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Номер 311, страница 88 - 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 3. Многогранники. Параграф 3. Правильные многогранники, дополнительные задачи - номер 311, страница 88.
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" => 744726 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "88" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/00-311" "field_display_title" => "311" "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" => 744380 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Многогранники" "field_branch_order" => "3" "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" => "63" "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" => 744380 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Многогранники" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1050} "field_page_start" => "63" "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" => 744380 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Многогранники" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1130 #items: array:1 [ 0 => App\Models\Term {#1047} ] #escapeWhenCastingToString: false } "field_page_start" => "63" "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" => 744380 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Многогранники" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1130} "field_page_start" => "63" "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" => 744385 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Правильные многогранники, 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" => "86" "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" => 744385 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Правильные многогранники, дополнительные задачи" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1137} "field_page_start" => "86" "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" => 938960 "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" => "744726" "type" => "task" ] "text" => "<p> <b>311.</b> Основанием пирамиды DABC является треугольник со сторонами <span class="long">АС = 13 см, </span> <span class="long">AB = 15 см, </span> <span class="long">СВ = 14 см. </span> Боковое ребро DA перпендикулярно к плоскости основания и равно 9 см. а) Найдите площадь полной поверхности пирамиды. б) Докажите, что основание перпендикуляра, проведённого из вершины А к плоскости грани BDC, лежит на высоте этой грани, и найдите длину этого перпендикуляра. </p>" "img" => array:1 [ 0 => array:5 [ "name" => "311-1.jpg" "alt" => null "width" => "1564" "height" => 398 "path" => "/media/geometrija_10/atanasjan-u23/0-00/311-1.webp?ts=1739436538" ] ] ] #original: array:7 [ "id" => 938960 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1100} "task" => array:2 [ "refs" => "744726" "type" => "task" ] "text" => "<p> <b>311.</b> Основанием пирамиды DABC является треугольник со сторонами <span class="long">АС = 13 см, </span> <span class="long">AB = 15 см, </span> <span class="long">СВ = 14 см. </span> Боковое ребро DA перпендикулярно к плоскости основания и равно 9 см. а) Найдите площадь полной поверхности пирамиды. б) Докажите, что основание перпендикуляра, проведённого из вершины А к плоскости грани BDC, лежит на высоте этой грани, и найдите длину этого перпендикуляра. </p>" "img" => array:1 [ 0 => array:5 [ "name" => "311-1.jpg" "alt" => null "width" => "1564" "height" => 398 "path" => "/media/geometrija_10/atanasjan-u23/0-00/311-1.webp?ts=1739436538" ] ] ] #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" => 941161 "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" => "744726" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "311-1.png" "alt" => null "width" => "694" "height" => 4072 "path" => "/media/geometrija_10/atanasjan-u23/2-00/311-1.webp?ts=1739441169" ] 1 => array:5 [ "name" => "311-2.png" "alt" => null "width" => "694" "height" => 4397 "path" => "/media/geometrija_10/atanasjan-u23/2-00/311-2.webp?ts=1739441169" ] ] ] #original: array:6 [ "id" => 941161 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1043} "task" => array:2 [ "refs" => "744726" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "311-1.png" "alt" => null "width" => "694" "height" => 4072 "path" => "/media/geometrija_10/atanasjan-u23/2-00/311-1.webp?ts=1739441169" ] 1 => array:5 [ "name" => "311-2.png" "alt" => null "width" => "694" "height" => 4397 "path" => "/media/geometrija_10/atanasjan-u23/2-00/311-2.webp?ts=1739441169" ] ] ] #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" => 940976 "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" => "744726" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "311-1.jpg" "alt" => null "width" => "1328" "height" => 2073 "path" => "/media/geometrija_10/atanasjan-u23/3-00/311-1.webp?ts=1739436785" ] 1 => array:5 [ "name" => "311-2.jpg" "alt" => null "width" => "1328" "height" => 793 "path" => "/media/geometrija_10/atanasjan-u23/3-00/311-2.webp?ts=1739436785" ] ] ] #original: array:6 [ "id" => 940976 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1094} "task" => array:2 [ "refs" => "744726" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "311-1.jpg" "alt" => null "width" => "1328" "height" => 2073 "path" => "/media/geometrija_10/atanasjan-u23/3-00/311-1.webp?ts=1739436785" ] 1 => array:5 [ "name" => "311-2.jpg" "alt" => null "width" => "1328" "height" => 793 "path" => "/media/geometrija_10/atanasjan-u23/3-00/311-2.webp?ts=1739436785" ] ] ] #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" => 1352039 "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" => "744726" "type" => "task" ] "text" => "<p><strong>а)</strong> Найдём площадь полной поверхности пирамиды $S_{полн}$. Она равна сумме площади основания $S_{ABC}$ и площадей боковых граней $S_{DAB}$, $S_{DAC}$ и $S_{DBC}$.</p><p>$S_{полн} = S_{ABC} + S_{DAB} + S_{DAC} + S_{DBC}$</p><p>1. Вычислим площадь основания $S_{ABC}$.<br>Основанием является треугольник со сторонами $AC=13$ см, $AB=15$ см, $CB=14$ см. Для нахождения площади воспользуемся формулой Герона.<br>Полупериметр $p$ треугольника $ABC$ равен:<br>$p = (13 + 15 + 14) / 2 = 42 / 2 = 21$ см.<br>Площадь основания:<br>$S_{ABC} = \sqrt{p(p-a)(p-b)(p-c)} = \sqrt{21(21-14)(21-13)(21-15)} = \sqrt{21 \cdot 7 \cdot 8 \cdot 6} = \sqrt{7056} = 84$ см<sup>2</sup>.</p><p>2. Вычислим площади боковых граней.<br>По условию, ребро $DA$ перпендикулярно плоскости основания $(ABC)$, следовательно, $DA \perp AB$ и $DA \perp AC$. Это означает, что треугольники $DAB$ и $DAC$ — прямоугольные.<br>Площадь грани $DAB$:<br>$S_{DAB} = \frac{1}{2} \cdot DA \cdot AB = \frac{1}{2} \cdot 9 \cdot 15 = 67,5$ см<sup>2</sup>.<br>Площадь грани $DAC$:<br>$S_{DAC} = \frac{1}{2} \cdot DA \cdot AC = \frac{1}{2} \cdot 9 \cdot 13 = 58,5$ см<sup>2</sup>.