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Номер 766, страница 189 - 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
Глава 7. Метод координат в пространстве. Движения. Параграф 3. Движения, дополнительные задачи - номер 766, страница 189.
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" => 745181 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "189" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/00-766" "field_display_title" => "766" "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" => 744405 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Метод координат в пространстве. Движения" "field_branch_order" => "7" "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" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "160" "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 {#1086 #items: array:1 [ 0 => App\Models\Book {#1049 #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 {#1051 #items: array:1 [ 0 => App\Models\Term {#1048 #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 {#1052 #items: array:2 [ 0 => 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" => 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 {#1054 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 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 {#1055 #items: array:1 [ 0 => App\Models\Term {#1056 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array: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 {#1057 #items: array:5 [ 0 => 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" => 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 {#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" => 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 {#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" => 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 {#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" => 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 {#1062 #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 {#1063 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1064 #items: array:1 [ 0 => App\Models\Term {#1065 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array: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 {#1066 #items: array:1 [ 0 => App\Models\Term {#1067 #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 {#1068 #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" => 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 {#1070 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1071 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1072 #items: array:1 [ 0 => App\Models\Term {#1073 #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 {#1074 #items: array:1 [ 0 => App\Models\Term {#1075 #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 {#1076 #items: array:1 [ 0 => App\Models\Term {#1077 #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 {#1078 #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 {#1079 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1080 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1081 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1082 #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" => 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 {#1051} "field_class" => Illuminate\Database\Eloquent\Collection {#1052} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1055} "field_author" => Illuminate\Database\Eloquent\Collection {#1057} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1063} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1064} "field_country" => Illuminate\Database\Eloquent\Collection {#1066} "field_city" => Illuminate\Database\Eloquent\Collection {#1068} "field_series" => Illuminate\Database\Eloquent\Collection {#1070} "field_umk" => Illuminate\Database\Eloquent\Collection {#1071} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1072} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1074} "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1076} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1078} "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 {#1079} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1080} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1081} "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1082} "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 {#1084 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 744405 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Метод координат в пространстве. Движения" "field_branch_order" => "7" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1050} "field_page_start" => "160" "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 {#1086} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1084} ] #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 {#1121 #items: array:2 [ 0 => App\Models\Branch {#1129 #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" => 744405 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Метод