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Номер 63, страница 50 - 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
2.3. Двугранный угол. Перпендикулярность плоскостей - номер 63, страница 50.
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" => 738534 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "50" "field_page_end" => null "field_url" => "/10-klass/geometrija/glazkov-tetrad/00-63" "field_display_title" => "63" "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" => 738459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "2.3. Двугранный угол. Перпендикулярность плоскостей" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1047 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "47" "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 {#1083 #items: array:1 [ 0 => App\Models\Book {#1048 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 261 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1050 #items: array:1 [ 0 => App\Models\Term {#1049 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6477 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Геометрия" "field_abbreviated_name" => null "field_cases" => array:6 [ "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 13:58:26" "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 {#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" => 5459 "created_at" => "2026-04-10 13:58:26" "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 13:58:26" "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 {#1053 #items: array:1 [ 0 => App\Models\Term {#1054 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5153 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #original: array:6 [ "id" => 5153 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1055 #items: array:3 [ 0 => App\Models\Term {#1056 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 4121 "created_at" => "2026-04-10 13:58:26" "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 13:58:26" "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 {#1057 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 6432 "created_at" => "2026-04-10 13:58:26" "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 13:58:26" "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 {#1058 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 3949 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Бутузов" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Валентин" "field_patronymic" => "Фёдорович" "field_surname_rp" => null ] #original: array:12 [ "id" => 3949 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Бутузов" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Валентин" "field_patronymic" => "Фёдорович" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1059 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1060 #items: array:1 [ 0 => 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:10 [ "id" => 6736 "created_at" => "2026-04-10 13:58:26" "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 13:58:26" "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 {#1062 #items: array:1 [ 0 => App\Models\Term {#1063 #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 [ "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 13:58:26" "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 {#1064 #items: array:1 [ 0 => App\Models\Term {#1065 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 15 "created_at" => "2026-04-10 13:58:26" "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 13:58:26" "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 {#1066 #items: array:1 [ 0 => App\Models\Term {#1067 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:8 [ "id" => 4728 "created_at" => "2026-04-10 13:58:26" "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 13:58:26" "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 {#1068 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1069 #items: array:1 [ 0 => App\Models\Term {#1070 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 41 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "базовый и углублённый" "field_cases" => array:6 [ "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 13:58:26" "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 {#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:10 [ "id" => 6706 "created_at" => "2026-04-10 13:58:26" "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 13:58:26" "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 {#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:6 [ "id" => 33 "created_at" => "2026-04-10 13:58:26" "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 13:58:26" "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 {#1075 #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?ts=1739295144" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_10/glazkov-rt/covers/cover1.webp?ts=1739295144" "alt" => "" "width" => "1200" "height" => "1661" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1076 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1077 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1078 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/geometrija/glazkov-tetrad" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1079 #items: array:1 [ 0 => App\Models\Term {#1080 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 51 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "коричневый" "field_cases" => array:6 [ "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 13:58:26" "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 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1050} "field_class" => Illuminate\Database\Eloquent\Collection {#1051} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1053} "field_author" => Illuminate\Database\Eloquent\Collection {#1055} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1059} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1060} "field_country" => Illuminate\Database\Eloquent\Collection {#1062} "field_city" => Illuminate\Database\Eloquent\Collection {#1064} "field_series" => Illuminate\Database\Eloquent\Collection {#1066} "field_umk" => Illuminate\Database\Eloquent\Collection {#1068} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1069} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1071} "field_publication_number" => "4" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1073} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1075} "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?ts=1739295144" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_10/glazkov-rt/covers/cover1.webp?ts=1739295144" "alt" => "" "width" => "1200" "height" => "1661" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1076} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1077} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1078} "field_url" => "/10-klass/geometrija/glazkov-tetrad" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1079} "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 {#1081 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 738459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "2.3. Двугранный угол. Перпендикулярность плоскостей" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1047} "field_page_start" => "47" "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 {#1083} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1081} ] #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 {#1038 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1042 #items: array:3 [ 0 => App\Models\Element {#1088 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 937142 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1097 #items: array:1 [ 0 => App\Models\Edition {#1089 #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 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1091 #items: array:1 [ 0 => App\Models\Term {#1090 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5153 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #original: array:6 [ "id" => 5153 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_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 {#1092 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1093 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1094 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 2220 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1091} "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 {#1092} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1093} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1094} "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" => "738534" "type" => "task" ] "text" => "<p><strong>63</strong> В треугольнике $ABC$ $AB = 13$ см, $BC = 14$ см, $AC = 15$ см. Точка $M$ удалена от прямых $AB$, $BC$ и $AC$ на 5 см. Найдите расстояние от точки $M$ до плоскости $ABC$, если известно, что ее проекция на эту плоскость лежит внутри треугольника.</p><p><strong>Решение.</strong></p><p>Пусть $MO$ — перпендикуляр к плоскости $ABC$, а $MN$, $MP$ и $MQ$ — перпендикуляры к прямым $AB$, $BC$ и $AC$. Требуется найти $MO$. По теореме, ______ имеем: $AB \perp ON$, $BC \perp$ ______ и $AC \perp$ ______. Итак, из точки $M$ проведены к плоскости $ABC$ перпендикуляр $MO$ и наклонные $MN$, $MP$ и $MQ$. По условию расстояния от точки $M$ до прямых $AB$, $BC$ и $AC$ равны, т. е. равны наклонные ______ , и ______ . Следовательно, потому равны и их проекции на эту плоскость: ______ . Таким образом, точка $O$ лежит внутри треугольника $ABC$ и равноудалена от ______ , поэтому она является ______ .