<br>Для вычисления площади грани $DBC$ найдем её высоту, проведённую из вершины $D$. Проведём в треугольнике основания $ABC$ высоту $AM_1$ к стороне $BC$. Площадь $S_{ABC} = \frac{1}{2} \cdot BC \cdot AM_1$, отсюда:<br>$AM_1 = \frac{2 \cdot S_{ABC}}{BC} = \frac{2 \cdot 84}{14} = 12$ см.<br>Так как $DA$ — перпендикуляр к плоскости $(ABC)$, $AM_1$ — проекция наклонной $DM_1$ на эту плоскость, и $AM_1 \perp BC$, то по теореме о трёх перпендикулярах наклонная $DM_1$ также перпендикулярна $BC$ ($DM_1 \perp BC$). Следовательно, $DM_1$ — высота треугольника $DBC$.<br>Рассмотрим треугольник $DAM_1$. Так как $DA \perp (ABC)$, то $DA \perp AM_1$, и треугольник $DAM_1$ является прямоугольным. По теореме Пифагора найдём гипотенузу $DM_1$:<br>$DM_1 = \sqrt{DA^2 + AM_1^2} = \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15$ см.<br>Теперь можем найти площадь грани $DBC$:<br>$S_{DBC} = \frac{1}{2} \cdot BC \cdot DM_1 = \frac{1}{2} \cdot 14 \cdot 15 = 105$ см<sup>2</sup>.</p><p>3. Вычислим площадь полной поверхности пирамиды.<br>$S_{полн} = S_{ABC} + S_{DAB} + S_{DAC} + S_{DBC} = 84 + 67,5 + 58,5 + 105 = 315$ см<sup>2</sup>.</p><p><strong>Ответ:</strong> $315$ см<sup>2</sup>.</p><hr><p><strong>б)</strong> <em>Доказательство.</em><br>Пусть $AK$ — перпендикуляр, проведённый из вершины $A$ к плоскости грани $BDC$. Требуется доказать, что основание перпендикуляра, точка $K$, лежит на высоте этой грани, проведённой из вершины $D$. Назовём эту высоту $DM_1$.<br>В плоскости основания $(ABC)$ проведем высоту $AM_1$ к стороне $BC$. Так как по условию $DA \perp (ABC)$, то $DA$ перпендикулярно любой прямой в этой плоскости, в том числе $DA \perp BC$.<br>Прямая $BC$ перпендикулярна двум пересекающимся прямым $AM_1$ и $DA$, которые лежат в плоскости $(DAM_1)$. Следовательно, прямая $BC$ перпендикулярна плоскости $(DAM_1)$.<br>Плоскость $(BDC)$ проходит через прямую $BC$, которая перпендикулярна плоскости $(DAM_1)$. По признаку перпендикулярности плоскостей, плоскость $(BDC)$ перпендикулярна плоскости $(DAM_1)$.<br>Линией пересечения этих двух взаимно перпендикулярных плоскостей является прямая $DM_1$.<br>Перпендикуляр $AK$, опущенный из точки $A$ (которая принадлежит плоскости $(DAM_1)$) на плоскость $(BDC)$, должен лежать в плоскости $(DAM_1)$, а его основание $K$ — на линии пересечения плоскостей, то есть на прямой $DM_1$.<br>Как было показано в пункте <strong>а)</strong>, отрезок $DM_1$ является высотой треугольника $DBC$.<br>Таким образом, доказано, что основание перпендикуляра, проведённого из вершины $A$ к плоскости грани $BDC$, лежит на высоте этой грани.</p><p><em>Нахождение длины перпендикуляра.</em><br>Искомая длина перпендикуляра $AK$ — это высота прямоугольного треугольника $DAM_1$, проведённая из вершины прямого угла $A$ к гипотенузе $DM_1$.<br>Из пункта <strong>а)</strong> мы знаем длины сторон треугольника $DAM_1$: катеты $DA = 9$ см, $AM_1 = 12$ см, и гипотенуза $DM_1 = 15$ см.<br>Площадь прямоугольного треугольника $DAM_1$ можно вычислить как $S_{DAM_1} = \frac{1}{2} \cdot DA \cdot AM_1$. Также площадь можно выразить через высоту к гипотенузе: $S_{DAM_1} = \frac{1}{2} \cdot DM_1 \cdot AK$.<br>Приравняв эти выражения, получим:<br>$DA \cdot AM_1 = DM_1 \cdot AK$<br>Отсюда выразим $AK$:<br>$AK = \frac{DA \cdot AM_1}{DM_1} = \frac{9 \cdot 12}{15} = \frac{108}{15} = \frac{36}{5} = 7,2$ см.</p><p><strong>Ответ:</strong> длина перпендикуляра равна $7,2$ см.