координат в пространстве. Движения" "field_branch_order" => "7" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1130 #items: array:1 [ 0 => App\Models\Term {#1047} ] #escapeWhenCastingToString: false } "field_page_start" => "160" "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 {#1131 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1132 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 744405 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Метод координат в пространстве. Движения" "field_branch_order" => "7" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1130} "field_page_start" => "160" "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 {#1131} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1132} ] #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 {#1133 #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" => 744410 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Движения, дополнительные задачи" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1137 #items: array:1 [ 0 => App\Models\Term {#1134 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => "§" ] #original: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => "§" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "187" "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 {#1136 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1135 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 744410 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Движения, дополнительные задачи" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1137} "field_page_start" => "187" "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 {#1136} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1135} ] #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 {#1111 #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" => 939810 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1100 #items: array:1 [ 0 => App\Models\Edition {#1108 #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 {#1106 …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 {#1107 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_process_formula" => "wiris" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1104 …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 {#1106 …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 {#1107 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_process_formula" => "wiris" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1104 …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" => "745181" "type" => "task" ] "text" => "<p> <b>766.</b> В правильной треугольной пирамиде <i>DABC</i> высота DO равна 3 см, а боковое ребро <i>DA</i> равно 5 см. Найдите: </p> <p> а) площадь полной поверхности пирамиды; </p> <p> б) объём пирамиды; </p> <p> в) угол между боковым ребром и плоскостью основания; </p> <p> г) угол наклона боковой грани к плоскости основания; </p> <p> д) скалярное произведение векторов <span class="long"><span class="fraction"><span>1</span><span>2</span></span>(<span class="arrtoptoleft">DB</span> + <span class="arrtoptoleft">DC</span>) <span class="arrtoptoleft">MA</span>,</span> где <i>M</i> — середина ребра BC; </p> <p> е) радиус шара, вписанного в пирамиду. </p>" "img" => array:1 [ 0 => array:5 [ "name" => "766-1.jpg" "alt" => null "width" => "1569" "height" => 515 "path" => "/media/geometrija_10/atanasjan-u23/0-00/766-1.webp?ts=1739436882" ] ] ] #original: array:7 [ "id" => 939810 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1100} "task" => array:2 [ "refs" => "745181" "type" => "task" ] "text" => "<p> <b>766.</b> В правильной треугольной пирамиде <i>DABC</i> высота DO равна 3 см, а боковое ребро <i>DA</i> равно 5 см. Найдите: </p> <p> а) площадь полной поверхности пирамиды; </p> <p> б) объём пирамиды; </p> <p> в) угол между боковым ребром и плоскостью основания; </p> <p> г) угол наклона боковой грани к плоскости основания; </p> <p> д) скалярное произведение векторов <span class="long"><span class="fraction"><span>1</span><span>2</span></span>(<span class="arrtoptoleft">DB</span> + <span class="arrtoptoleft">DC</span>) <span class="arrtoptoleft">MA</span>,</span> где <i>M</i> — середина ребра BC; </p> <p> е) радиус шара, вписанного в пирамиду. </p>" "img" => array:1 [ 0 => array:5 [ "name" => "766-1.jpg" "alt" => null "width" => "1569" "height" => 515 "path" => "/media/geometrija_10/atanasjan-u23/0-00/766-1.webp?ts=1739436882" ] ] ] #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 {#1096 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 942710 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1043 #items: array:1 [ 0 => App\Models\Edition {#1095 #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 {#1092 …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 {#1089 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1090 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_content_source" => null ] #original: 