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Номер 81, страница 60 - гдз по геометрии 10 класс рабочая тетрадь Глазков, Юдина
Авторы: Глазков Ю. А., Юдина И. И., Бутузов В. Ф.
Тип: рабочая тетрадь
Серия: мгу - школе
Издательство: Просвещение
Год издания: 2020 - 2026
Уровень обучения: базовый и углублённый
Цвет обложки: коричневый в сеточку
ISBN: 978-5-09-097573-5
Популярные ГДЗ в 10 классе
3.2. Пирамида - номер 81, страница 60.
App\Models\Task {#1003 // 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" => 738552 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "field_page_start" => "60" "field_page_end" => "61" "field_url" => "/10-klass/geometrija/glazkov-tetrad/00-81" "field_display_title" => "81" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1009 #items: array:1 [ 0 => App\Models\Term {#1008 #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 00:54:11" "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 00:54:11" "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 {#1007 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1018 #items: array:1 [ 0 => App\Models\Branch {#1017 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:23 [ "id" => 738461 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "field_display_title" => "3.2. Пирамида" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1019 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "59" "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" => [] "book" => Illuminate\Database\Eloquent\Collection {#1055 #items: array:1 [ 0 => App\Models\Book {#1020 #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" => 261 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1022 #items: array:1 [ 0 => App\Models\Term {#1021 #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 00:54:11" "updated_at" => null "name" => "Геометрия" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "геометрию" "field_creative_case" => "геометрией" "field_dative_case" => "геометрии" "field_genitive_case" => "геометрии" "field_nominative_case" => "геометрия" "field_prepositional_case" => "геометрии" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "geometrija" ] #original: array:10 [ "id" => 6477 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "name" => "Геометрия" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "геометрию" "field_creative_case" => "геометрией" "field_dative_case" => "геометрии" "field_genitive_case" => "геометрии" "field_nominative_case" => "геометрия" "field_prepositional_case" => "геометрии" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "geometrija" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1023 #items: array:1 [ 0 => App\Models\Term {#1024 #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 00:54:11" "updated_at" => null "name" => "10" "field_cases" => array:6 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "field_translit" => "desjatyj" ] #original: array:6 [ "id" => 5459 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "name" => "10" "field_cases" => array:6 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "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 {#1025 #items: array:1 [ 0 => App\Models\Term {#1026 #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 00:54:11" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #original: array:6 [ "id" => 5153 "created_at" => "2026-04-10 00:54:11" "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 {#1027 #items: array:3 [ 0 => App\Models\Term {#1028 #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" => 4121 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "name" => "Глазков" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Юрий" "field_patronymic" => "Александрович" "field_surname_rp" => null ] #original: array:12 [ "id" => 4121 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "name" => "Глазков" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Юрий" "field_patronymic" => "Александрович" "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 {#1029 #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" => 6432 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "name" => "Юдина" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Ирина" "field_patronymic" => "Игоревна" "field_surname_rp" => null ] #original: array:12 [ "id" => 6432 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "name" => "Юдина" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Ирина" "field_patronymic" => "Игоревна" "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 {#1030 #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 00:54:11" "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 00:54:11" "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 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1031 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1032 #items: array:1 [ 0 => App\Models\Term {#1033 #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" => 6736 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "name" => "рабочая тетрадь" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "рабочую тетрадь" "field_creative_case" => "рабочей тетрадью" "field_dative_case" => "рабочей тетради" "field_genitive_case" => "рабочей тетради" "field_nominative_case" => "рабочая тетрадь" "field_prepositional_case" => "рабочей