</p>" ] #original: array:6 [ "id" => 1352039 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1122} "task" => array:2 [ "refs" => "744726" "type" => "task" ] "text" => "<p><strong>а)</strong> Найдём площадь полной поверхности пирамиды $S_{полн}$. Она равна сумме площади основания $S_{ABC}$ и площадей боковых граней $S_{DAB}$, $S_{DAC}$ и $S_{DBC}$.</p><p>$S_{полн} = S_{ABC} + S_{DAB} + S_{DAC} + S_{DBC}$</p><p>1. Вычислим площадь основания $S_{ABC}$.<br>Основанием является треугольник со сторонами $AC=13$ см, $AB=15$ см, $CB=14$ см. Для нахождения площади воспользуемся формулой Герона.<br>Полупериметр $p$ треугольника $ABC$ равен:<br>$p = (13 + 15 + 14) / 2 = 42 / 2 = 21$ см.<br>Площадь основания:<br>$S_{ABC} = \sqrt{p(p-a)(p-b)(p-c)} = \sqrt{21(21-14)(21-13)(21-15)} = \sqrt{21 \cdot 7 \cdot 8 \cdot 6} = \sqrt{7056} = 84$ см<sup>2</sup>.</p><p>2. Вычислим площади боковых граней.<br>По условию, ребро $DA$ перпендикулярно плоскости основания $(ABC)$, следовательно, $DA \perp AB$ и $DA \perp AC$. Это означает, что треугольники $DAB$ и $DAC$ — прямоугольные.<br>Площадь грани $DAB$:<br>$S_{DAB} = \frac{1}{2} \cdot DA \cdot AB = \frac{1}{2} \cdot 9 \cdot 15 = 67,5$ см<sup>2</sup>.<br>Площадь грани $DAC$:<br>$S_{DAC} = \frac{1}{2} \cdot DA \cdot AC = \frac{1}{2} \cdot 9 \cdot 13 = 58,5$ см<sup>2</sup>.<br>Для вычисления площади грани $DBC$ найдем её высоту, проведённую из вершины $D$. Проведём в треугольнике основания $ABC$ высоту $AM_1$ к стороне $BC$. Площадь $S_{ABC} = \frac{1}{2} \cdot BC \cdot AM_1$, отсюда:<br>$AM_1 = \frac{2 \cdot S_{ABC}}{BC} = \frac{2 \cdot 84}{14} = 12$ см.<br>Так как $DA$ — перпендикуляр к плоскости $(ABC)$, $AM_1$ — проекция наклонной $DM_1$ на эту плоскость, и $AM_1 \perp BC$, то по теореме о трёх перпендикулярах наклонная $DM_1$ также перпендикулярна $BC$ ($DM_1 \perp BC$). Следовательно, $DM_1$ — высота треугольника $DBC$.<br>Рассмотрим треугольник $DAM_1$. Так как $DA \perp (ABC)$, то $DA \perp AM_1$, и треугольник $DAM_1$ является прямоугольным. По теореме Пифагора найдём гипотенузу $DM_1$:<br>$DM_1 = \sqrt{DA^2 + AM_1^2} = \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15$ см.<br>Теперь можем найти площадь грани $DBC$:<br>$S_{DBC} = \frac{1}{2} \cdot BC \cdot DM_1 = \frac{1}{2} \cdot 14 \cdot 15 = 105$ см<sup>2</sup>.</p><p>3. Вычислим площадь полной поверхности пирамиды.<br>$S_{полн} = S_{ABC} + S_{DAB} + S_{DAC} + S_{DBC} = 84 + 67,5 + 58,5 + 105 = 315$ см<sup>2</sup>.</p><p><strong>Ответ:</strong> $315$ см<sup>2</sup>.</p><hr><p><strong>б)</strong> <em>Доказательство.</em><br>Пусть $AK$ — перпендикуляр, проведённый из вершины $A$ к плоскости грани $BDC$. Требуется доказать, что основание перпендикуляра, точка $K$, лежит на высоте этой грани, проведённой из вершины $D$. Назовём эту высоту $DM_1$.<br>В плоскости основания $(ABC)$ проведем высоту $AM_1$ к стороне $BC$. Так как по условию $DA \perp (ABC)$, то $DA$ перпендикулярно любой прямой в этой плоскости, в том числе $DA \perp BC$.<br>Прямая $BC$ перпендикулярна двум пересекающимся прямым $AM_1$ и $DA$, которые лежат в плоскости $(DAM_1)$. Следовательно, прямая $BC$ перпендикулярна плоскости $(DAM_1)$.<br>Плоскость $(BDC)$ проходит через прямую $BC$, которая перпендикулярна плоскости $(DAM_1)$. По признаку перпендикулярности плоскостей, плоскость $(BDC)$ перпендикулярна плоскости $(DAM_1)$.<br>Линией пересечения этих двух взаимно перпендикулярных плоскостей является прямая $DM_1$.