</p><p>Радиус $ON$ этой окружности найдем, используя формулу $S = pr$, где $S$ — площадь треугольника $ABC$, $p$ — его ______ , $r = ON$. По формуле Герона $S = $ ______ см$^2$, следовательно, $r = \frac{S}{p} = $ ______ (см) = ______ см.</p><p>Итак, $NO = $ ______ см.</p><p>Треугольник $MON$ ______ , поскольку $MO \perp ABC$, и потому $MO \perp ON$. Так как $MN = $ ______ , $ON = $ ______ , то из треугольника $MON$ находим: $MO = $ ______ см.</p><p><strong>Ответ.</strong></p><p>______ см.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "63-1.jpg" "alt" => null "width" => "1646" "height" => 2006 "path" => "/media/geometrija_10/glazkov-rt/0-00/63-1.webp?ts=1739375737" ] ] ] #original: array:7 [ "id" => 937142 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1097} "task" => array:2 [ "refs" => "738534" "type" => "task" ] "text" => "<p><strong>63</strong> В треугольнике $ABC$ $AB = 13$ см, $BC = 14$ см, $AC = 15$ см. Точка $M$ удалена от прямых $AB$, $BC$ и $AC$ на 5 см. Найдите расстояние от точки $M$ до плоскости $ABC$, если известно, что ее проекция на эту плоскость лежит внутри треугольника.</p><p><strong>Решение.</strong></p><p>Пусть $MO$ — перпендикуляр к плоскости $ABC$, а $MN$, $MP$ и $MQ$ — перпендикуляры к прямым $AB$, $BC$ и $AC$. Требуется найти $MO$. По теореме, ______ имеем: $AB \perp ON$, $BC \perp$ ______ и $AC \perp$ ______. Итак, из точки $M$ проведены к плоскости $ABC$ перпендикуляр $MO$ и наклонные $MN$, $MP$ и $MQ$. По условию расстояния от точки $M$ до прямых $AB$, $BC$ и $AC$ равны, т. е. равны наклонные ______ , и ______ . Следовательно, потому равны и их проекции на эту плоскость: ______ . Таким образом, точка $O$ лежит внутри треугольника $ABC$ и равноудалена от ______ , поэтому она является ______ .</p><p>Радиус $ON$ этой окружности найдем, используя формулу $S = pr$, где $S$ — площадь треугольника $ABC$, $p$ — его ______ , $r = ON$. По формуле Герона $S = $ ______ см$^2$, следовательно, $r = \frac{S}{p} = $ ______ (см) = ______ см.</p><p>Итак, $NO = $ ______ см.</p><p>Треугольник $MON$ ______ , поскольку $MO \perp ABC$, и потому $MO \perp ON$. Так как $MN = $ ______ , $ON = $ ______ , то из треугольника $MON$ находим: $MO = $ ______ см.</p><p><strong>Ответ.</strong></p><p>______ см.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "63-1.jpg" "alt" => null "width" => "1646" "height" => 2006 "path" => "/media/geometrija_10/glazkov-rt/0-00/63-1.webp?ts=1739375737" ] ] ] #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 {#1095 #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" => 937252 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1105 #items: array:1 [ 0 => App\Models\Edition {#1096 #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 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1099 #items: array:1 [ 0 => App\Models\Term {#1098 #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 13:58:26" "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 13:58:26" "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 {#1100 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1101 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1102 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 2221 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1099} "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 {#1100} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1101} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1102} "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" => "738534" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "63-1.png" "alt" => null "width" => "694" "height" => 4423 "path" => "/media/geometrija_10/glazkov-rt/1-00/63-1.webp?ts=1739375872" ] ] ] #original: array:6 [ "id" => 937252 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1105} "task" => array:2 [ "refs" => "738534" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "63-1.png" "alt" => null "width" => "694" "height" => 4423 "path" => "/media/geometrija_10/glazkov-rt/1-00/63-1.webp?ts=1739375872" ] ] ] #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 {#1103 #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" => 1434326 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1113 #items: array:1 [ 0 => App\Models\Edition {#1104 #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 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1107 #items: array:1 [ 0 => App\Models\Term {#1106 #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 13:58:26" "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 