тетради" ] "field_plural_form" => "рабочие тетради" "field_short_name" => "рт" "field_short_name_foreign" => null "field_translit" => "rabochaya tetrad" ] #original: array:10 [ "id" => 6736 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "name" => "рабочая тетрадь" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "рабочую тетрадь" "field_creative_case" => "рабочей тетрадью" "field_dative_case" => "рабочей тетради" "field_genitive_case" => "рабочей тетради" "field_nominative_case" => "рабочая тетрадь" "field_prepositional_case" => "рабочей тетради" ] "field_plural_form" => "рабочие тетради" "field_short_name" => "рт" "field_short_name_foreign" => null "field_translit" => "rabochaya tetrad" ] #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 {#1034 #items: array:1 [ 0 => App\Models\Term {#1035 #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 00:54:11" "updated_at" => null "name" => "Россия" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "rossiya" ] #original: array:6 [ "id" => 9 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "name" => "Россия" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "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 {#1036 #items: array:1 [ 0 => App\Models\Term {#1037 #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 00:54:11" "updated_at" => null "name" => "Москва" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "moskva" ] #original: array:6 [ "id" => 15 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "name" => "Москва" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "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 {#1038 #items: array:1 [ 0 => App\Models\Term {#1039 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:8 [ "id" => 4728 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "name" => "мгу - школе" "field_cases" => null "field_foreign_lang_series" => null "field_transcription_eng" => null "field_translit" => "mgu - shkole" ] #original: array:8 [ "id" => 4728 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "name" => "мгу - школе" "field_cases" => null "field_foreign_lang_series" => null "field_transcription_eng" => null "field_translit" => "mgu - shkole" ] #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_umk" => Illuminate\Database\Eloquent\Collection {#1040 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1041 #items: array:1 [ 0 => App\Models\Term {#1042 #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 00:54:11" "updated_at" => null "name" => "базовый и углублённый" "field_cases" => array:6 [ "field_accusative_case" => "базовый и углубленный" "field_creative_case" => "базовым и углубленным" "field_dative_case" => "базовому и углубленному" "field_genitive_case" => "базового и углубленного" "field_nominative_case" => "базовый и углубленный" "field_prepositional_case" => "базовом и углубленном" ] "field_translit" => "bazovyy i uglublennyy" ] #original: array:6 [ "id" => 41 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "name" => "базовый и углублённый" "field_cases" => array:6 [ "field_accusative_case" => "базовый и углубленный" "field_creative_case" => "базовым и углубленным" "field_dative_case" => "базовому и углубленному" "field_genitive_case" => "базового и углубленного" "field_nominative_case" => "базовый и углубленный" "field_prepositional_case" => "базовом и углубленном" ] "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 {#1043 #items: array:1 [ 0 => App\Models\Term {#1044 #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 00:54:11" "updated_at" => null "name" => "ФГОС (старый)" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_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 00:54:11" "updated_at" => null "name" => "ФГОС (старый)" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_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" => "4" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1045 #items: array:1 [ 0 => App\Models\Term {#1046 #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 00:54:11" "updated_at" => null "name" => "стереотипное" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "stereotipnoe" ] #original: array:6 [ "id" => 33 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "name" => "стереотипное" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "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 {#1047 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => "в сеточку" "field_publication_year" => "2020" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное пособие" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "110" "field_priority" => "14" "field_default_folder" => "/geometrija_10/glazkov-rt/" "field_isbn" => "978-5-09-097573-5" "field_cover" => array:1 [ 0 => "/media/geometrija_10/glazkov-rt/covers/cover1.webp" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_10/glazkov-rt/covers/cover1.webp" "alt" => "" "width" => "1200" "height" => "1661" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1048 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1049 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1050 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/geometrija/glazkov-tetrad" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1051 #items: array:1 [ 0 => App\Models\Term {#1052 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 