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 {#1092 …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 {#1089 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1090 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1085 …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" => "745181" "type" => "task" ] "img" => array:6 [ 0 => array:5 [ "name" => "766-1.png" "alt" => null "width" => "694" "height" => 3250 "path" => "/media/geometrija_10/atanasjan-u23/2-00/766-1.webp?ts=1739441669" ] 1 => array:5 [ "name" => "766-2.png" "alt" => null "width" => "694" "height" => 2701 "path" => "/media/geometrija_10/atanasjan-u23/2-00/766-2.webp?ts=1739441669" ] 2 => array:5 [ "name" => "766-3.png" "alt" => null "width" => "694" "height" => 1776 "path" => "/media/geometrija_10/atanasjan-u23/2-00/766-3.webp?ts=1739441669" ] 3 => array:5 [ "name" => "766-4.png" "alt" => null "width" => "694" "height" => 2837 "path" => "/media/geometrija_10/atanasjan-u23/2-00/766-4.webp?ts=1739441669" ] 4 => array:5 [ "name" => "766-5.png" "alt" => null "width" => "694" "height" => 2663 "path" => "/media/geometrija_10/atanasjan-u23/2-00/766-5.webp?ts=1739441669" ] 5 => array:5 [ "name" => "766-6.png" "alt" => null "width" => "694" "height" => 3747 "path" => "/media/geometrija_10/atanasjan-u23/2-00/766-6.webp?ts=1739441669" ] ] ] #original: array:6 [ "id" => 942710 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1043} "task" => array:2 [ "refs" => "745181" "type" => "task" ] "img" => array:6 [ 0 => array:5 [ "name" => "766-1.png" "alt" => null "width" => "694" "height" => 3250 "path" => "/media/geometrija_10/atanasjan-u23/2-00/766-1.webp?ts=1739441669" ] 1 => array:5 [ "name" => "766-2.png" "alt" => null "width" => "694" "height" => 2701 "path" => "/media/geometrija_10/atanasjan-u23/2-00/766-2.webp?ts=1739441669" ] 2 => array:5 [ "name" => "766-3.png" "alt" => null "width" => "694" "height" => 1776 "path" => "/media/geometrija_10/atanasjan-u23/2-00/766-3.webp?ts=1739441669" ] 3 => array:5 [ "name" => "766-4.png" "alt" => null "width" => "694" "height" => 2837 "path" => "/media/geometrija_10/atanasjan-u23/2-00/766-4.webp?ts=1739441669" ] 4 => array:5 [ "name" => "766-5.png" "alt" => null "width" => "694" "height" => 2663 "path" => "/media/geometrija_10/atanasjan-u23/2-00/766-5.webp?ts=1739441669" ] 5 => array:5 [ "name" => "766-6.png" "alt" => null "width" => "694" "height" => 3747 "path" => "/media/geometrija_10/atanasjan-u23/2-00/766-6.webp?ts=1739441669" ] ] ] #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 {#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:6 [ "id" => 942484 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1094 #items: array:1 [ 0 => App\Models\Edition {#1103 #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" => 2318 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 3" "field_order" => "4" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1098 …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" => "3-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_content_source" => null ] #original: array:21 [ "id" => 2318 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 3" "field_order" => "4" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1098 …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" => "3-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1091 …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" => "745181" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "766-1.jpg" "alt" => null "width" => "1413" "height" => 1354 "path" => "/media/geometrija_10/atanasjan-u23/3-00/766-1.webp?ts=1739437220" ] ] ] #original: array:6 [ "id" => 942484 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1094} "task" => array:2 [ "refs" => "745181" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "766-1.jpg" "alt" => null "width" => "1413" "height" => 1354 "path" => "/media/geometrija_10/atanasjan-u23/3-00/766-1.webp?ts=1739437220" ] ] ] #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 {#1088 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1352494 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1122 #items: array:1 [ 0 => App\Models\Edition {#1087 #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 {#1039 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1120 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1119 …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" => 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 {#1039 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1120 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1119 …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" => "745181" "type" => "task" ] "text" => "<p>Дано: правильная треугольная пирамида $DABC$, высота $DO = 3$ см, боковое ребро $DA = 5$ см.</p><p>Так как пирамида правильная, в основании лежит равносторонний треугольник $ABC$, а высота $DO$ проецируется в его центр $O$. Точка $O$ является центром вписанной и описанной окружностей треугольника $ABC$.