13:58:26" "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 {#1108 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1109 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1110 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5337 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1107} "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 {#1108} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1109} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1110} "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" => "738534" "type" => "task" ] "text" => "<p>Пусть $MO$ — перпендикуляр, опущенный из точки $M$ на плоскость треугольника $ABC$. Тогда длина отрезка $MO$ — это искомое расстояние от точки $M$ до плоскости $ABC$.</p> <p>По условию, точка $M$ равноудалена от прямых $AB$, $BC$ и $AC$. Расстояние от точки до прямой — это длина перпендикуляра. Проведем из точки $M$ перпендикуляры $MN$, $MP$ и $MQ$ к прямым $AB$, $BC$ и $AC$ соответственно. По условию, $MN = MP = MQ = 5$ см. Эти отрезки являются наклонными к плоскости $ABC$.</p> <p>Отрезки $ON$, $OP$ и $OQ$ являются проекциями этих наклонных на плоскость $ABC$. Так как наклонные, проведенные из одной точки, равны, то равны и их проекции. Следовательно, $ON = OP = OQ$.</p> <p>Поскольку точка $O$ (проекция точки $M$) лежит внутри треугольника и равноудалена от его сторон ($ON \perp AB$, $OP \perp BC$, $OQ \perp AC$), то $O$ является центром окружности, вписанной в треугольник $ABC$. Длина отрезков $ON$, $OP$ и $OQ$ равна радиусу этой окружности ($r$).</p> <p><strong>Найдем радиус вписанной окружности $r$.</strong></p> <p>Радиус вписанной окружности можно найти по формуле $r = \frac{S}{p}$, где $S$ — площадь треугольника, а $p$ — его полупериметр.</p> <p>1. Вычислим полупериметр треугольника $ABC$ со сторонами $a=13$ см, $b=14$ см, $c=15$ см: $p = \frac{a+b+c}{2} = \frac{13+14+15}{2} = \frac{42}{2} = 21$ см.</p> <p>2. Вычислим площадь треугольника по формуле Герона: $S = \sqrt{p(p-a)(p-b)(p-c)} = \sqrt{21(21-13)(21-14)(21-15)}$ $S = \sqrt{21 \cdot 8 \cdot 7 \cdot 6} = \sqrt{(3 \cdot 7) \cdot (2^3) \cdot 7 \cdot (2 \cdot 3)} = \sqrt{2^4 \cdot 3^2 \cdot 7^2} = 2^2 \cdot 3 \cdot 7 = 84$ см².</p> <p>3. Теперь найдем радиус вписанной окружности: $r = \frac{S}{p} = \frac{84}{21} = 4$ см. Таким образом, $ON = 4$ см.</p> <p><strong>Найдем расстояние от точки $M$ до плоскости $ABC$.</strong></p> <p>Рассмотрим треугольник $MON$. Так как $MO$ — перпендикуляр к плоскости $ABC$, а $ON$ лежит в этой плоскости, то $MO \perp ON$. Следовательно, треугольник $MON$ — прямоугольный.</p> <p>В этом треугольнике:</p><ul> <li>$MN$ — гипотенуза, $MN = 5$ см (расстояние от $M$ до стороны $AB$).</li> <li>$ON$ — катет, $ON = r = 4$ см (радиус вписанной окружности).</li> <li>$MO$ — катет, искомое расстояние.</li> </ul><p>По теореме Пифагора: $MO^2 + ON^2 = MN^2$ $MO = \sqrt{MN^2 - ON^2} = \sqrt{5^2 - 4^2} = \sqrt{25 - 16} = \sqrt{9} = 3$ см.</p> <p>Ответ: 3 см.</p>" ] #original: array:6 [ "id" => 1434326 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1113} "task" => array:2 [ "refs" => "738534" "type" => "task" ] "text" => "<p>Пусть $MO$ — перпендикуляр, опущенный из точки $M$ на плоскость треугольника $ABC$. Тогда длина отрезка $MO$ — это искомое расстояние от точки $M$ до плоскости $ABC$.</p> <p>По условию, точка $M$ равноудалена от прямых $AB$, $BC$ и $AC$. Расстояние от точки до прямой — это длина перпендикуляра. Проведем из точки $M$ перпендикуляры $MN$, $MP$ и $MQ$ к прямым $AB$, $BC$ и $AC$ соответственно. По условию, $MN = MP = MQ = 5$ см. Эти отрезки являются наклонными к плоскости $ABC$.</p> <p>Отрезки $ON$, $OP$ и $OQ$ являются проекциями этих наклонных на плоскость $ABC$. Так как наклонные, проведенные из одной точки, равны, то равны и их проекции. Следовательно, $ON = OP = OQ$.</p> <p>Поскольку точка $O$ (проекция точки $M$) лежит внутри треугольника и равноудалена от его сторон ($ON \perp AB$, $OP \perp BC$, $OQ \perp AC$), то $O$ является центром окружности, вписанной в треугольник $ABC$. Длина отрезков $ON$, $OP$ и $OQ$ равна радиусу этой окружности ($r$).</p> <p><strong>Найдем радиус вписанной окружности $r$.</strong></p> <p>Радиус вписанной окружности можно найти по формуле $r = \frac{S}{p}$, где $S$ — площадь треугольника, а $p$ — его полупериметр.</p> <p>1. Вычислим полупериметр треугольника $ABC$ со сторонами $a=13$ см, $b=14$ см, $c=15$ см: $p = \frac{a+b+c}{2} = \frac{13+14+15}{2} = \frac{42}{2} = 21$ см.</p> <p>2. Вычислим площадь треугольника по формуле Герона: $S = \sqrt{p(p-a)(p-b)(p-c)} = \sqrt{21(21-13)(21-14)(21-15)}$ $S = \sqrt{21 \cdot 8 \cdot 7 \cdot 6} = \sqrt{(3 \cdot 7) \cdot (2^3) \cdot 7 \cdot (2 \cdot 3)} = \sqrt{2^4 \cdot 3^2 \cdot 7^2} = 2^2 \cdot 3 \cdot 7 = 84$ см².