51 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "name" => "коричневый" "field_cases" => array:6 [ "field_accusative_case" => "коричневый" "field_creative_case" => "коричневым" "field_dative_case" => "коричневому" "field_genitive_case" => "коричневого" "field_nominative_case" => "коричневый" "field_prepositional_case" => "коричневом" ] "field_translit" => "korichnevyy" ] #original: array:6 [ "id" => 51 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "name" => "коричневый" "field_cases" => array:6 [ "field_accusative_case" => "коричневый" "field_creative_case" => "коричневым" "field_dative_case" => "коричневому" "field_genitive_case" => "коричневого" "field_nominative_case" => "коричневый" "field_prepositional_case" => "коричневом" ] "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" => 261 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1022} "field_class" => Illuminate\Database\Eloquent\Collection {#1023} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1025} "field_author" => Illuminate\Database\Eloquent\Collection {#1027} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1031} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1032} "field_country" => Illuminate\Database\Eloquent\Collection {#1034} "field_city" => Illuminate\Database\Eloquent\Collection {#1036} "field_series" => Illuminate\Database\Eloquent\Collection {#1038} "field_umk" => Illuminate\Database\Eloquent\Collection {#1040} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1041} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1043} "field_publication_number" => "4" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1045} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1047} "field_under_the_edition_degree" => null "field_cover_description" => "в сеточку" "field_publication_year" => "2020" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное пособие" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "110" "field_priority" => "14" "field_default_folder" => "/geometrija_10/glazkov-rt/" "field_isbn" => "978-5-09-097573-5" "field_cover" => array:1 [ 0 => "/media/geometrija_10/glazkov-rt/covers/cover1.webp" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_10/glazkov-rt/covers/cover1.webp" "alt" => "" "width" => "1200" "height" => "1661" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1048} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1049} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1050} "field_url" => "/10-klass/geometrija/glazkov-tetrad" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1051} "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 {#1053 #items: [] #escapeWhenCastingToString: false } ] #original: array:23 [ "id" => 738461 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "field_display_title" => "3.2. Пирамида" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1019} "field_page_start" => "59" "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" => [] "book" => Illuminate\Database\Eloquent\Collection {#1055} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1053} ] #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 {#1010 #items: array:1 [ 0 => App\Models\Branch {#1017} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1066 #items: array:3 [ 0 => App\Models\Element {#1076 #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" => 937160 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1085 #items: array:1 [ 0 => App\Models\Edition {#1077 #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" => 2220 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1079 #items: array:1 [ 0 => App\Models\Term {#1078 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5153 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #original: array:6 [ "id" => 5153 "created_at" => "2026-04-10 00:54:11" "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_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 {#1080 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1081 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1082 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 2220 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1079} "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 {#1080} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1081} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1082} "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" => "738552" "type" => "task" ] "text" => "<p><strong>81</strong> Все боковые ребра пирамиды равны между собой. Докажите, что:</p><p>а) высота пирамиды проходит через центр окружности, описанной около основания;</p><p>б) все боковые ребра составляют равные углы с плоскостью основания.</p><p><strong>Доказательство.</strong></p><p>а) Пусть основание пирамиды — многоугольник $A_1A_2...A_n$, отрезок $PO$ — высота пирамиды. Тогда отрезки $OA_1$, $OA_2$, ..., $OA_n$ — проекции боковых ребер $PA_1, PA_2$, ..., $PA_n$ на плоскость основания. Так как</p><p>$PA_1 = PA_2 = ... = PA_n$, то $OA_1 = OA_2 = \dots = OA_n$. Следовательно, точка $O$ равноудалена от вершин многоугольника $A_1A_2...A_n$, поэтому она является центром окружности, описанной около основания пирамиды.</p><p>б) $ \triangle A_1PO = \triangle A_2PO = \dots = \triangle A_nPO $ (по гипотенузе и катету), следовательно, $ \angle PA_1O = \angle PA_2O = \dots = \angle PA_nO$, что и требовалось доказать.