</p><p>Рассмотрим прямоугольный треугольник $DOA$. $DO$ — катет (высота пирамиды), $DA$ — гипотенуза (боковое ребро), $OA$ — второй катет. $OA$ является радиусом описанной окружности $R$ основания $ABC$.</p><p>По теореме Пифагора:</p><p>$OA = \sqrt{DA^2 - DO^2} = \sqrt{5^2 - 3^2} = \sqrt{25 - 9} = \sqrt{16} = 4$ см.</p><p>Итак, $R = OA = 4$ см.</p><p>Для равностороннего треугольника со стороной $a$ радиус описанной окружности $R = \frac{a}{\sqrt{3}}$. Отсюда найдем сторону основания $a$:</p><p>$a = R \cdot \sqrt{3} = 4\sqrt{3}$ см.</p><p>Радиус вписанной окружности $r$ для равностороннего треугольника равен $r = \frac{R}{2}$.</p><p>$r = OM = \frac{4}{2} = 2$ см, где $M$ — середина стороны $BC$.</p><p><strong>а) площадь полной поверхности пирамиды;</strong></p><p>Площадь полной поверхности пирамиды $S_{\text{полн}}$ равна сумме площади основания $S_{\text{осн}}$ и площади боковой поверхности $S_{\text{бок}}$.</p><p>1. Площадь основания (равностороннего треугольника):</p><p>$S_{\text{осн}} = \frac{a^2\sqrt{3}}{4} = \frac{(4\sqrt{3})^2\sqrt{3}}{4} = \frac{16 \cdot 3 \cdot \sqrt{3}}{4} = 12\sqrt{3}$ см?.</p><p>2. Площадь боковой поверхности. $S_{\text{бок}} = 3 \cdot S_{\triangle DBC}$. Для нахождения площади боковой грани $DBC$ нужна ее высота (апофема пирамиды) $DM$. Найдем $DM$ из прямоугольного треугольника $DOM$ (где $M$ — середина $BC$):</p><p>$DM = \sqrt{DO^2 + OM^2} = \sqrt{3^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13}$ см.</p><p>Площадь одной боковой грани:</p><p>$S_{\triangle DBC} = \frac{1}{2} \cdot BC \cdot DM = \frac{1}{2} \cdot 4\sqrt{3} \cdot \sqrt{13} = 2\sqrt{39}$ см?.</p><p>Площадь всей боковой поверхности:</p><p>$S_{\text{бок}} = 3 \cdot 2\sqrt{39} = 6\sqrt{39}$ см?.</p><p>3. Площадь полной поверхности:</p><p>$S_{\text{полн}} = S_{\text{осн}} + S_{\text{бок}} = 12\sqrt{3} + 6\sqrt{39} = 6\sqrt{3}(2 + \sqrt{13})$ см?.</p><p><strong>Ответ:</strong> $12\sqrt{3} + 6\sqrt{39}$ см?.</p><p><strong>б) объём пирамиды;</strong></p><p>Объём пирамиды вычисляется по формуле $V = \frac{1}{3} S_{\text{осн}} \cdot H$, где $H=DO$.</p><p>$V = \frac{1}{3} \cdot 12\sqrt{3} \cdot 3 = 12\sqrt{3}$ см?.</p><p><strong>Ответ:</strong> $12\sqrt{3}$ см?.</p><p><strong>в) угол между боковым ребром и плоскостью основания;</strong></p><p>Угол между боковым ребром (например, $DA$) и плоскостью основания ($ABC$) — это угол между ребром и его проекцией на эту плоскость. Проекцией ребра $DA$ является отрезок $OA$. Искомый угол — это $\angle DAO$ в прямоугольном треугольнике $DOA$.</p><p>Мы знаем все стороны этого треугольника: $DO=3$, $OA=4$, $DA=5$.</p><p>$\sin(\angle DAO) = \frac{DO}{DA} = \frac{3}{5}$.</p><p>Следовательно, угол равен $\arcsin\left(\frac{3}{5}\right)$.</p><p><strong>Ответ:</strong> $\arcsin\left(\frac{3}{5}\right)$.</p><p><strong>г) угол наклона боковой грани к плоскости основания;</strong></p><p>Угол наклона боковой грани (например, $DBC$) к плоскости основания ($ABC$) — это линейный угол двугранного угла, который образуется перпендикулярами, проведенными к линии их пересечения ($BC$). Такими перпендикулярами являются апофема $DM$ и радиус вписанной окружности $OM$. Искомый угол — это $\angle DMO$ в прямоугольном треугольнике $DOM$.</p><p>Мы знаем катеты этого треугольника: $DO=3$, $OM=2$.</p><p>$\text{tg}(\angle DMO) = \frac{DO}{OM} = \frac{3}{2}$.</p><p>Следовательно, угол равен $\text{arctg}\left(\frac{3}{2}\right)$.</p><p><strong>Ответ:</strong> $\text{arctg}\left(\frac{3}{2}\right)$.</p><p><strong>д) скалярное произведение векторов $\frac{1}{2}(\overrightarrow{DB} + \overrightarrow{DC})\overrightarrow{MA}$, где $M$ — середина ребра $BC$;</strong></p><p>1. Упростим первый вектор. По правилу сложения векторов, выходящих из одной точки, вектор $\overrightarrow{DB} + \overrightarrow{DC}$ равен $2\overrightarrow{DM}$, где $M$ — середина отрезка $BC$.</p><p>Таким образом, $\frac{1}{2}(\overrightarrow{DB} + \overrightarrow{DC}) = \frac{1}{2}(2\overrightarrow{DM}) = \overrightarrow{DM}$.</p><p>2. Нам нужно найти скалярное произведение $\overrightarrow{DM} \cdot \overrightarrow{MA}$.</p><p>По определению, скалярное произведение равно произведению длин векторов на косинус угла между ними: $\overrightarrow{DM} \cdot \overrightarrow{MA} = |\overrightarrow{DM}| \cdot |\overrightarrow{MA}| \cdot \cos(\angle DMA)$.</p><p>3. Найдем длины векторов и косинус угла между ними. Рассмотрим треугольник $DMA$.</p><p>Длина $|\overrightarrow{DM}|$ — это апофема, мы ее нашли в пункте а): $|\overrightarrow{DM}| = \sqrt{13}$ см.</p><p>Длина $|\overrightarrow{MA}|$ — это медиана (и высота) равностороннего треугольника $ABC$. $|\overrightarrow{MA}| = \frac{a\sqrt{3}}{2} = \frac{4\sqrt{3} \cdot \sqrt{3}}{2} = \frac{4 \cdot 3}{2} = 6$ см.</p><p>Длина ребра $|\overrightarrow{DA}|$ дана по условию: $|\overrightarrow{DA}| = 5$ см.</p><p>4. По теореме косинусов для треугольника $DMA$:</p><p>$DA^2 = DM^2 + MA^2 - 2 \cdot DM \cdot MA \cdot \cos(\angle DMA)$.</p><p>$5^2 = (\sqrt{13})^2 + 6^2 - 2 \cdot \sqrt{13} \cdot 6 \cdot \cos(\angle DMA)$.