</p> <p>3. Теперь найдем радиус вписанной окружности: $r = \frac{S}{p} = \frac{84}{21} = 4$ см. Таким образом, $ON = 4$ см.</p> <p><strong>Найдем расстояние от точки $M$ до плоскости $ABC$.</strong></p> <p>Рассмотрим треугольник $MON$. Так как $MO$ — перпендикуляр к плоскости $ABC$, а $ON$ лежит в этой плоскости, то $MO \perp ON$. Следовательно, треугольник $MON$ — прямоугольный.</p> <p>В этом треугольнике:</p><ul> <li>$MN$ — гипотенуза, $MN = 5$ см (расстояние от $M$ до стороны $AB$).</li> <li>$ON$ — катет, $ON = r = 4$ см (радиус вписанной окружности).</li> <li>$MO$ — катет, искомое расстояние.</li> </ul><p>По теореме Пифагора: $MO^2 + ON^2 = MN^2$ $MO = \sqrt{MN^2 - ON^2} = \sqrt{5^2 - 4^2} = \sqrt{25 - 16} = \sqrt{9} = 3$ см.</p> <p>Ответ: 3 см.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "738535" "type" => "task" ] "previous" => array:2 [ "refs" => "738533" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1085 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1082 #items: array:1 [ 0 => App\Models\BookPage {#1040 #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" => 971697 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "50" "field_url" => "/10-klass/geometrija/glazkov-tetrad/page-50" "field_display_title" => "50" "field_folder" => "1" "field_image_name" => "50" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1043 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1111 #items: array:1 [ 0 => App\Models\Book {#1048} ] #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 {#1112 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1123 #items: array:1 [ 0 => App\Models\Element {#1122 #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" => 1147813 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1124 #items: array:1 [ 0 => App\Models\Edition {#1089} ] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "971697" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "50-1.jpg" "alt" => null "width" => "1962" "height" => 2806 "path" => "/media/geometrija_10/glazkov-rt/0-1/50-1.webp?ts=1745219749" ] ] ] #original: array:6 [ "id" => 1147813 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1124} "book_page" => array:2 [ "refs" => "971697" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "50-1.jpg" "alt" => null "width" => "1962" "height" => 2806 "path" => "/media/geometrija_10/glazkov-rt/0-1/50-1.webp?ts=1745219749" ] ] ] #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" => "971698" "type" => "book_page" ] "previous" => array:2 [ "refs" => "971696" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1189 #items: array:2 [ 0 => App\Models\Task {#1197 #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" => 738533 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "49" "field_page_end" => "50" "field_url" => "/10-klass/geometrija/glazkov-tetrad/00-62" "field_display_title" => "62" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1198 #items: array:1 [ 0 => App\Models\Term {#1036} ] #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 {#1199 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1200 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1201 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1204 #items: array:3 [ 0 => App\Models\Element {#1213 #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" => 937141 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1214 #items: array:1 [ 0 => App\Models\Edition {#1089} ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "738533" "type" => "task" ] "text" => "<p><strong>62</strong> Докажите, что любая точка прямой, которая проходит через центр окружности, описанной около многоугольника, и перпендикулярна к плоскости многоугольника, равноудалена от вершин этого многоугольника (задача 200 учебника).</p><p><strong>Доказательство.</strong> Пусть прямая $a$ проходит через центр $O$ окружности, описанной около многоугольника $A_1A_2...A_n$, и перпендикулярна к плоскости $\alpha$ этого многоугольника. Ясно, что точка $O$ равноудалена от вершин многоугольника, так как является центром описанной около него окружности: $OA_1 = OA_2 = ... = OA_n$. Пусть $M$ — произвольная точка прямой $a$, отличная от точки $O$. Тогда</p><p>$MO$ — перпендикуляр, $MA_1$, $MA_2$, ..., $MA_n$ —</p><p>проведенные из точки — к —, а $OA_1$,</p><p>$OA_2$, ..., $OA_n$ — проекции наклонных на —. Так</p><p>как проекции равны, то равны и —. Т. е.</p><p>—. Таким образом, любая точка прямой $a$</p><p>равноудалена от —.