</p>" "img" => array:2 [ 0 => array:5 [ "name" => "81-1.jpg" "alt" => null "width" => "1671" "height" => 695 "path" => "/media/geometrija_10/glazkov-rt/0-00/81-1.webp?ts=1739375753" ] 1 => array:5 [ "name" => "81-2.jpg" "alt" => null "width" => "1671" "height" => 782 "path" => "/media/geometrija_10/glazkov-rt/0-00/81-2.webp?ts=1739375753" ] ] ] #original: array:7 [ "id" => 937160 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1085} "task" => array:2 [ "refs" => "738552" "type" => "task" ] "text" => "<p><strong>81</strong> Все боковые ребра пирамиды равны между собой. Докажите, что:</p><p>а) высота пирамиды проходит через центр окружности, описанной около основания;</p><p>б) все боковые ребра составляют равные углы с плоскостью основания.</p><p><strong>Доказательство.</strong></p><p>а) Пусть основание пирамиды — многоугольник $A_1A_2...A_n$, отрезок $PO$ — высота пирамиды. Тогда отрезки $OA_1$, $OA_2$, ..., $OA_n$ — проекции боковых ребер $PA_1, PA_2$, ..., $PA_n$ на плоскость основания. Так как</p><p>$PA_1 = PA_2 = ... = PA_n$, то $OA_1 = OA_2 = \dots = OA_n$. Следовательно, точка $O$ равноудалена от вершин многоугольника $A_1A_2...A_n$, поэтому она является центром окружности, описанной около основания пирамиды.</p><p>б) $ \triangle A_1PO = \triangle A_2PO = \dots = \triangle A_nPO $ (по гипотенузе и катету), следовательно, $ \angle PA_1O = \angle PA_2O = \dots = \angle PA_nO$, что и требовалось доказать.</p>" "img" => array:2 [ 0 => array:5 [ "name" => "81-1.jpg" "alt" => null "width" => "1671" "height" => 695 "path" => "/media/geometrija_10/glazkov-rt/0-00/81-1.webp?ts=1739375753" ] 1 => array:5 [ "name" => "81-2.jpg" "alt" => null "width" => "1671" "height" => 782 "path" => "/media/geometrija_10/glazkov-rt/0-00/81-2.webp?ts=1739375753" ] ] ] #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 {#1083 #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" => 937270 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1093 #items: array:1 [ 0 => App\Models\Edition {#1084 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 2221 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1087 #items: array:1 [ 0 => App\Models\Term {#1086 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6696 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "name" => "Skysmart" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "skysmart" ] #original: array:6 [ "id" => 6696 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "name" => "Skysmart" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "skysmart" ] #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_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 {#1088 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1089 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1090 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 2221 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1087} "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 {#1088} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1089} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1090} "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" => "738552" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "81-1.png" "alt" => null "width" => "694" "height" => 2463 "path" => "/media/geometrija_10/glazkov-rt/1-00/81-1.webp?ts=1739375889" ] 1 => array:5 [ "name" => "81-2.png" "alt" => null "width" => "694" "height" => 3285 "path" => "/media/geometrija_10/glazkov-rt/1-00/81-2.webp?ts=1739375889" ] ] ] #original: array:6 [ "id" => 937270 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1093} "task" => array:2 [ "refs" => "738552" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "81-1.png" "alt" => null "width" => "694" "height" => 2463 "path" => "/media/geometrija_10/glazkov-rt/1-00/81-1.webp?ts=1739375889" ] 1 => array:5 [ "name" => "81-2.png" "alt" => null "width" => "694" "height" => 3285 "path" => "/media/geometrija_10/glazkov-rt/1-00/81-2.webp?ts=1739375889" ] ] ] #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 {#1091 #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" => 1434344 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1101 #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" => 5337 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1095 #items: array:1 [ 0 => App\Models\Term {#1094 #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" => 7012 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "name" => "Gemini 2.5 Pro" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "gemini 2.5 pro" ] #original: array:6 [ "id" => 7012 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "name" => "Gemini 2.5 Pro" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "gemini 2.5 pro" ] #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_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" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1096 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1097 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1098 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5337 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1095} "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" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1096} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1097} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1098} "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" => "738552" "type" => "task" ] "text" => "<strong>а)</strong> <p>Пусть основание пирамиды — многоугольник $A_1A_2...A_n$, а отрезок $PO$ — ее высота. Тогда отрезки $OA_1, OA_2, ..., OA_n$ являются проекциями боковых ребер $PA_1, PA_2, ..., PA_n$ на плоскость основания. Рассмотрим прямоугольные треугольники $\Delta A_1PO, \Delta A_2PO, ..., \Delta A_nPO$. Они являются прямоугольными, так как высота $PO$ перпендикулярна плоскости основания ($PO \perp$ пл. $A_1...A_n$), а значит и любой прямой в этой плоскости, проходящей через точку $O$.