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Номер 11, страница 174 - гдз по геометрии 10 класс учебник Мерзляк, Номировский
Авторы: Мерзляк А. Г., Номировский Д. А., Полонский В. Б., Якир М. С.
Тип: Учебник
Издательство: Просвещение, Вентана-граф
Год издания: 2017 - 2026
Уровень обучения: базовый
Цвет обложки: оранжевый
ISBN: 978-5-360-07142-6
Математика: алгебра и начала математического анализа, геометрия
Популярные ГДЗ в 10 классе
Глава 4. Многогранники. Параграф 19. Усечённая пирамида. Упражнения - номер 11, страница 174.
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" => 733472 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "174" "field_page_end" => null "field_url" => "/10-klass/geometrija/merzlyak-uchebnik/19-11" "field_display_title" => "11" "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" => 732721 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Многогранники" "field_branch_order" => "4" "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" => "147" "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 {#1084 #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" => 1851 "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:1 [ 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 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1054 #items: array:2 [ 0 => App\Models\Term {#1055 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5153 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #original: array:6 [ "id" => 5153 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => 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" => 5174 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Вентана-граф" "field_cases" => null "field_translit" => "ventana-graf" ] #original: array:6 [ "id" => 5174 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Вентана-граф" "field_cases" => null "field_translit" => "ventana-graf" ] #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:4 [ 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" => 5164 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мерзляк" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Аркадий" "field_patronymic" => "Григорьевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 5164 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мерзляк" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Аркадий" "field_patronymic" => "Григорьевич" "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" => 5370 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Номировский" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Дмитрий" "field_patronymic" => "Анатольевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 5370 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Номировский" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Дмитрий" "field_patronymic" => "Анатольевич" "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" => 5506 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Полонский" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Виталий" "field_patronymic" => "Борисович" "field_surname_rp" => null ] #original: array:12 [ "id" => 5506 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Полонский" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Виталий" "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" => 6445 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Якир" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Михаил" "field_patronymic" => "Семёнович" "field_surname_rp" => null ] #original: array:12 [ "id" => 6445 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Якир" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Михаил" "field_patronymic" => "Семёнович" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1062 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1063 #items: array:1 [ 0 => App\Models\Term {#1064 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1065 #items: array:1 [ 0 => App\Models\Term {#1066 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 9 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Россия" "field_cases" => array:6 [ …6] "field_translit" => "rossiya" ] #original: array:6 [ "id" => 9 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Россия" "field_cases" => array:6 [ …6] "field_translit" => "rossiya" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1067 #items: array:1 [ 0 => App\Models\Term {#1068 