<br>Перпендикуляр $AK$, опущенный из точки $A$ (которая принадлежит плоскости $(DAM_1)$) на плоскость $(BDC)$, должен лежать в плоскости $(DAM_1)$, а его основание $K$ — на линии пересечения плоскостей, то есть на прямой $DM_1$.<br>Как было показано в пункте <strong>а)</strong>, отрезок $DM_1$ является высотой треугольника $DBC$.<br>Таким образом, доказано, что основание перпендикуляра, проведённого из вершины $A$ к плоскости грани $BDC$, лежит на высоте этой грани.</p><p><em>Нахождение длины перпендикуляра.</em><br>Искомая длина перпендикуляра $AK$ — это высота прямоугольного треугольника $DAM_1$, проведённая из вершины прямого угла $A$ к гипотенузе $DM_1$.<br>Из пункта <strong>а)</strong> мы знаем длины сторон треугольника $DAM_1$: катеты $DA = 9$ см, $AM_1 = 12$ см, и гипотенуза $DM_1 = 15$ см.<br>Площадь прямоугольного треугольника $DAM_1$ можно вычислить как $S_{DAM_1} = \frac{1}{2} \cdot DA \cdot AM_1$. Также площадь можно выразить через высоту к гипотенузе: $S_{DAM_1} = \frac{1}{2} \cdot DM_1 \cdot AK$.<br>Приравняв эти выражения, получим:<br>$DA \cdot AM_1 = DM_1 \cdot AK$<br>Отсюда выразим $AK$:<br>$AK = \frac{DA \cdot AM_1}{DM_1} = \frac{9 \cdot 12}{15} = \frac{108}{15} = \frac{36}{5} = 7,2$ см.</p><p><strong>Ответ:</strong> длина перпендикуляра равна $7,2$ см.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "744727" "type" => "task" ] "previous" => array:2 [ "refs" => "744725" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1112 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "page" => 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№311 (с. 88)
Условие. №311 (с. 88)
скриншот условия
311. Основанием пирамиды DABC является треугольник со сторонами АС = 13 см, AB = 15 см, СВ = 14 см. Боковое ребро DA перпендикулярно к плоскости основания и равно 9 см. а) Найдите площадь полной поверхности пирамиды. б) Докажите, что основание перпендикуляра, проведённого из вершины А к плоскости грани BDC, лежит на высоте этой грани, и найдите длину этого перпендикуляра.
Решение 2. №311 (с. 88)
Решение 3. №311 (с. 88)
Решение 6. №311 (с. 88)
а) Найдём площадь полной поверхности пирамиды $S_{полн}$. Она равна сумме площади основания $S_{ABC}$ и площадей боковых граней $S_{DAB}$, $S_{DAC}$ и $S_{DBC}$.
$S_{полн} = S_{ABC} + S_{DAB} + S_{DAC} + S_{DBC}$
1. Вычислим площадь основания $S_{ABC}$.
Основанием является треугольник со сторонами $AC=13$ см, $AB=15$ см, $CB=14$ см. Для нахождения площади воспользуемся формулой Герона.
Полупериметр $p$ треугольника $ABC$ равен:
$p = (13 + 15 + 14) / 2 = 42 / 2 = 21$ см.
Площадь основания:
$S_{ABC} = \sqrt{p(p-a)(p-b)(p-c)} = \sqrt{21(21-14)(21-13)(21-15)} = \sqrt{21 \cdot 7 \cdot 8 \cdot 6} = \sqrt{7056} = 84$ см2.
2. Вычислим площади боковых граней.
По условию, ребро $DA$ перпендикулярно плоскости основания $(ABC)$, следовательно, $DA \perp AB$ и $DA \perp AC$. Это означает, что треугольники $DAB$ и $DAC$ — прямоугольные.
Площадь грани $DAB$:
$S_{DAB} = \frac{1}{2} \cdot DA \cdot AB = \frac{1}{2} \cdot 9 \cdot 15 = 67,5$ см2.
Площадь грани $DAC$:
$S_{DAC} = \frac{1}{2} \cdot DA \cdot AC = \frac{1}{2} \cdot 9 \cdot 13 = 58,5$ см2.
Для вычисления площади грани $DBC$ найдем её высоту, проведённую из вершины $D$. Проведём в треугольнике основания $ABC$ высоту $AM_1$ к стороне $BC$. Площадь $S_{ABC} = \frac{1}{2} \cdot BC \cdot AM_1$, отсюда:
$AM_1 = \frac{2 \cdot S_{ABC}}{BC} = \frac{2 \cdot 84}{14} = 12$ см.