</p><p>$25 = 13 + 36 - 12\sqrt{13} \cdot \cos(\angle DMA)$.</p><p>$25 = 49 - 12\sqrt{13} \cdot \cos(\angle DMA)$.</p><p>$12\sqrt{13} \cdot \cos(\angle DMA) = 49 - 25 = 24$.</p><p>$\cos(\angle DMA) = \frac{24}{12\sqrt{13}} = \frac{2}{\sqrt{13}}$.</p><p>5. Вычисляем скалярное произведение:</p><p>$\overrightarrow{DM} \cdot \overrightarrow{MA} = |\overrightarrow{DM}| \cdot |\overrightarrow{MA}| \cdot \cos(\angle DMA) = \sqrt{13} \cdot 6 \cdot \frac{2}{\sqrt{13}} = 6 \cdot 2 = 12$.</p><p><strong>Ответ:</strong> $12$.</p><p><strong>е) радиус шара, вписанного в пирамиду.</strong></p><p>Радиус $r_{\text{впис}}$ вписанного шара можно найти по формуле, связывающей объем пирамиды $V$ и площадь ее полной поверхности $S_{\text{полн}}$:</p><p>$V = \frac{1}{3} r_{\text{впис}} \cdot S_{\text{полн}}$.</p><p>Отсюда $r_{\text{впис}} = \frac{3V}{S_{\text{полн}}}$.</p><p>Из предыдущих пунктов имеем:</p><p>$V = 12\sqrt{3}$ см?.</p><p>$S_{\text{полн}} = 12\sqrt{3} + 6\sqrt{39}$ см?.</p><p>Подставляем значения:</p><p>$r_{\text{впис}} = \frac{3 \cdot 12\sqrt{3}}{12\sqrt{3} + 6\sqrt{39}} = \frac{36\sqrt{3}}{6\sqrt{3}(2 + \sqrt{13})} = \frac{6}{2 + \sqrt{13}}$.</p><p>Избавимся от иррациональности в знаменателе, умножив числитель и знаменатель на сопряженное выражение $(\sqrt{13} - 2)$:</p><p>$r_{\text{впис}} = \frac{6(\sqrt{13} - 2)}{(2 + \sqrt{13})(\sqrt{13} - 2)} = \frac{6(\sqrt{13} - 2)}{(\sqrt{13})^2 - 2^2} = \frac{6(\sqrt{13} - 2)}{13 - 4} = \frac{6(\sqrt{13} - 2)}{9} = \frac{2(\sqrt{13} - 2)}{3}$ см.</p><p><strong>Ответ:</strong> $\frac{2(\sqrt{13} - 2)}{3}$ см.</p>" ] #original: array:6 [ "id" => 1352494 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1122} "task" => array:2 [ "refs" => "745181" "type" => "task" ] "text" => "<p>Дано: правильная треугольная пирамида $DABC$, высота $DO = 3$ см, боковое ребро $DA = 5$ см.</p><p>Так как пирамида правильная, в основании лежит равносторонний треугольник $ABC$, а высота $DO$ проецируется в его центр $O$. Точка $O$ является центром вписанной и описанной окружностей треугольника $ABC$.</p><p>Рассмотрим прямоугольный треугольник $DOA$. $DO$ — катет (высота пирамиды), $DA$ — гипотенуза (боковое ребро), $OA$ — второй катет. $OA$ является радиусом описанной окружности $R$ основания $ABC$.</p><p>По теореме Пифагора:</p><p>$OA = \sqrt{DA^2 - DO^2} = \sqrt{5^2 - 3^2} = \sqrt{25 - 9} = \sqrt{16} = 4$ см.</p><p>Итак, $R = OA = 4$ см.</p><p>Для равностороннего треугольника со стороной $a$ радиус описанной окружности $R = \frac{a}{\sqrt{3}}$. Отсюда найдем сторону основания $a$:</p><p>$a = R \cdot \sqrt{3} = 4\sqrt{3}$ см.</p><p>Радиус вписанной окружности $r$ для равностороннего треугольника равен $r = \frac{R}{2}$.</p><p>$r = OM = \frac{4}{2} = 2$ см, где $M$ — середина стороны $BC$.</p><p><strong>а) площадь полной поверхности пирамиды;</strong></p><p>Площадь полной поверхности пирамиды $S_{\text{полн}}$ равна сумме площади основания $S_{\text{осн}}$ и площади боковой поверхности $S_{\text{бок}}$.</p><p>1. Площадь основания (равностороннего треугольника):</p><p>$S_{\text{осн}} = \frac{a^2\sqrt{3}}{4} = \frac{(4\sqrt{3})^2\sqrt{3}}{4} = \frac{16 \cdot 3 \cdot \sqrt{3}}{4} = 12\sqrt{3}$ см?.</p><p>2. Площадь боковой поверхности. $S_{\text{бок}} = 3 \cdot S_{\triangle DBC}$. Для нахождения площади боковой грани $DBC$ нужна ее высота (апофема пирамиды) $DM$. Найдем $DM$ из прямоугольного треугольника $DOM$ (где $M$ — середина $BC$):</p><p>$DM = \sqrt{DO^2 + OM^2} = \sqrt{3^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13}$ см.</p><p>Площадь одной боковой грани:</p><p>$S_{\triangle DBC} = \frac{1}{2} \cdot BC \cdot DM = \frac{1}{2} \cdot 4\sqrt{3} \cdot \sqrt{13} = 2\sqrt{39}$ см?.</p><p>Площадь всей боковой поверхности:</p><p>$S_{\text{бок}} = 3 \cdot 2\sqrt{39} = 6\sqrt{39}$ см?.</p><p>3. Площадь полной поверхности:</p><p>$S_{\text{полн}} = S_{\text{осн}} + S_{\text{бок}} = 12\sqrt{3} + 6\sqrt{39} = 6\sqrt{3}(2 + \sqrt{13})$ см?.</p><p><strong>Ответ:</strong> $12\sqrt{3} + 6\sqrt{39}$ см?.</p><p><strong>б) объём пирамиды;</strong></p><p>Объём пирамиды вычисляется по формуле $V = \frac{1}{3} S_{\text{осн}} \cdot H$, где $H=DO$.</p><p>$V = \frac{1}{3} \cdot 12\sqrt{3} \cdot 3 = 12\sqrt{3}$ см?.</p><p><strong>Ответ:</strong> $12\sqrt{3}$ см?.</p><p><strong>в) угол между боковым ребром и плоскостью основания;</strong></p><p>Угол между боковым ребром (например, $DA$) и плоскостью основания ($ABC$) — это угол между ребром и его проекцией на эту плоскость. Проекцией ребра $DA$ является отрезок $OA$. Искомый угол — это $\angle DAO$ в прямоугольном треугольнике $DOA$.</p><p>Мы знаем все стороны этого треугольника: $DO=3$, $OA=4$, $DA=5$.</p><p>$\sin(\angle DAO) = \frac{DO}{DA} = \frac{3}{5}$.</p><p>Следовательно, угол равен $\arcsin\left(\frac{3}{5}\right)$.</p><p><strong>Ответ:</strong> $\arcsin\left(\frac{3}{5}\right)$.