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 15 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Москва" "field_cases" => array:6 [ …6] "field_translit" => "moskva" ] #original: array:6 [ "id" => 15 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Москва" "field_cases" => array:6 [ …6] "field_translit" => "moskva" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1069 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1070 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1071 #items: array:1 [ 0 => App\Models\Term {#1072 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 38 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "базовый" "field_cases" => array:6 [ …6] "field_translit" => "bazovyy" ] #original: array:6 [ "id" => 38 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "базовый" "field_cases" => array:6 [ …6] "field_translit" => "bazovyy" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1073 #items: array:1 [ 0 => App\Models\Term {#1074 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6706 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "ФГОС (старый)" "field_cases" => array:6 [ …6] "field_color" => null "field_end_year" => "2022" "field_start_year" => "2019" "field_translit" => "fgos-old" "field_type" => null ] #original: array:10 [ "id" => 6706 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "ФГОС (старый)" "field_cases" => array:6 [ …6] "field_color" => null "field_end_year" => "2022" "field_start_year" => "2019" "field_translit" => "fgos-old" "field_type" => null ] #changes: [] #casts: [] 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"branch_parent" => Illuminate\Database\Eloquent\Collection {#1129} ] #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 {#1130 #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" => 732731 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Усечённая пирамида" "field_branch_order" => "19" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1134 #items: array:1 [ 0 => App\Models\Term {#1131 #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" => "171" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1133 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1132 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 732731 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Усечённая пирамида" "field_branch_order" => "19" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1134} "field_page_start" => "171" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1133} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1132} ] #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\Branch {#1135 #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" => 732733 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Упражнения" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1136 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "173" "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 {#1137 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1146 #items: array:1 [ 0 => App\Models\Branch {#1145 #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" => 732731 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Усечённая пирамида" "field_branch_order" => "19" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1147 …2} "field_page_start" => "171" "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 {#1148 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1149 …2} ] #original: array:24 [ "id" => 732731 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Усечённая пирамида" "field_branch_order" => "19" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1147 …2} "field_page_start" => "171" "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 {#1148 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1149 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 732733 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Упражнения" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1136} "field_page_start" => "173" "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 {#1137} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1146} ] #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 {#1114 #items: array:3 [ 0 => App\Models\Element {#1107 #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" => 931902 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1097 #items: array:1 [ 0 => App\Models\Edition {#1105 #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" => 2217 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1103 …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" => "margarita" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1104 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1102 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1101 …2} "field_content_source" => null ] #original: array:21 [ "id" => 2217 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1103 …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" => "margarita" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1104 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1102 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1101 …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" => "733472" "type" => "task" ] "text" => "<p><strong>19.11.</strong> Боковое ребро $BB_1$ усечённой пирамиды $ABCA_1B_1C_1$ перпендикулярно плоскости основания, $BB_1 = 4$ см, $AB = BC = 16$ см, $A_1B_1 = B_1C_1 = 10$ см, $\angle ABC = 120^\circ$. Найдите площадь боковой поверхности пирамиды.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "11-1.png" "alt" => null "width" => "497" "height" => 114 "path" => "/media/geometrija_10/merzljak/0-19/11-1.webp?ts=1739301668" ] ] ] #original: array:7 [ "id" => 931902 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1097} "task" => array:2 [ "refs" => "733472" "type" => "task" ] "text" => "<p><strong>19.11.</strong> Боковое ребро $BB_1$ усечённой пирамиды $ABCA_1B_1C_1$ перпендикулярно плоскости основания, $BB_1 = 4$ см, $AB = BC = 16$ см, $A_1B_1 = B_1C_1 = 10$ см, $\angle ABC = 120^\circ$. Найдите площадь боковой поверхности пирамиды.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "11-1.png" "alt" => null "width" => "497" "height" => 114 "path" => "/media/geometrija_10/merzljak/0-19/11-1.webp?ts=1739301668" ] ] ] #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 {#1099 #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" => 932735 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1091 #items: array:1 [ 0 => App\Models\Edition {#1100 #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" => 2218 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 1" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1095 …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" => "margarita" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1096 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1094 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_content_source" => null ] #original: array:21 [ "id" => 2218 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 1" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1095 …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" => "margarita" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1096 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1094 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "733472" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "11-1.png" "alt" => null "width" => "694" "height" => 3134 "path" => "/media/geometrija_10/merzljak/1-19/11-1.webp?ts=1739302569" ] ] ] #original: array:6 [ "id" => 932735 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1091} "task" => array:2 [ "refs" => "733472" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "11-1.png" "alt" => null "width" => "694" "height" => 3134 "path" => "/media/geometrija_10/merzljak/1-19/11-1.webp?ts=1739302569" ] ] ] #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 {#1093 #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" => 1445771 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1041 #items: array:1 [ 0 => App\Models\Edition {#1092 #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" => 5367 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 3" "field_order" => "4" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1086 …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" => "3-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1083 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5367 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 3" "field_order" => "4" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1086 …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" => "3-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1083 …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" => "733472" "type" => "task" ] "text" => "<p>Площадь боковой поверхности усеченной пирамиды $ABCA_1B_1C_1$ равна сумме площадей ее боковых граней. Боковыми гранями являются трапеции $ABB_1A_1$, $BCC_1B_1$ и $ACC_1A_1$.