Так как $DA$ — перпендикуляр к плоскости $(ABC)$, $AM_1$ — проекция наклонной $DM_1$ на эту плоскость, и $AM_1 \perp BC$, то по теореме о трёх перпендикулярах наклонная $DM_1$ также перпендикулярна $BC$ ($DM_1 \perp BC$). Следовательно, $DM_1$ — высота треугольника $DBC$.
Рассмотрим треугольник $DAM_1$. Так как $DA \perp (ABC)$, то $DA \perp AM_1$, и треугольник $DAM_1$ является прямоугольным. По теореме Пифагора найдём гипотенузу $DM_1$:
$DM_1 = \sqrt{DA^2 + AM_1^2} = \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15$ см.
Теперь можем найти площадь грани $DBC$:
$S_{DBC} = \frac{1}{2} \cdot BC \cdot DM_1 = \frac{1}{2} \cdot 14 \cdot 15 = 105$ см2.
3. Вычислим площадь полной поверхности пирамиды.
$S_{полн} = S_{ABC} + S_{DAB} + S_{DAC} + S_{DBC} = 84 + 67,5 + 58,5 + 105 = 315$ см2.
Ответ: $315$ см2.
б) Доказательство.
Пусть $AK$ — перпендикуляр, проведённый из вершины $A$ к плоскости грани $BDC$. Требуется доказать, что основание перпендикуляра, точка $K$, лежит на высоте этой грани, проведённой из вершины $D$. Назовём эту высоту $DM_1$.
В плоскости основания $(ABC)$ проведем высоту $AM_1$ к стороне $BC$. Так как по условию $DA \perp (ABC)$, то $DA$ перпендикулярно любой прямой в этой плоскости, в том числе $DA \perp BC$.
Прямая $BC$ перпендикулярна двум пересекающимся прямым $AM_1$ и $DA$, которые лежат в плоскости $(DAM_1)$. Следовательно, прямая $BC$ перпендикулярна плоскости $(DAM_1)$.
Плоскость $(BDC)$ проходит через прямую $BC$, которая перпендикулярна плоскости $(DAM_1)$. По признаку перпендикулярности плоскостей, плоскость $(BDC)$ перпендикулярна плоскости $(DAM_1)$.
Линией пересечения этих двух взаимно перпендикулярных плоскостей является прямая $DM_1$.
Перпендикуляр $AK$, опущенный из точки $A$ (которая принадлежит плоскости $(DAM_1)$) на плоскость $(BDC)$, должен лежать в плоскости $(DAM_1)$, а его основание $K$ — на линии пересечения плоскостей, то есть на прямой $DM_1$.
Как было показано в пункте а), отрезок $DM_1$ является высотой треугольника $DBC$.
Таким образом, доказано, что основание перпендикуляра, проведённого из вершины $A$ к плоскости грани $BDC$, лежит на высоте этой грани.
Нахождение длины перпендикуляра.
Искомая длина перпендикуляра $AK$ — это высота прямоугольного треугольника $DAM_1$, проведённая из вершины прямого угла $A$ к гипотенузе $DM_1$.
Из пункта а) мы знаем длины сторон треугольника $DAM_1$: катеты $DA = 9$ см, $AM_1 = 12$ см, и гипотенуза $DM_1 = 15$ см.
Площадь прямоугольного треугольника $DAM_1$ можно вычислить как $S_{DAM_1} = \frac{1}{2} \cdot DA \cdot AM_1$. Также площадь можно выразить через высоту к гипотенузе: $S_{DAM_1} = \frac{1}{2} \cdot DM_1 \cdot AK$.
Приравняв эти выражения, получим:
$DA \cdot AM_1 = DM_1 \cdot AK$
Отсюда выразим $AK$:
$AK = \frac{DA \cdot AM_1}{DM_1} = \frac{9 \cdot 12}{15} = \frac{108}{15} = \frac{36}{5} = 7,2$ см.
Ответ: длина перпендикуляра равна $7,2$ см.
Другие задания:
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