</p><p><strong>г) угол наклона боковой грани к плоскости основания;</strong></p><p>Угол наклона боковой грани (например, $DBC$) к плоскости основания ($ABC$) — это линейный угол двугранного угла, который образуется перпендикулярами, проведенными к линии их пересечения ($BC$). Такими перпендикулярами являются апофема $DM$ и радиус вписанной окружности $OM$. Искомый угол — это $\angle DMO$ в прямоугольном треугольнике $DOM$.</p><p>Мы знаем катеты этого треугольника: $DO=3$, $OM=2$.</p><p>$\text{tg}(\angle DMO) = \frac{DO}{OM} = \frac{3}{2}$.</p><p>Следовательно, угол равен $\text{arctg}\left(\frac{3}{2}\right)$.</p><p><strong>Ответ:</strong> $\text{arctg}\left(\frac{3}{2}\right)$.</p><p><strong>д) скалярное произведение векторов $\frac{1}{2}(\overrightarrow{DB} + \overrightarrow{DC})\overrightarrow{MA}$, где $M$ — середина ребра $BC$;</strong></p><p>1. Упростим первый вектор. По правилу сложения векторов, выходящих из одной точки, вектор $\overrightarrow{DB} + \overrightarrow{DC}$ равен $2\overrightarrow{DM}$, где $M$ — середина отрезка $BC$.</p><p>Таким образом, $\frac{1}{2}(\overrightarrow{DB} + \overrightarrow{DC}) = \frac{1}{2}(2\overrightarrow{DM}) = \overrightarrow{DM}$.</p><p>2. Нам нужно найти скалярное произведение $\overrightarrow{DM} \cdot \overrightarrow{MA}$.</p><p>По определению, скалярное произведение равно произведению длин векторов на косинус угла между ними: $\overrightarrow{DM} \cdot \overrightarrow{MA} = |\overrightarrow{DM}| \cdot |\overrightarrow{MA}| \cdot \cos(\angle DMA)$.</p><p>3. Найдем длины векторов и косинус угла между ними. Рассмотрим треугольник $DMA$.</p><p>Длина $|\overrightarrow{DM}|$ — это апофема, мы ее нашли в пункте а): $|\overrightarrow{DM}| = \sqrt{13}$ см.</p><p>Длина $|\overrightarrow{MA}|$ — это медиана (и высота) равностороннего треугольника $ABC$. $|\overrightarrow{MA}| = \frac{a\sqrt{3}}{2} = \frac{4\sqrt{3} \cdot \sqrt{3}}{2} = \frac{4 \cdot 3}{2} = 6$ см.</p><p>Длина ребра $|\overrightarrow{DA}|$ дана по условию: $|\overrightarrow{DA}| = 5$ см.</p><p>4. По теореме косинусов для треугольника $DMA$:</p><p>$DA^2 = DM^2 + MA^2 - 2 \cdot DM \cdot MA \cdot \cos(\angle DMA)$.</p><p>$5^2 = (\sqrt{13})^2 + 6^2 - 2 \cdot \sqrt{13} \cdot 6 \cdot \cos(\angle DMA)$.</p><p>$25 = 13 + 36 - 12\sqrt{13} \cdot \cos(\angle DMA)$.</p><p>$25 = 49 - 12\sqrt{13} \cdot \cos(\angle DMA)$.</p><p>$12\sqrt{13} \cdot \cos(\angle DMA) = 49 - 25 = 24$.</p><p>$\cos(\angle DMA) = \frac{24}{12\sqrt{13}} = \frac{2}{\sqrt{13}}$.</p><p>5. Вычисляем скалярное произведение:</p><p>$\overrightarrow{DM} \cdot \overrightarrow{MA} = |\overrightarrow{DM}| \cdot |\overrightarrow{MA}| \cdot \cos(\angle DMA) = \sqrt{13} \cdot 6 \cdot \frac{2}{\sqrt{13}} = 6 \cdot 2 = 12$.</p><p><strong>Ответ:</strong> $12$.</p><p><strong>е) радиус шара, вписанного в пирамиду.</strong></p><p>Радиус $r_{\text{впис}}$ вписанного шара можно найти по формуле, связывающей объем пирамиды $V$ и площадь ее полной поверхности $S_{\text{полн}}$:</p><p>$V = \frac{1}{3} r_{\text{впис}} \cdot S_{\text{полн}}$.</p><p>Отсюда $r_{\text{впис}} = \frac{3V}{S_{\text{полн}}}$.</p><p>Из предыдущих пунктов имеем:</p><p>$V = 12\sqrt{3}$ см?.</p><p>$S_{\text{полн}} = 12\sqrt{3} + 6\sqrt{39}$ см?.</p><p>Подставляем значения:</p><p>$r_{\text{впис}} = \frac{3 \cdot 12\sqrt{3}}{12\sqrt{3} + 6\sqrt{39}} = \frac{36\sqrt{3}}{6\sqrt{3}(2 + \sqrt{13})} = \frac{6}{2 + \sqrt{13}}$.</p><p>Избавимся от иррациональности в знаменателе, умножив числитель и знаменатель на сопряженное выражение $(\sqrt{13} - 2)$:</p><p>$r_{\text{впис}} = \frac{6(\sqrt{13} - 2)}{(2 + \sqrt{13})(\sqrt{13} - 2)} = \frac{6(\sqrt{13} - 2)}{(\sqrt{13})^2 - 2^2} = \frac{6(\sqrt{13} - 2)}{13 - 4} = \frac{6(\sqrt{13} - 2)}{9} = \frac{2(\sqrt{13} - 2)}{3}$ см.</p><p><strong>Ответ:</strong> $\frac{2(\sqrt{13} - 2)}{3}$ см.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "745182" "type" => "task" ] "previous" => array:2 [ "refs" => "745180" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1112 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1042 #items: array:1 [ 0 => 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#dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "933972" "type" => "book_page" ] "previous" => array:2 [ "refs" => "933970" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1229 #items: array:5 [ 0 => App\Models\Task {#1241 #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" => 745178 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "189" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/00-763" "field_display_title" => "763" "field_outside_task" => null "field_task_type" => 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№766 (с. 189)
Условие. №766 (с. 189)
скриншот условия
766. В правильной треугольной пирамиде DABC высота DO равна 3 см, а боковое ребро DA равно 5 см. Найдите:
а) площадь полной поверхности пирамиды;
б) объём пирамиды;
в) угол между боковым ребром и плоскостью основания;
г) угол наклона боковой грани к плоскости основания;
д) скалярное произведение векторов 12(DB + DC) MA, где M — середина ребра BC;
е) радиус шара, вписанного в пирамиду.