</p><p><strong>1. Найдем площади граней $ABB_1A_1$ и $BCC_1B_1$</strong></p><p>По условию, боковое ребро $BB_1$ перпендикулярно плоскости основания $ABC$. Это означает, что ребро $BB_1$ перпендикулярно любой прямой в этой плоскости, проходящей через точку $B$. Следовательно, $BB_1 \perp AB$ и $BB_1 \perp BC$.</p><p>Поскольку плоскости оснований $ABC$ и $A_1B_1C_1$ параллельны, ребро $BB_1$ также перпендикулярно прямым $A_1B_1$ и $B_1C_1$.</p><p>Таким образом, грани $ABB_1A_1$ и $BCC_1B_1$ являются прямоугольными трапециями, где $BB_1$ — их высота.</p><p>Дано: $AB = BC = 16$ см, $A_1B_1 = B_1C_1 = 10$ см, $BB_1 = 4$ см.</p><p>Площадь трапеции $ABB_1A_1$ вычисляется по формуле:</p><p>$S_{ABB_1A_1} = \frac{AB + A_1B_1}{2} \cdot BB_1 = \frac{16 + 10}{2} \cdot 4 = \frac{26}{2} \cdot 4 = 13 \cdot 4 = 52$ см<sup>2</sup>.</p><p>Аналогично, площадь трапеции $BCC_1B_1$:</p><p>$S_{BCC_1B_1} = \frac{BC + B_1C_1}{2} \cdot BB_1 = \frac{16 + 10}{2} \cdot 4 = \frac{26}{2} \cdot 4 = 13 \cdot 4 = 52$ см<sup>2</sup>.</p><p><strong>2. Найдем площадь грани $ACC_1A_1$</strong></p><p>Грань $ACC_1A_1$ является трапецией с основаниями $AC$ и $A_1C_1$.</p><p>Найдем длины оснований $AC$ и $A_1C_1$ с помощью теоремы косинусов для треугольников $ABC$ и $A_1B_1C_1$ соответственно.</p><p>В треугольнике $\triangle ABC$ ($AB = BC = 16$, $\angle ABC = 120^\circ$):</p><p>$AC^2 = AB^2 + BC^2 - 2 \cdot AB \cdot BC \cdot \cos(\angle ABC)$</p><p>$AC^2 = 16^2 + 16^2 - 2 \cdot 16 \cdot 16 \cdot \cos(120^\circ) = 256 + 256 - 512 \cdot (-\frac{1}{2}) = 512 + 256 = 768$</p><p>$AC = \sqrt{768} = \sqrt{256 \cdot 3} = 16\sqrt{3}$ см.</p><p>Треугольник $\triangle A_1B_1C_1$ подобен треугольнику $\triangle ABC$, поэтому $\angle A_1B_1C_1 = \angle ABC = 120^\circ$.</p><p>В треугольнике $\triangle A_1B_1C_1$ ($A_1B_1 = B_1C_1 = 10$):</p><p>$A_1C_1^2 = A_1B_1^2 + B_1C_1^2 - 2 \cdot A_1B_1 \cdot B_1C_1 \cdot \cos(\angle A_1B_1C_1)$</p><p>$A_1C_1^2 = 10^2 + 10^2 - 2 \cdot 10 \cdot 10 \cdot \cos(120^\circ) = 100 + 100 - 200 \cdot (-\frac{1}{2}) = 200 + 100 = 300$</p><p>$A_1C_1 = \sqrt{300} = \sqrt{100 \cdot 3} = 10\sqrt{3}$ см.</p><p>Теперь найдем длины боковых сторон трапеции $AA_1$ и $CC_1$. Рассмотрим прямоугольную трапецию $ABB_1A_1$. Проведем высоту $A_1H$ из точки $A_1$ на основание $AB$. Получим прямоугольный треугольник $\triangle AHA_1$, где $A_1H = BB_1 = 4$ см, а $AH = AB - B_1H = AB - A_1B_1 = 16 - 10 = 6$ см. По теореме Пифагора:</p><p>$AA_1^2 = AH^2 + A_1H^2 = 6^2 + 4^2 = 36 + 16 = 52$</p><p>$AA_1 = \sqrt{52} = 2\sqrt{13}$ см.</p><p>Аналогично для трапеции $BCC_1B_1$ находим $CC_1 = 2\sqrt{13}$ см. Так как $AA_1 = CC_1$, трапеция $ACC_1A_1$ — равнобедренная.</p><p>Найдем высоту $h$ трапеции $ACC_1A_1$. Проведем высоты из вершин $A_1$ и $C_1$ на основание $AC$. Они отсекут на большем основании отрезок, равный $\frac{AC - A_1C_1}{2}$.</p><p>$\frac{16\sqrt{3} - 10\sqrt{3}}{2} = \frac{6\sqrt{3}}{2} = 3\sqrt{3}$ см.</p><p>Высота $h$ является катетом в прямоугольном треугольнике, где гипотенуза — боковая сторона $AA_1$, а другой катет — найденный отрезок. По теореме Пифагора:</p><p>$h^2 = AA_1^2 - (3\sqrt{3})^2 = (2\sqrt{13})^2 - 27 = 52 - 27 = 25$</p><p>$h = \sqrt{25} = 5$ см.</p><p>Теперь можем найти площадь трапеции $ACC_1A_1$:</p><p>$S_{ACC_1A_1} = \frac{AC + A_1C_1}{2} \cdot h = \frac{16\sqrt{3} + 10\sqrt{3}}{2} \cdot 5 = \frac{26\sqrt{3}}{2} \cdot 5 = 13\sqrt{3} \cdot 5 = 65\sqrt{3}$ см<sup>2</sup>.</p><p><strong>3. Найдем площадь боковой поверхности пирамиды</strong></p><p>Площадь боковой поверхности $S_{бок}$ равна сумме площадей боковых граней:</p><p>$S_{бок} = S_{ABB_1A_1} + S_{BCC_1B_1} + S_{ACC_1A_1} = 52 + 52 + 65\sqrt{3} = 104 + 65\sqrt{3}$ см<sup>2</sup>.</p><p><strong>Ответ:</strong> $104 + 65\sqrt{3}$ см<sup>2</sup>.</p>" ] #original: array:6 [ "id" => 1445771 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1041} "task" => array:2 [ "refs" => "733472" "type" => "task" ] "text" => "<p>Площадь боковой поверхности усеченной пирамиды $ABCA_1B_1C_1$ равна сумме площадей ее боковых граней. Боковыми гранями являются трапеции $ABB_1A_1$, $BCC_1B_1$ и $ACC_1A_1$.