</p> <p>По условию задачи все боковые ребра равны, то есть $PA_1 = PA_2 = ... = PA_n$. Эти ребра служат гипотенузами для данных прямоугольных треугольников. Катет $PO$ у этих треугольников является общим. По теореме Пифагора для любого из этих треугольников ($i=1, 2, ..., n$): $OA_i^2 = PA_i^2 - PO^2$. Так как гипотенузы $PA_i$ равны и катет $PO$ общий, то равны и вторые катеты $OA_i$: $OA_1 = OA_2 = ... = OA_n$.</p> <p>Следовательно, точка $O$ (основание высоты) равноудалена от всех вершин многоугольника $A_1A_2...A_n$. По определению, точка, равноудаленная от всех вершин многоугольника, является центром описанной около него окружности. Таким образом, высота пирамиды проходит через центр окружности, описанной около основания.</p> <p>Ответ: Утверждение а) доказано.</p> <strong>б)</strong> <p>Углом между боковым ребром (наклонной) и плоскостью основания является угол между этим ребром и его проекцией на плоскость. Для ребра $PA_i$ это угол $\angle PA_iO$.</p> <p>Рассмотрим прямоугольные треугольники $\Delta A_1PO, \Delta A_2PO, ..., \Delta A_nPO$. Как было показано в пункте а), у них равны гипотенузы ($PA_1 = PA_2 = ... = PA_n$) и есть общий катет $PO$. Следовательно, эти треугольники равны по гипотенузе и катету: $\Delta A_1PO = \Delta A_2PO = ... = \Delta A_nPO$.</p> <p>Из равенства треугольников следует равенство их соответствующих углов. В нашем случае это означает, что $\angle PA_1O = \angle PA_2O = ... = \angle PA_nO$. Это и есть углы, которые боковые ребра составляют с плоскостью основания. Таким образом, все боковые ребра составляют равные углы с плоскостью основания, что и требовалось доказать.</p> <p>Ответ: Утверждение б) доказано.</p>" ] #original: array:6 [ "id" => 1434344 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1101} "task" => array:2 [ "refs" => "738552" "type" => "task" ] "text" => "<strong>а)</strong> <p>Пусть основание пирамиды — многоугольник $A_1A_2...A_n$, а отрезок $PO$ — ее высота. Тогда отрезки $OA_1, OA_2, ..., OA_n$ являются проекциями боковых ребер $PA_1, PA_2, ..., PA_n$ на плоскость основания. Рассмотрим прямоугольные треугольники $\Delta A_1PO, \Delta A_2PO, ..., \Delta A_nPO$. Они являются прямоугольными, так как высота $PO$ перпендикулярна плоскости основания ($PO \perp$ пл. $A_1...A_n$), а значит и любой прямой в этой плоскости, проходящей через точку $O$.</p> <p>По условию задачи все боковые ребра равны, то есть $PA_1 = PA_2 = ... = PA_n$. Эти ребра служат гипотенузами для данных прямоугольных треугольников. Катет $PO$ у этих треугольников является общим. По теореме Пифагора для любого из этих треугольников ($i=1, 2, ..., n$): $OA_i^2 = PA_i^2 - PO^2$. Так как гипотенузы $PA_i$ равны и катет $PO$ общий, то равны и вторые катеты $OA_i$: $OA_1 = OA_2 = ... = OA_n$.</p> <p>Следовательно, точка $O$ (основание высоты) равноудалена от всех вершин многоугольника $A_1A_2...A_n$. По определению, точка, равноудаленная от всех вершин многоугольника, является центром описанной около него окружности. Таким образом, высота пирамиды проходит через центр окружности, описанной около основания.</p> <p>Ответ: Утверждение а) доказано.</p> <strong>б)</strong> <p>Углом между боковым ребром (наклонной) и плоскостью основания является угол между этим ребром и его проекцией на плоскость. Для ребра $PA_i$ это угол $\angle PA_iO$.</p> <p>Рассмотрим прямоугольные треугольники $\Delta A_1PO, \Delta A_2PO, ..., \Delta A_nPO$. Как было показано в пункте а), у них равны гипотенузы ($PA_1 = PA_2 = ... = PA_n$) и есть общий катет $PO$. Следовательно, эти треугольники равны по гипотенузе и катету: $\Delta A_1PO = \Delta A_2PO = ... = \Delta A_nPO$.</p> <p>Из равенства треугольников следует равенство их соответствующих углов. В нашем случае это означает, что $\angle PA_1O = \angle PA_2O = ... = \angle PA_nO$. Это и есть углы, которые боковые ребра составляют с плоскостью основания. Таким образом, все боковые ребра составляют равные углы с плоскостью основания, что и требовалось доказать.</p> <p>Ответ: Утверждение б) доказано.