</p>" "img" => array:2 [ 0 => array:5 [ "name" => "62-1.jpg" "alt" => null "width" => "1646" "height" => 1054 "path" => "/media/geometrija_10/glazkov-rt/0-00/62-1.webp?ts=1739375737" ] 1 => array:5 [ "name" => "62-2.jpg" "alt" => null "width" => "1646" "height" => 493 "path" => "/media/geometrija_10/glazkov-rt/0-00/62-2.webp?ts=1739375737" ] ] ] #original: array:7 [ "id" => 937141 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1214} "task" => array:2 [ "refs" => "738533" "type" => "task" ] "text" => "<p><strong>62</strong> Докажите, что любая точка прямой, которая проходит через центр окружности, описанной около многоугольника, и перпендикулярна к плоскости многоугольника, равноудалена от вершин этого многоугольника (задача 200 учебника).</p><p><strong>Доказательство.</strong> Пусть прямая $a$ проходит через центр $O$ окружности, описанной около многоугольника $A_1A_2...A_n$, и перпендикулярна к плоскости $\alpha$ этого многоугольника. Ясно, что точка $O$ равноудалена от вершин многоугольника, так как является центром описанной около него окружности: $OA_1 = OA_2 = ... = OA_n$. Пусть $M$ — произвольная точка прямой $a$, отличная от точки $O$. Тогда</p><p>$MO$ — перпендикуляр, $MA_1$, $MA_2$, ..., $MA_n$ —</p><p>проведенные из точки — к —, а $OA_1$,</p><p>$OA_2$, ..., $OA_n$ — проекции наклонных на —. Так</p><p>как проекции равны, то равны и —. Т. е.</p><p>—. Таким образом, любая точка прямой $a$</p><p>равноудалена от —.</p>" "img" => array:2 [ 0 => array:5 [ "name" => "62-1.jpg" "alt" => null "width" => "1646" "height" => 1054 "path" => "/media/geometrija_10/glazkov-rt/0-00/62-1.webp?ts=1739375737" ] 1 => array:5 [ "name" => "62-2.jpg" "alt" => null "width" => "1646" "height" => 493 "path" => "/media/geometrija_10/glazkov-rt/0-00/62-2.webp?ts=1739375737" ] ] ] #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 {#1215 #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" => 937251 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1216 #items: array:1 [ 0 => App\Models\Edition {#1096} ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "738533" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "62-1.png" "alt" => null "width" => "694" "height" => 2126 "path" => "/media/geometrija_10/glazkov-rt/1-00/62-1.webp?ts=1739375872" ] ] ] #original: array:6 [ "id" => 937251 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1216} "task" => array:2 [ "refs" => "738533" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "62-1.png" "alt" => null "width" => "694" "height" => 2126 "path" => "/media/geometrija_10/glazkov-rt/1-00/62-1.webp?ts=1739375872" ] ] ] #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 {#1217 #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" => 1434325 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1218 #items: array:1 [ 0 => App\Models\Edition {#1104} ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "738533" "type" => "task" ] "text" => "<p><strong>Доказательство</strong></p><p>Пусть прямая $a$ проходит через центр $O$ окружности, описанной около многоугольника $A_1A_2...A_n$, и перпендикулярна к плоскости $\alpha$ этого многоугольника. Ясно, что точка $O$ равноудалена от вершин многоугольника, так как является центром описанной около него окружности: $OA_1 = OA_2 = ... = OA_n$.</p><p>Пусть $M$ — произвольная точка прямой $a$, отличная от точки $O$. Тогда $MO$ — перпендикуляр, $MA_1, MA_2, ..., MA_n$ — <strong>наклонные</strong>, проведенные из точки <strong>$M$</strong> к <strong>плоскости $\alpha$</strong>, а $OA_1, OA_2, ..., OA_n$ — проекции наклонных на <strong>плоскость $\alpha$</strong>. Так как проекции равны ($OA_1 = OA_2 = ... = OA_n$), то равны и <strong>сами наклонные</strong>, т. е. <strong>$MA_1 = MA_2 = ... = MA_n$</strong>. Таким образом, любая точка прямой $a$ равноудалена от <strong>вершин этого многоугольника</strong>.</p><p>Это можно также показать с помощью теоремы Пифагора. Для любого $i$ от 1 до $n$, треугольник $\triangle MOA_i$ является прямоугольным, так как $MO \perp \alpha$ и $OA_i \subset \alpha$. Катет $MO$ является общим для всех таких треугольников, а катеты $OA_i$ равны как радиусы описанной окружности. Следовательно, гипотенузы $MA_i$ также равны:</p><p>$MA_i = \sqrt{MO^2 + OA_i^2}$</p><p>Поскольку $MO$ и $OA_i$ (как радиус) имеют постоянную длину для всех $i$, то и $MA_i$ равны между собой. Утверждение доказано.</p><p><strong>Ответ:</strong> Утверждение, представленное в задаче, доказано. Любая точка прямой, которая проходит через центр окружности, описанной около многоугольника, и перпендикулярна к его плоскости, действительно равноудалена от вершин этого многоугольника.</p>" ] #original: array:6 [ "id" => 1434325 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1218} "task" => array:2 [ "refs" => "738533" "type" => "task" ] "text" => "<p><strong>Доказательство</strong></p><p>Пусть прямая $a$ проходит через центр $O$ окружности, описанной около многоугольника $A_1A_2...A_n$, и перпендикулярна к плоскости $\alpha$ этого многоугольника. Ясно, что точка $O$ равноудалена от вершин многоугольника, так как является центром описанной около него окружности: $OA_1 = OA_2 = ... = OA_n$.