Решение 2. №766 (с. 189)
Решение 3. №766 (с. 189)
Решение 6. №766 (с. 189)
Дано: правильная треугольная пирамида $DABC$, высота $DO = 3$ см, боковое ребро $DA = 5$ см.
Так как пирамида правильная, в основании лежит равносторонний треугольник $ABC$, а высота $DO$ проецируется в его центр $O$. Точка $O$ является центром вписанной и описанной окружностей треугольника $ABC$.
Рассмотрим прямоугольный треугольник $DOA$. $DO$ — катет (высота пирамиды), $DA$ — гипотенуза (боковое ребро), $OA$ — второй катет. $OA$ является радиусом описанной окружности $R$ основания $ABC$.
По теореме Пифагора:
$OA = \sqrt{DA^2 - DO^2} = \sqrt{5^2 - 3^2} = \sqrt{25 - 9} = \sqrt{16} = 4$ см.
Итак, $R = OA = 4$ см.
Для равностороннего треугольника со стороной $a$ радиус описанной окружности $R = \frac{a}{\sqrt{3}}$. Отсюда найдем сторону основания $a$:
$a = R \cdot \sqrt{3} = 4\sqrt{3}$ см.
Радиус вписанной окружности $r$ для равностороннего треугольника равен $r = \frac{R}{2}$.
$r = OM = \frac{4}{2} = 2$ см, где $M$ — середина стороны $BC$.
а) площадь полной поверхности пирамиды;
Площадь полной поверхности пирамиды $S_{\text{полн}}$ равна сумме площади основания $S_{\text{осн}}$ и площади боковой поверхности $S_{\text{бок}}$.
1. Площадь основания (равностороннего треугольника):
$S_{\text{осн}} = \frac{a^2\sqrt{3}}{4} = \frac{(4\sqrt{3})^2\sqrt{3}}{4} = \frac{16 \cdot 3 \cdot \sqrt{3}}{4} = 12\sqrt{3}$ см?.
2. Площадь боковой поверхности. $S_{\text{бок}} = 3 \cdot S_{\triangle DBC}$. Для нахождения площади боковой грани $DBC$ нужна ее высота (апофема пирамиды) $DM$. Найдем $DM$ из прямоугольного треугольника $DOM$ (где $M$ — середина $BC$):
$DM = \sqrt{DO^2 + OM^2} = \sqrt{3^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13}$ см.
Площадь одной боковой грани:
$S_{\triangle DBC} = \frac{1}{2} \cdot BC \cdot DM = \frac{1}{2} \cdot 4\sqrt{3} \cdot \sqrt{13} = 2\sqrt{39}$ см?.
Площадь всей боковой поверхности:
$S_{\text{бок}} = 3 \cdot 2\sqrt{39} = 6\sqrt{39}$ см?.
3. Площадь полной поверхности:
$S_{\text{полн}} = S_{\text{осн}} + S_{\text{бок}} = 12\sqrt{3} + 6\sqrt{39} = 6\sqrt{3}(2 + \sqrt{13})$ см?.
Ответ: $12\sqrt{3} + 6\sqrt{39}$ см?.
б) объём пирамиды;
Объём пирамиды вычисляется по формуле $V = \frac{1}{3} S_{\text{осн}} \cdot H$, где $H=DO$.
$V = \frac{1}{3} \cdot 12\sqrt{3} \cdot 3 = 12\sqrt{3}$ см?.
Ответ: $12\sqrt{3}$ см?.
в) угол между боковым ребром и плоскостью основания;
Угол между боковым ребром (например, $DA$) и плоскостью основания ($ABC$) — это угол между ребром и его проекцией на эту плоскость. Проекцией ребра $DA$ является отрезок $OA$. Искомый угол — это $\angle DAO$ в прямоугольном треугольнике $DOA$.
Мы знаем все стороны этого треугольника: $DO=3$, $OA=4$, $DA=5$.
$\sin(\angle DAO) = \frac{DO}{DA} = \frac{3}{5}$.
Следовательно, угол равен $\arcsin\left(\frac{3}{5}\right)$.
Ответ: $\arcsin\left(\frac{3}{5}\right)$.