</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" => "738553" "type" => "task" ] "previous" => array:2 [ "refs" => "738551" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1073 #items: array:1 [ 0 => App\Models\Book {#1020} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1060 #items: array:1 [ 0 => App\Models\BookPage {#1062 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 971707 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "field_page_start" => "60" "field_url" => "/10-klass/geometrija/glazkov-tetrad/page-60" "field_display_title" => "60" "field_folder" => "1" "field_image_name" => "60" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1061 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1011 #items: array:1 [ 0 => App\Models\Book {#1020} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1014 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1074 #items: array:1 [ 0 => App\Models\Element {#1072 #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" => 1147823 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1070 #items: array:1 [ 0 => App\Models\Edition {#1077} ] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "971707" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "60-1.jpg" "alt" => null "width" => "1956" "height" => 2802 "path" => "/media/geometrija_10/glazkov-rt/0-1/60-1.webp?ts=1745219763" ] ] ] #original: array:6 [ "id" => 1147823 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1070} "book_page" => array:2 [ "refs" => "971707" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "60-1.jpg" "alt" => null "width" => "1956" "height" => 2802 "path" => "/media/geometrija_10/glazkov-rt/0-1/60-1.webp?ts=1745219763" ] ] ] #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" => "971708" "type" => "book_page" ] "previous" => array:2 [ "refs" => "971706" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1161 #items: array:2 [ 0 => App\Models\Task {#1169 #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" => 738551 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "field_page_start" => "59" "field_page_end" => "60" "field_url" => "/10-klass/geometrija/glazkov-tetrad/00-80" "field_display_title" => "80" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1170 #items: array:1 [ 0 => App\Models\Term {#1008} ] #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 {#1171 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1172 #items: array:1 [ 0 => App\Models\Branch {#1017} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1173 #items: array:1 [ 0 => App\Models\Branch {#1017} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1191 #items: array:3 [ 0 => App\Models\Element {#1201 #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" => 937159 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1202 #items: array:1 [ 0 => App\Models\Edition {#1077} ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "738551" "type" => "task" ] "text" => "<p><strong>80</strong> Основание пирамиды — прямоугольник $ABCD$, $AB = 18$ м, $BC = 10$ м, высота пирамиды проходит через точку пересечения диагоналей основания и равна 12 м. Найдите площадь полной поверхности пирамиды.</p><p>Решение.</p><p><strong>1)</strong> Площадь полной поверхности пирамиды вычисляется по формуле $S_{полн} = $ __________ + ______. Так как основание пирамиды —</p><p>со сторонами 10 м и ______, то</p><p>$S_{осн} = $ ______ $\cdot$ ______ = ______ ($м^2$).</p><p><strong>2)</strong> Чтобы найти площадь боковой</p><p>пирамиды, вычислим площади ее ______ граней.</p><p>В прямоугольнике $ABCD$ $AC$ ______ $BD$,</p><p>диагонали ______ в точке $O$,</p><p>поэтому $AO = BO = $ ______ $=$ ______. Отрезок $MO$ — высота пирамиды, значит,</p><p>$MO$ — ______ к плоскости основания, и отрезки $AO, BO$,</p><p>______, $DO$ — проекции наклонных $AM$,</p><p>______ и ______ на плоскость основания. Следовательно, $AM = BM = $ ______ $ =$ ______</p><p>$=$ ______ и $\triangle ABM = \triangle $ ______, а $\triangle BCM = $</p><p>$=$ ______ (по трем ______),</p><p>поэтому $S_{ABM}$ ____ $S_{CDM}$ и $S_{BCM}$ ____ $S_{ADM}$.</p><p><strong>3)</strong> Пусть $MK \perp AB$, тогда $OK$ ____ $AB$ (обратная теорема о ____</p><p>перпендикулярах) и $OK = $ ______ $BC = 0,5 \cdot $ ______ $=$ ______ (м). Аналогично</p><p>если $MN \perp BC$, то $ON = $ ______ $AB = 0,5 \cdot $ ______ $=$ ______ (м).</p><p>Поскольку $MO \perp ABC$, то $MO$ ____ $OK$, а значит, $MK = \sqrt{MO^2 + \_\_\_\_\_\_}$</p><p>$= \sqrt{\_\_\_\_\_\_ + 5^2} = \sqrt{ \_\_\_\_\_\_ } = $______ (м).</p><p>Аналогично $MN = \sqrt{\_\_\_\_\_\_ + ON^2} = \sqrt{12^2 + \_\_\_\__\_}$ = $\sqrt{\_\_\_\_\_\_} = $______ (м).</p><p>Итак, $S_{ABM} = 0,5AB \cdot $ ______ $=$ ______ $\cdot 18 \cdot $ ______ $=$ ______ ($м^2$), $S_{BCM} = $</p><p>______. Отсюда получаем:</p><p>$S_{бок} = 2(S_{ABM} + $______) $=$ ______ $\cdot$ (______ $+$ ______) $=$ ______ ($м^2$),</p><p>$S_{полн} = $ ______ $+$ ______ $=$ ______ ($м^2$).</p><p>О т в е т .</p>" "img" => array:2 [ 0 => array:5 [ "name" => "80-1.jpg" "alt" => null "width" => "1646" "height" => 732 "path" => "/media/geometrija_10/glazkov-rt/0-00/80-1.webp?ts=1739375753" ] 1 => array:5 [ "name" => "80-2.jpg" "alt" => null "width" => "1671" "height" => 1807 "path" => "/media/geometrija_10/glazkov-rt/0-00/80-2.webp?ts=1739375753" ] ] ] #original: array:7 [ "id" => 937159 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1202} "task" => array:2 [ "refs" => "738551" "type" => "task" ] "text" => "<p><strong>80</strong> Основание пирамиды — прямоугольник $ABCD$, $AB = 18$ м, $BC = 10$ м, высота пирамиды проходит через точку пересечения диагоналей основания и равна 12 м. Найдите площадь полной поверхности пирамиды.</p><p>Решение.</p><p><strong>1)</strong> Площадь полной поверхности пирамиды вычисляется по формуле $S_{полн} = $ __________ + ______. Так как основание пирамиды —</p><p>со сторонами 10 м и ______, то</p><p>$S_{осн} = $ ______ $\cdot$ ______ = ______ ($м^2$).