</p><p>Пусть $M$ — произвольная точка прямой $a$, отличная от точки $O$. Тогда $MO$ — перпендикуляр, $MA_1, MA_2, ..., MA_n$ — <strong>наклонные</strong>, проведенные из точки <strong>$M$</strong> к <strong>плоскости $\alpha$</strong>, а $OA_1, OA_2, ..., OA_n$ — проекции наклонных на <strong>плоскость $\alpha$</strong>. Так как проекции равны ($OA_1 = OA_2 = ... = OA_n$), то равны и <strong>сами наклонные</strong>, т. е. <strong>$MA_1 = MA_2 = ... = MA_n$</strong>. Таким образом, любая точка прямой $a$ равноудалена от <strong>вершин этого многоугольника</strong>.</p><p>Это можно также показать с помощью теоремы Пифагора. Для любого $i$ от 1 до $n$, треугольник $\triangle MOA_i$ является прямоугольным, так как $MO \perp \alpha$ и $OA_i \subset \alpha$. Катет $MO$ является общим для всех таких треугольников, а катеты $OA_i$ равны как радиусы описанной окружности. Следовательно, гипотенузы $MA_i$ также равны:</p><p>$MA_i = \sqrt{MO^2 + OA_i^2}$</p><p>Поскольку $MO$ и $OA_i$ (как радиус) имеют постоянную длину для всех $i$, то и $MA_i$ равны между собой. Утверждение доказано.</p><p><strong>Ответ:</strong> Утверждение, представленное в задаче, доказано. Любая точка прямой, которая проходит через центр окружности, описанной около многоугольника, и перпендикулярна к его плоскости, действительно равноудалена от вершин этого многоугольника.</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" => "738534" "type" => "task" ] "previous" => array:2 [ "refs" => "738532" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1210 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "971696" "type" => "book_page" ] ] #original: array:24 [ "id" => 738533 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "49" "field_page_end" => "50" "field_url" => 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#observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Task {#1203 #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" => 738534 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "50" "field_page_end" => null "field_url" => "/10-klass/geometrija/glazkov-tetrad/00-63" "field_display_title" => "63" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1202 #items: array:1 [ 0 => App\Models\Term {#1036} ] #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 {#1209 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1211 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1207 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1205 #items: array:3 [ 0 => App\Models\Element {#1088} 1 => App\Models\Element {#1095} 2 => App\Models\Element {#1103} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "738535" "type" => "task" ] "previous" => array:2 [ "refs" => "738533" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1208 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "971697" "type" => "book_page" ] ] #original: array:24 [ "id" => 738534 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "50" "field_page_end" => null "field_url" => "/10-klass/geometrija/glazkov-tetrad/00-63" "field_display_title" => "63" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1202} "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 {#1209} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1211} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1207} "content" => Illuminate\Database\Eloquent\Collection {#1205} "next" => array:2 [ "refs" => "738535" "type" => "task" ] "previous" => array:2 [ "refs" => "738533" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1208} "page" 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№63 (с. 50)
Условие. №63 (с. 50)
скриншот условия
63 В треугольнике $ABC$ $AB = 13$ см, $BC = 14$ см, $AC = 15$ см. Точка $M$ удалена от прямых $AB$, $BC$ и $AC$ на 5 см. Найдите расстояние от точки $M$ до плоскости $ABC$, если известно, что ее проекция на эту плоскость лежит внутри треугольника.
Решение.
Пусть $MO$ — перпендикуляр к плоскости $ABC$, а $MN$, $MP$ и $MQ$ — перпендикуляры к прямым $AB$, $BC$ и $AC$. Требуется найти $MO$. По теореме, ______ имеем: $AB \perp ON$, $BC \perp$ ______ и $AC \perp$ ______. Итак, из точки $M$ проведены к плоскости $ABC$ перпендикуляр $MO$ и наклонные $MN$, $MP$ и $MQ$. По условию расстояния от точки $M$ до прямых $AB$, $BC$ и $AC$ равны, т. е. равны наклонные ______ , и ______ . Следовательно, потому равны и их проекции на эту плоскость: ______ . Таким образом, точка $O$ лежит внутри треугольника $ABC$ и равноудалена от ______ , поэтому она является ______ .