</p><p><strong>1. Найдем площади граней $ABB_1A_1$ и $BCC_1B_1$</strong></p><p>По условию, боковое ребро $BB_1$ перпендикулярно плоскости основания $ABC$. Это означает, что ребро $BB_1$ перпендикулярно любой прямой в этой плоскости, проходящей через точку $B$. Следовательно, $BB_1 \perp AB$ и $BB_1 \perp BC$.</p><p>Поскольку плоскости оснований $ABC$ и $A_1B_1C_1$ параллельны, ребро $BB_1$ также перпендикулярно прямым $A_1B_1$ и $B_1C_1$.</p><p>Таким образом, грани $ABB_1A_1$ и $BCC_1B_1$ являются прямоугольными трапециями, где $BB_1$ — их высота.</p><p>Дано: $AB = BC = 16$ см, $A_1B_1 = B_1C_1 = 10$ см, $BB_1 = 4$ см.</p><p>Площадь трапеции $ABB_1A_1$ вычисляется по формуле:</p><p>$S_{ABB_1A_1} = \frac{AB + A_1B_1}{2} \cdot BB_1 = \frac{16 + 10}{2} \cdot 4 = \frac{26}{2} \cdot 4 = 13 \cdot 4 = 52$ см<sup>2</sup>.</p><p>Аналогично, площадь трапеции $BCC_1B_1$:</p><p>$S_{BCC_1B_1} = \frac{BC + B_1C_1}{2} \cdot BB_1 = \frac{16 + 10}{2} \cdot 4 = \frac{26}{2} \cdot 4 = 13 \cdot 4 = 52$ см<sup>2</sup>.</p><p><strong>2. Найдем площадь грани $ACC_1A_1$</strong></p><p>Грань $ACC_1A_1$ является трапецией с основаниями $AC$ и $A_1C_1$.</p><p>Найдем длины оснований $AC$ и $A_1C_1$ с помощью теоремы косинусов для треугольников $ABC$ и $A_1B_1C_1$ соответственно.</p><p>В треугольнике $\triangle ABC$ ($AB = BC = 16$, $\angle ABC = 120^\circ$):</p><p>$AC^2 = AB^2 + BC^2 - 2 \cdot AB \cdot BC \cdot \cos(\angle ABC)$</p><p>$AC^2 = 16^2 + 16^2 - 2 \cdot 16 \cdot 16 \cdot \cos(120^\circ) = 256 + 256 - 512 \cdot (-\frac{1}{2}) = 512 + 256 = 768$</p><p>$AC = \sqrt{768} = \sqrt{256 \cdot 3} = 16\sqrt{3}$ см.</p><p>Треугольник $\triangle A_1B_1C_1$ подобен треугольнику $\triangle ABC$, поэтому $\angle A_1B_1C_1 = \angle ABC = 120^\circ$.</p><p>В треугольнике $\triangle A_1B_1C_1$ ($A_1B_1 = B_1C_1 = 10$):</p><p>$A_1C_1^2 = A_1B_1^2 + B_1C_1^2 - 2 \cdot A_1B_1 \cdot B_1C_1 \cdot \cos(\angle A_1B_1C_1)$</p><p>$A_1C_1^2 = 10^2 + 10^2 - 2 \cdot 10 \cdot 10 \cdot \cos(120^\circ) = 100 + 100 - 200 \cdot (-\frac{1}{2}) = 200 + 100 = 300$</p><p>$A_1C_1 = \sqrt{300} = \sqrt{100 \cdot 3} = 10\sqrt{3}$ см.</p><p>Теперь найдем длины боковых сторон трапеции $AA_1$ и $CC_1$. Рассмотрим прямоугольную трапецию $ABB_1A_1$. Проведем высоту $A_1H$ из точки $A_1$ на основание $AB$. Получим прямоугольный треугольник $\triangle AHA_1$, где $A_1H = BB_1 = 4$ см, а $AH = AB - B_1H = AB - A_1B_1 = 16 - 10 = 6$ см. По теореме Пифагора:</p><p>$AA_1^2 = AH^2 + A_1H^2 = 6^2 + 4^2 = 36 + 16 = 52$</p><p>$AA_1 = \sqrt{52} = 2\sqrt{13}$ см.</p><p>Аналогично для трапеции $BCC_1B_1$ находим $CC_1 = 2\sqrt{13}$ см. Так как $AA_1 = CC_1$, трапеция $ACC_1A_1$ — равнобедренная.</p><p>Найдем высоту $h$ трапеции $ACC_1A_1$. Проведем высоты из вершин $A_1$ и $C_1$ на основание $AC$. Они отсекут на большем основании отрезок, равный $\frac{AC - A_1C_1}{2}$.</p><p>$\frac{16\sqrt{3} - 10\sqrt{3}}{2} = \frac{6\sqrt{3}}{2} = 3\sqrt{3}$ см.</p><p>Высота $h$ является катетом в прямоугольном треугольнике, где гипотенуза — боковая сторона $AA_1$, а другой катет — найденный отрезок. По теореме Пифагора:</p><p>$h^2 = AA_1^2 - (3\sqrt{3})^2 = (2\sqrt{13})^2 - 27 = 52 - 27 = 25$</p><p>$h = \sqrt{25} = 5$ см.</p><p>Теперь можем найти площадь трапеции $ACC_1A_1$:</p><p>$S_{ACC_1A_1} = \frac{AC + A_1C_1}{2} \cdot h = \frac{16\sqrt{3} + 10\sqrt{3}}{2} \cdot 5 = \frac{26\sqrt{3}}{2} \cdot 5 = 13\sqrt{3} \cdot 5 = 65\sqrt{3}$ см<sup>2</sup>.</p><p><strong>3. Найдем площадь боковой поверхности пирамиды</strong></p><p>Площадь боковой поверхности $S_{бок}$ равна сумме площадей боковых граней:</p><p>$S_{бок} = S_{ABB_1A_1} + S_{BCC_1B_1} + S_{ACC_1A_1} = 52 + 52 + 65\sqrt{3} = 104 + 65\sqrt{3}$ см<sup>2</sup>.</p><p><strong>Ответ:</strong> $104 + 65\sqrt{3}$ см<sup>2</sup>.</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" => "733473" "type" => "task" ] "previous" => array:2 [ "refs" => "733471" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1110 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1113 #items: array:1 [ 0 => 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"edition_groups" => Illuminate\Database\Eloquent\Collection {#1043 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1119 #items: array:1 [ 0 => App\Models\Element {#1121 #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" => 1147729 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1139 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1147729 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1139 