</p><p><strong>2)</strong> Чтобы найти площадь боковой</p><p>пирамиды, вычислим площади ее ______ граней.</p><p>В прямоугольнике $ABCD$ $AC$ ______ $BD$,</p><p>диагонали ______ в точке $O$,</p><p>поэтому $AO = BO = $ ______ $=$ ______. Отрезок $MO$ — высота пирамиды, значит,</p><p>$MO$ — ______ к плоскости основания, и отрезки $AO, BO$,</p><p>______, $DO$ — проекции наклонных $AM$,</p><p>______ и ______ на плоскость основания. Следовательно, $AM = BM = $ ______ $ =$ ______</p><p>$=$ ______ и $\triangle ABM = \triangle $ ______, а $\triangle BCM = $</p><p>$=$ ______ (по трем ______),</p><p>поэтому $S_{ABM}$ ____ $S_{CDM}$ и $S_{BCM}$ ____ $S_{ADM}$.</p><p><strong>3)</strong> Пусть $MK \perp AB$, тогда $OK$ ____ $AB$ (обратная теорема о ____</p><p>перпендикулярах) и $OK = $ ______ $BC = 0,5 \cdot $ ______ $=$ ______ (м). Аналогично</p><p>если $MN \perp BC$, то $ON = $ ______ $AB = 0,5 \cdot $ ______ $=$ ______ (м).</p><p>Поскольку $MO \perp ABC$, то $MO$ ____ $OK$, а значит, $MK = \sqrt{MO^2 + \_\_\_\_\_\_}$</p><p>$= \sqrt{\_\_\_\_\_\_ + 5^2} = \sqrt{ \_\_\_\_\_\_ } = $______ (м).</p><p>Аналогично $MN = \sqrt{\_\_\_\_\_\_ + ON^2} = \sqrt{12^2 + \_\_\_\__\_}$ = $\sqrt{\_\_\_\_\_\_} = $______ (м).</p><p>Итак, $S_{ABM} = 0,5AB \cdot $ ______ $=$ ______ $\cdot 18 \cdot $ ______ $=$ ______ ($м^2$), $S_{BCM} = $</p><p>______. Отсюда получаем:</p><p>$S_{бок} = 2(S_{ABM} + $______) $=$ ______ $\cdot$ (______ $+$ ______) $=$ ______ ($м^2$),</p><p>$S_{полн} = $ ______ $+$ ______ $=$ ______ ($м^2$).</p><p>О т в е т .</p>" "img" => array:2 [ 0 => array:5 [ "name" => "80-1.jpg" "alt" => null "width" => "1646" "height" => 732 "path" => "/media/geometrija_10/glazkov-rt/0-00/80-1.webp?ts=1739375753" ] 1 => array:5 [ "name" => "80-2.jpg" "alt" => null "width" => "1671" "height" => 1807 "path" => "/media/geometrija_10/glazkov-rt/0-00/80-2.webp?ts=1739375753" ] ] ] #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 {#1203 #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" => 937269 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1204 #items: array:1 [ 0 => App\Models\Edition {#1084} ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "738551" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "80-1.png" "alt" => null "width" => "694" "height" => 7001 "path" => "/media/geometrija_10/glazkov-rt/1-00/80-1.webp?ts=1739375889" ] ] ] #original: array:6 [ "id" => 937269 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1204} "task" => array:2 [ "refs" => "738551" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "80-1.png" "alt" => null "width" => "694" "height" => 7001 "path" => "/media/geometrija_10/glazkov-rt/1-00/80-1.webp?ts=1739375889" ] ] ] #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 {#1205 #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" => 1434343 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1206 #items: array:1 [ 0 => App\Models\Edition {#1092} ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "738551" "type" => "task" ] "text" => "<p><strong>1)</strong> Площадь полной поверхности пирамиды вычисляется по формуле $S_{полн} = S_{бок} + S_{осн}$. Так как основание пирамиды — прямоугольник со сторонами 10 м и 18 м, то $S_{осн} = 10 \cdot 18 = 180$ (м²).<br>Ответ: $S_{полн} = S_{бок} + S_{осн}$; 18; $10 \cdot 18 = 180$.</p><p><strong>2)</strong> Чтобы найти площадь боковой поверхности пирамиды, вычислим площади ее боковых граней.<br>В прямоугольнике ABCD $AC = BD$, диагонали равны и пересекаются в точке О, поэтому $AO = BO = CO = DO$. Отрезок $MO$ — высота пирамиды, значит, $MO$ перпендикулярен к плоскости основания, и отрезки $AO, BO, CO, DO$ — проекции наклонных $AM, BM, CM$ и $DM$ на плоскость основания. Следовательно, $AM = BM = CM = DM$ и $\Delta ABM = \Delta CDM$, а $\Delta BCM = \Delta ADM$ (по трем сторонам), поэтому $S_{ABM} = S_{CDM}$ и $S_{BCM} = S_{ADM}$.<br>Ответ: поверхности; боковых; =; равны и пересекаются; $CO = DO$; перпендикулярен; $CO$; $BM, CM, DM$; $CM = DM$; $CDM$; $ADM$; сторонам; =; =.</p><p><strong>3)</strong> Пусть $MK \perp AB$, тогда $OK \perp AB$ (обратная теорема о трех перпендикулярах) и $OK = 0,5 \cdot BC = 0,5 \cdot 10 = 5$ (м). Аналогично если $MN \perp BC$, то $ON = 0,5 \cdot AB = 0,5 \cdot 18 = 9$ (м).<br>Поскольку $MO \perp ABC$, то $MO \perp OK$, а значит, $MK=\sqrt{MO^2 + OK^2} = \sqrt{12^2 + 5^2} = \sqrt{144+25} = \sqrt{169} = 13$ (м).<br>Аналогично $MN = \sqrt{MO^2 + ON^2} = \sqrt{12^2 + 9^2} = \sqrt{144+81} = \sqrt{225} = 15$ (м).<br>Итак, $S_{ABM} = 0,5AB \cdot MK = 0,5 \cdot 18 \cdot 13 = 117$ (м²), $S_{BCM} = 0,5BC \cdot MN = 0,5 \cdot 10 \cdot 15 = 75$ (м²). Отсюда получаем:<br>$S_{бок} = 2(S_{ABM} + S_{BCM}) = 2 \cdot (117 + 75) = 384$ (м²),<br>$S_{полн} = S_{осн} + S_{бок} = 180 + 384 = 564$ (м²).<br>Ответ: $\perp$; трех; $0,5 \cdot 10 = 5$; $0,5 \cdot 18 = 9$; $\perp$; $OK^2$; $12^2$; $\sqrt{169} = 13$; $9^2$; $\sqrt{225} = 15$; $MK$; $0,5 \cdot 13 = 117$; $S_{BCM} = 0,5 \cdot 10 \cdot 15 = 75$; $S_{BCM}$; 2; 117 + 75; 384; $S_{осн} + S_{бок}$; $180 + 384 = 564$.</p>" ] #original: array:6 [ "id" => 1434343 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1206} "task" => array:2 [ "refs" => "738551" "type" => "task" ] "text" => "<p><strong>1)</strong> Площадь полной поверхности пирамиды вычисляется по формуле $S_{полн} = S_{бок} + S_{осн}$. Так как основание пирамиды — прямоугольник со сторонами 10 м и 18 м, то $S_{осн} = 10 \cdot 18 = 180$ (м²).