Радиус $ON$ этой окружности найдем, используя формулу $S = pr$, где $S$ — площадь треугольника $ABC$, $p$ — его ______ , $r = ON$. По формуле Герона $S = $ ______ см$^2$, следовательно, $r = \frac{S}{p} = $ ______ (см) = ______ см.
Итак, $NO = $ ______ см.
Треугольник $MON$ ______ , поскольку $MO \perp ABC$, и потому $MO \perp ON$. Так как $MN = $ ______ , $ON = $ ______ , то из треугольника $MON$ находим: $MO = $ ______ см.
Ответ.
______ см.
Решение. №63 (с. 50)
Решение 2. №63 (с. 50)
Пусть $MO$ — перпендикуляр, опущенный из точки $M$ на плоскость треугольника $ABC$. Тогда длина отрезка $MO$ — это искомое расстояние от точки $M$ до плоскости $ABC$.
По условию, точка $M$ равноудалена от прямых $AB$, $BC$ и $AC$. Расстояние от точки до прямой — это длина перпендикуляра. Проведем из точки $M$ перпендикуляры $MN$, $MP$ и $MQ$ к прямым $AB$, $BC$ и $AC$ соответственно. По условию, $MN = MP = MQ = 5$ см. Эти отрезки являются наклонными к плоскости $ABC$.
Отрезки $ON$, $OP$ и $OQ$ являются проекциями этих наклонных на плоскость $ABC$. Так как наклонные, проведенные из одной точки, равны, то равны и их проекции. Следовательно, $ON = OP = OQ$.
Поскольку точка $O$ (проекция точки $M$) лежит внутри треугольника и равноудалена от его сторон ($ON \perp AB$, $OP \perp BC$, $OQ \perp AC$), то $O$ является центром окружности, вписанной в треугольник $ABC$. Длина отрезков $ON$, $OP$ и $OQ$ равна радиусу этой окружности ($r$).
Найдем радиус вписанной окружности $r$.
Радиус вписанной окружности можно найти по формуле $r = \frac{S}{p}$, где $S$ — площадь треугольника, а $p$ — его полупериметр.
1. Вычислим полупериметр треугольника $ABC$ со сторонами $a=13$ см, $b=14$ см, $c=15$ см: $p = \frac{a+b+c}{2} = \frac{13+14+15}{2} = \frac{42}{2} = 21$ см.
2. Вычислим площадь треугольника по формуле Герона: $S = \sqrt{p(p-a)(p-b)(p-c)} = \sqrt{21(21-13)(21-14)(21-15)}$ $S = \sqrt{21 \cdot 8 \cdot 7 \cdot 6} = \sqrt{(3 \cdot 7) \cdot (2^3) \cdot 7 \cdot (2 \cdot 3)} = \sqrt{2^4 \cdot 3^2 \cdot 7^2} = 2^2 \cdot 3 \cdot 7 = 84$ см².
3. Теперь найдем радиус вписанной окружности: $r = \frac{S}{p} = \frac{84}{21} = 4$ см. Таким образом, $ON = 4$ см.
Найдем расстояние от точки $M$ до плоскости $ABC$.
Рассмотрим треугольник $MON$. Так как $MO$ — перпендикуляр к плоскости $ABC$, а $ON$ лежит в этой плоскости, то $MO \perp ON$. Следовательно, треугольник $MON$ — прямоугольный.
В этом треугольнике:
- $MN$ — гипотенуза, $MN = 5$ см (расстояние от $M$ до стороны $AB$).
- $ON$ — катет, $ON = r = 4$ см (радиус вписанной окружности).
- $MO$ — катет, искомое расстояние.
По теореме Пифагора: $MO^2 + ON^2 = MN^2$ $MO = \sqrt{MN^2 - ON^2} = \sqrt{5^2 - 4^2} = \sqrt{25 - 16} = \sqrt{9} = 3$ см.
Ответ: 3 см.
Другие задания:
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