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "971613" "type" => "book_page" ] "previous" => array:2 [ "refs" => "971611" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1227 #items: array:10 [ 0 => App\Models\Task {#1243 #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" => 733464 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "173" "field_page_end" => "174" "field_url" => "/10-klass/geometrija/merzlyak-uchebnik/19-3" "field_display_title" => "3" "field_outside_task" => "0" "field_task_type" => 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{#1244 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1245 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1246 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1247 …2} "content" => Illuminate\Database\Eloquent\Collection {#1250 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1256 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Task {#1249 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 733465 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "174" "field_page_end" => null "field_url" => "/10-klass/geometrija/merzlyak-uchebnik/19-4" "field_display_title" => "4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1248 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1255 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1257 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1253 …2} "content" => Illuminate\Database\Eloquent\Collection {#1252 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1270 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 733465 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "174" "field_page_end" => null "field_url" => "/10-klass/geometrija/merzlyak-uchebnik/19-4" "field_display_title" => "4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1248 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1255 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1257 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1253 …2} "content" => Illuminate\Database\Eloquent\Collection {#1252 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1270 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Task {#1254 #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" => 733466 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "174" "field_page_end" => null "field_url" => "/10-klass/geometrija/merzlyak-uchebnik/19-5" "field_display_title" => "5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1251 …2} "field_metatags_title" => null 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false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 3 => App\Models\Task {#1268 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 4 => App\Models\Task {#1282 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] 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№11 (с. 174)
Условие. №11 (с. 174)
скриншот условия
19.11. Боковое ребро $BB_1$ усечённой пирамиды $ABCA_1B_1C_1$ перпендикулярно плоскости основания, $BB_1 = 4$ см, $AB = BC = 16$ см, $A_1B_1 = B_1C_1 = 10$ см, $\angle ABC = 120^\circ$. Найдите площадь боковой поверхности пирамиды.
Решение 1. №11 (с. 174)
Решение 3. №11 (с. 174)
Площадь боковой поверхности усеченной пирамиды $ABCA_1B_1C_1$ равна сумме площадей ее боковых граней. Боковыми гранями являются трапеции $ABB_1A_1$, $BCC_1B_1$ и $ACC_1A_1$.
1. Найдем площади граней $ABB_1A_1$ и $BCC_1B_1$
По условию, боковое ребро $BB_1$ перпендикулярно плоскости основания $ABC$. Это означает, что ребро $BB_1$ перпендикулярно любой прямой в этой плоскости, проходящей через точку $B$. Следовательно, $BB_1 \perp AB$ и $BB_1 \perp BC$.
Поскольку плоскости оснований $ABC$ и $A_1B_1C_1$ параллельны, ребро $BB_1$ также перпендикулярно прямым $A_1B_1$ и $B_1C_1$.
Таким образом, грани $ABB_1A_1$ и $BCC_1B_1$ являются прямоугольными трапециями, где $BB_1$ — их высота.
Дано: $AB = BC = 16$ см, $A_1B_1 = B_1C_1 = 10$ см, $BB_1 = 4$ см.
Площадь трапеции $ABB_1A_1$ вычисляется по формуле:
$S_{ABB_1A_1} = \frac{AB + A_1B_1}{2} \cdot BB_1 = \frac{16 + 10}{2} \cdot 4 = \frac{26}{2} \cdot 4 = 13 \cdot 4 = 52$ см2.
Аналогично, площадь трапеции $BCC_1B_1$:
$S_{BCC_1B_1} = \frac{BC + B_1C_1}{2} \cdot BB_1 = \frac{16 + 10}{2} \cdot 4 = \frac{26}{2} \cdot 4 = 13 \cdot 4 = 52$ см2.
2. Найдем площадь грани $ACC_1A_1$
Грань $ACC_1A_1$ является трапецией с основаниями $AC$ и $A_1C_1$.