<br>Ответ: $S_{полн} = S_{бок} + S_{осн}$; 18; $10 \cdot 18 = 180$.</p><p><strong>2)</strong> Чтобы найти площадь боковой поверхности пирамиды, вычислим площади ее боковых граней.<br>В прямоугольнике ABCD $AC = BD$, диагонали равны и пересекаются в точке О, поэтому $AO = BO = CO = DO$. Отрезок $MO$ — высота пирамиды, значит, $MO$ перпендикулярен к плоскости основания, и отрезки $AO, BO, CO, DO$ — проекции наклонных $AM, BM, CM$ и $DM$ на плоскость основания. Следовательно, $AM = BM = CM = DM$ и $\Delta ABM = \Delta CDM$, а $\Delta BCM = \Delta ADM$ (по трем сторонам), поэтому $S_{ABM} = S_{CDM}$ и $S_{BCM} = S_{ADM}$.<br>Ответ: поверхности; боковых; =; равны и пересекаются; $CO = DO$; перпендикулярен; $CO$; $BM, CM, DM$; $CM = DM$; $CDM$; $ADM$; сторонам; =; =.</p><p><strong>3)</strong> Пусть $MK \perp AB$, тогда $OK \perp AB$ (обратная теорема о трех перпендикулярах) и $OK = 0,5 \cdot BC = 0,5 \cdot 10 = 5$ (м). Аналогично если $MN \perp BC$, то $ON = 0,5 \cdot AB = 0,5 \cdot 18 = 9$ (м).<br>Поскольку $MO \perp ABC$, то $MO \perp OK$, а значит, $MK=\sqrt{MO^2 + OK^2} = \sqrt{12^2 + 5^2} = \sqrt{144+25} = \sqrt{169} = 13$ (м).<br>Аналогично $MN = \sqrt{MO^2 + ON^2} = \sqrt{12^2 + 9^2} = \sqrt{144+81} = \sqrt{225} = 15$ (м).<br>Итак, $S_{ABM} = 0,5AB \cdot MK = 0,5 \cdot 18 \cdot 13 = 117$ (м²), $S_{BCM} = 0,5BC \cdot MN = 0,5 \cdot 10 \cdot 15 = 75$ (м²). Отсюда получаем:<br>$S_{бок} = 2(S_{ABM} + S_{BCM}) = 2 \cdot (117 + 75) = 384$ (м²),<br>$S_{полн} = S_{осн} + S_{бок} = 180 + 384 = 564$ (м²).<br>Ответ: $\perp$; трех; $0,5 \cdot 10 = 5$; $0,5 \cdot 18 = 9$; $\perp$; $OK^2$; $12^2$; $\sqrt{169} = 13$; $9^2$; $\sqrt{225} = 15$; $MK$; $0,5 \cdot 13 = 117$; $S_{BCM} = 0,5 \cdot 10 \cdot 15 = 75$; $S_{BCM}$; 2; 117 + 75; 384; $S_{осн} + S_{бок}$; $180 + 384 = 564$.</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" => "738552" "type" => "task" ] "previous" => array:2 [ "refs" => "738550" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1198 #items: array:1 [ 0 => App\Models\Book {#1020} ] #escapeWhenCastingToString: false } 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№81 (с. 60)
Условие. №81 (с. 60)
скриншот условия
81 Все боковые ребра пирамиды равны между собой. Докажите, что:
а) высота пирамиды проходит через центр окружности, описанной около основания;
б) все боковые ребра составляют равные углы с плоскостью основания.
Доказательство.
а) Пусть основание пирамиды — многоугольник $A_1A_2...A_n$, отрезок $PO$ — высота пирамиды. Тогда отрезки $OA_1$, $OA_2$, ..., $OA_n$ — проекции боковых ребер $PA_1, PA_2$, ..., $PA_n$ на плоскость основания. Так как
$PA_1 = PA_2 = ... = PA_n$, то $OA_1 = OA_2 = \dots = OA_n$. Следовательно, точка $O$ равноудалена от вершин многоугольника $A_1A_2...A_n$, поэтому она является центром окружности, описанной около основания пирамиды.
б) $ \triangle A_1PO = \triangle A_2PO = \dots = \triangle A_nPO $ (по гипотенузе и катету), следовательно, $ \angle PA_1O = \angle PA_2O = \dots = \angle PA_nO$, что и требовалось доказать.
Решение. №81 (с. 60)
Решение 2. №81 (с. 60)
а)
Пусть основание пирамиды — многоугольник $A_1A_2...A_n$, а отрезок $PO$ — ее высота. Тогда отрезки $OA_1, OA_2, ..., OA_n$ являются проекциями боковых ребер $PA_1, PA_2, ..., PA_n$ на плоскость основания. Рассмотрим прямоугольные треугольники $\Delta A_1PO, \Delta A_2PO, ..., \Delta A_nPO$. Они являются прямоугольными, так как высота $PO$ перпендикулярна плоскости основания ($PO \perp$ пл. $A_1...A_n$), а значит и любой прямой в этой плоскости, проходящей через точку $O$.
По условию задачи все боковые ребра равны, то есть $PA_1 = PA_2 = ... = PA_n$. Эти ребра служат гипотенузами для данных прямоугольных треугольников. Катет $PO$ у этих треугольников является общим. По теореме Пифагора для любого из этих треугольников ($i=1, 2, ..., n$): $OA_i^2 = PA_i^2 - PO^2$. Так как гипотенузы $PA_i$ равны и катет $PO$ общий, то равны и вторые катеты $OA_i$: $OA_1 = OA_2 = ... = OA_n$.
Следовательно, точка $O$ (основание высоты) равноудалена от всех вершин многоугольника $A_1A_2...A_n$. По определению, точка, равноудаленная от всех вершин многоугольника, является центром описанной около него окружности. Таким образом, высота пирамиды проходит через центр окружности, описанной около основания.
Ответ: Утверждение а) доказано.
б)Углом между боковым ребром (наклонной) и плоскостью основания является угол между этим ребром и его проекцией на плоскость. Для ребра $PA_i$ это угол $\angle PA_iO$.
Рассмотрим прямоугольные треугольники $\Delta A_1PO, \Delta A_2PO, ..., \Delta A_nPO$. Как было показано в пункте а), у них равны гипотенузы ($PA_1 = PA_2 = ... = PA_n$) и есть общий катет $PO$. Следовательно, эти треугольники равны по гипотенузе и катету: $\Delta A_1PO = \Delta A_2PO = ... = \Delta A_nPO$.
Из равенства треугольников следует равенство их соответствующих углов. В нашем случае это означает, что $\angle PA_1O = \angle PA_2O = ... = \angle PA_nO$. Это и есть углы, которые боковые ребра составляют с плоскостью основания. Таким образом, все боковые ребра составляют равные углы с плоскостью основания, что и требовалось доказать.
Ответ: Утверждение б) доказано.
Другие задания:
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