABO_2$ является прямоугольным.
В прямоугольном треугольнике $ABO_2$ тангенс угла $AO_2B$ равен отношению противолежащего катета $AB$ к прилежащему катету $O_2B$:$\text{tg}(\angle AO_2B) = \frac{AB}{O_2B}$
Длина отрезка общей внешней касательной $AB$ для двух окружностей, касающихся внешне с радиусами $r_1$ и $r_2$, вычисляется по формуле $AB = 2\sqrt{r_1 r_2}$.Подставляем это выражение и $O_2B = r_2$:$\text{tg}(\angle AO_2B) = \frac{2\sqrt{r_1 r_2}}{r_2} = 2\sqrt{\frac{r_1 r_2}{r_2^2}} = 2\sqrt{\frac{r_1}{r_2}}$.
Мы уже нашли, что $\frac{r_1}{r_2} = \frac{1}{4}$. Подставляем это значение в формулу для тангенса:$\text{tg}(\angle AO_2B) = 2\sqrt{\frac{1}{4}} = 2 \cdot \frac{1}{2} = 1$.
Угол в прямоугольном треугольнике, тангенс которого равен 1, составляет $45^\circ$.Следовательно, $\angle AO_2B = 45^\circ$.
Ответ: $45^\circ$.
Другие задания:
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