Найдем длины оснований $AC$ и $A_1C_1$ с помощью теоремы косинусов для треугольников $ABC$ и $A_1B_1C_1$ соответственно.
В треугольнике $\triangle ABC$ ($AB = BC = 16$, $\angle ABC = 120^\circ$):
$AC^2 = AB^2 + BC^2 - 2 \cdot AB \cdot BC \cdot \cos(\angle ABC)$
$AC^2 = 16^2 + 16^2 - 2 \cdot 16 \cdot 16 \cdot \cos(120^\circ) = 256 + 256 - 512 \cdot (-\frac{1}{2}) = 512 + 256 = 768$
$AC = \sqrt{768} = \sqrt{256 \cdot 3} = 16\sqrt{3}$ см.
Треугольник $\triangle A_1B_1C_1$ подобен треугольнику $\triangle ABC$, поэтому $\angle A_1B_1C_1 = \angle ABC = 120^\circ$.
В треугольнике $\triangle A_1B_1C_1$ ($A_1B_1 = B_1C_1 = 10$):
$A_1C_1^2 = A_1B_1^2 + B_1C_1^2 - 2 \cdot A_1B_1 \cdot B_1C_1 \cdot \cos(\angle A_1B_1C_1)$
$A_1C_1^2 = 10^2 + 10^2 - 2 \cdot 10 \cdot 10 \cdot \cos(120^\circ) = 100 + 100 - 200 \cdot (-\frac{1}{2}) = 200 + 100 = 300$
$A_1C_1 = \sqrt{300} = \sqrt{100 \cdot 3} = 10\sqrt{3}$ см.
Теперь найдем длины боковых сторон трапеции $AA_1$ и $CC_1$. Рассмотрим прямоугольную трапецию $ABB_1A_1$. Проведем высоту $A_1H$ из точки $A_1$ на основание $AB$. Получим прямоугольный треугольник $\triangle AHA_1$, где $A_1H = BB_1 = 4$ см, а $AH = AB - B_1H = AB - A_1B_1 = 16 - 10 = 6$ см. По теореме Пифагора:
$AA_1^2 = AH^2 + A_1H^2 = 6^2 + 4^2 = 36 + 16 = 52$
$AA_1 = \sqrt{52} = 2\sqrt{13}$ см.
Аналогично для трапеции $BCC_1B_1$ находим $CC_1 = 2\sqrt{13}$ см. Так как $AA_1 = CC_1$, трапеция $ACC_1A_1$ — равнобедренная.
Найдем высоту $h$ трапеции $ACC_1A_1$. Проведем высоты из вершин $A_1$ и $C_1$ на основание $AC$. Они отсекут на большем основании отрезок, равный $\frac{AC - A_1C_1}{2}$.
$\frac{16\sqrt{3} - 10\sqrt{3}}{2} = \frac{6\sqrt{3}}{2} = 3\sqrt{3}$ см.
Высота $h$ является катетом в прямоугольном треугольнике, где гипотенуза — боковая сторона $AA_1$, а другой катет — найденный отрезок. По теореме Пифагора:
$h^2 = AA_1^2 - (3\sqrt{3})^2 = (2\sqrt{13})^2 - 27 = 52 - 27 = 25$
$h = \sqrt{25} = 5$ см.
Теперь можем найти площадь трапеции $ACC_1A_1$:
$S_{ACC_1A_1} = \frac{AC + A_1C_1}{2} \cdot h = \frac{16\sqrt{3} + 10\sqrt{3}}{2} \cdot 5 = \frac{26\sqrt{3}}{2} \cdot 5 = 13\sqrt{3} \cdot 5 = 65\sqrt{3}$ см2.
3. Найдем площадь боковой поверхности пирамиды
Площадь боковой поверхности $S_{бок}$ равна сумме площадей боковых граней:
$S_{бок} = S_{ABB_1A_1} + S_{BCC_1B_1} + S_{ACC_1A_1} = 52 + 52 + 65\sqrt{3} = 104 + 65\sqrt{3}$ см2.
Ответ: $104 + 65\sqrt{3}$ см2.
Другие задания:
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