г) угол наклона боковой грани к плоскости основания;
Угол наклона боковой грани (например, $DBC$) к плоскости основания ($ABC$) — это линейный угол двугранного угла, который образуется перпендикулярами, проведенными к линии их пересечения ($BC$). Такими перпендикулярами являются апофема $DM$ и радиус вписанной окружности $OM$. Искомый угол — это $\angle DMO$ в прямоугольном треугольнике $DOM$.
Мы знаем катеты этого треугольника: $DO=3$, $OM=2$.
$\text{tg}(\angle DMO) = \frac{DO}{OM} = \frac{3}{2}$.
Следовательно, угол равен $\text{arctg}\left(\frac{3}{2}\right)$.
Ответ: $\text{arctg}\left(\frac{3}{2}\right)$.
д) скалярное произведение векторов $\frac{1}{2}(\overrightarrow{DB} + \overrightarrow{DC})\overrightarrow{MA}$, где $M$ — середина ребра $BC$;
1. Упростим первый вектор. По правилу сложения векторов, выходящих из одной точки, вектор $\overrightarrow{DB} + \overrightarrow{DC}$ равен $2\overrightarrow{DM}$, где $M$ — середина отрезка $BC$.
Таким образом, $\frac{1}{2}(\overrightarrow{DB} + \overrightarrow{DC}) = \frac{1}{2}(2\overrightarrow{DM}) = \overrightarrow{DM}$.
2. Нам нужно найти скалярное произведение $\overrightarrow{DM} \cdot \overrightarrow{MA}$.
По определению, скалярное произведение равно произведению длин векторов на косинус угла между ними: $\overrightarrow{DM} \cdot \overrightarrow{MA} = |\overrightarrow{DM}| \cdot |\overrightarrow{MA}| \cdot \cos(\angle DMA)$.
3. Найдем длины векторов и косинус угла между ними. Рассмотрим треугольник $DMA$.
Длина $|\overrightarrow{DM}|$ — это апофема, мы ее нашли в пункте а): $|\overrightarrow{DM}| = \sqrt{13}$ см.
Длина $|\overrightarrow{MA}|$ — это медиана (и высота) равностороннего треугольника $ABC$. $|\overrightarrow{MA}| = \frac{a\sqrt{3}}{2} = \frac{4\sqrt{3} \cdot \sqrt{3}}{2} = \frac{4 \cdot 3}{2} = 6$ см.
Длина ребра $|\overrightarrow{DA}|$ дана по условию: $|\overrightarrow{DA}| = 5$ см.
4. По теореме косинусов для треугольника $DMA$:
$DA^2 = DM^2 + MA^2 - 2 \cdot DM \cdot MA \cdot \cos(\angle DMA)$.
$5^2 = (\sqrt{13})^2 + 6^2 - 2 \cdot \sqrt{13} \cdot 6 \cdot \cos(\angle DMA)$.
$25 = 13 + 36 - 12\sqrt{13} \cdot \cos(\angle DMA)$.
$25 = 49 - 12\sqrt{13} \cdot \cos(\angle DMA)$.
$12\sqrt{13} \cdot \cos(\angle DMA) = 49 - 25 = 24$.
$\cos(\angle DMA) = \frac{24}{12\sqrt{13}} = \frac{2}{\sqrt{13}}$.
5. Вычисляем скалярное произведение:
$\overrightarrow{DM} \cdot \overrightarrow{MA} = |\overrightarrow{DM}| \cdot |\overrightarrow{MA}| \cdot \cos(\angle DMA) = \sqrt{13} \cdot 6 \cdot \frac{2}{\sqrt{13}} = 6 \cdot 2 = 12$.
Ответ: $12$.
е) радиус шара, вписанного в пирамиду.
Радиус $r_{\text{впис}}$ вписанного шара можно найти по формуле, связывающей объем пирамиды $V$ и площадь ее полной поверхности $S_{\text{полн}}$:
$V = \frac{1}{3} r_{\text{впис}} \cdot S_{\text{полн}}$.
Отсюда $r_{\text{впис}} = \frac{3V}{S_{\text{полн}}}$.
Из предыдущих пунктов имеем:
$V = 12\sqrt{3}$ см?.
$S_{\text{полн}} = 12\sqrt{3} + 6\sqrt{39}$ см?.
Подставляем значения:
$r_{\text{впис}} = \frac{3 \cdot 12\sqrt{3}}{12\sqrt{3} + 6\sqrt{39}} = \frac{36\sqrt{3}}{6\sqrt{3}(2 + \sqrt{13})} = \frac{6}{2 + \sqrt{13}}$.
Избавимся от иррациональности в знаменателе, умножив числитель и знаменатель на сопряженное выражение $(\sqrt{13} - 2)$:
$r_{\text{впис}} = \frac{6(\sqrt{13} - 2)}{(2 + \sqrt{13})(\sqrt{13} - 2)} = \frac{6(\sqrt{13} - 2)}{(\sqrt{13})^2 - 2^2} = \frac{6(\sqrt{13} - 2)}{13 - 4} = \frac{6(\sqrt{13} - 2)}{9} = \frac{2(\sqrt{13} - 2)}{3}$ см.
Ответ: $\frac{2(\sqrt{13} - 2)}{3}$ см.
Другие задания:
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