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Номер 50, страница 42 - 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
2.2. Перпендикуляр и наклонные. Угол между прямой и плоскостью - номер 50, страница 42.
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" => 738521 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "42" "field_page_end" => "43" "field_url" => "/10-klass/geometrija/glazkov-tetrad/00-50" "field_display_title" => "50" "field_outside_task" => "0" "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" => 738458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "2.2. Перпендикуляр и наклонные. Угол между прямой и плоскостью" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1047 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "42" "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 {#1057 #items: array:1 [ 0 => App\Models\Book {#1056 #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 {#1061 #items: array:1 [ 0 => App\Models\Term {#1058 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array: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 {#1063 #items: array:1 [ 0 => App\Models\Term {#1060 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array: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 {#1065 #items: array:1 [ 0 => App\Models\Term {#1062 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array: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 {#1066 #items: array:3 [ 0 => App\Models\Term {#1076 #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 {#1077 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array: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 {#1075 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array: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 {#1072 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1073 #items: array:1 [ 0 => App\Models\Term {#1059 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array: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 {#1067 #items: array:1 [ 0 => App\Models\Term {#1071 #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 {#1068 #items: array:1 [ 0 => App\Models\Term {#1069 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 15 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Москва" "field_cases" => array:6 [ "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 {#1079 #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: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 {#1078 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1082 #items: array:1 [ 0 => App\Models\Term {#1064 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array: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 {#1084 #items: array:1 [ 0 => App\Models\Term {#1081 #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 {#1086 #items: array:1 [ 0 => App\Models\Term {#1083 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 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 {#1085 #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 {#1080 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1087 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1088 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/geometrija/glazkov-tetrad" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1098 #items: array:1 [ 0 => App\Models\Term {#1097 #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 {#1061} "field_class" => Illuminate\Database\Eloquent\Collection {#1063} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1065} "field_author" => Illuminate\Database\Eloquent\Collection {#1066} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1072} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1073} "field_country" => Illuminate\Database\Eloquent\Collection {#1067} "field_city" => Illuminate\Database\Eloquent\Collection {#1068} "field_series" => Illuminate\Database\Eloquent\Collection {#1079} "field_umk" => Illuminate\Database\Eloquent\Collection {#1078} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1082} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1084} "field_publication_number" => "4" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1086} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1085} "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 {#1080} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1087} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1088} "field_url" => "/10-klass/geometrija/glazkov-tetrad" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1098} "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 {#1049 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 738458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "2.2. Перпендикуляр и наклонные. Угол между прямой и плоскостью" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1047} "field_page_start" => "42" "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 {#1057} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1049} ] #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 {#1052 #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" => 937129 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1100 #items: array:1 [ 0 => App\Models\Edition {#1090 #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 {#1089 #items: array:1 [ 0 => App\Models\Term {#1091 #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 {#1094 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1092 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1095 #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 {#1089} "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 {#1094} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1092} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1095} "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" => "738521" "type" => "task" ] "text" => "<p><strong>50</strong> Концы отрезка отстоят от плоскости $\alpha$ на расстояниях 1 см и 4 см. Найдите расстояние от середины отрезка до плоскости $\alpha$ (задача 142 учебника).</p><p>Решение. Рассмотрим два случая:</p><p>1) концы отрезка находятся по одну сторону от плоскости $\alpha$;</p><p>2) концы отрезка находятся по разные стороны от плоскости $\alpha$.</p><p><strong>1)</strong> Пусть отрезок $AB$ расположен по одну сторону от плоскости $\alpha$</p><p>1) Пусть отрезок $AB$ расположен по одну сторону от плоскости $\alpha$ (см. рис. а), $AA_1 \perp \alpha$, $AA_1 = 1$ см, $BB_1 \perp \alpha$, $BB_1 = 4$ см. Так как $AA_1 \perp \alpha$ и $BB_1 \perp \alpha$, то $AA_1 \parallel \text{—}$, и поэтому четырехугольник $A_1ABB_1$ — —. Проведем в ней среднюю линию $PP_1$, тогда $PP_1 \parallel \text{—}$, и так как $AA_1 \perp \alpha$, то и $PP_1 \perp \text{—}$. Следовательно, длина отрезка $PP_1$ и есть искомое расстояние от середины отрезка $AB$ до плоскости $\alpha$, $PP_1 = \frac{1}{2} \text{\_\_\_\_\_\_\_} = \text{\_\_\_\_\_\_\_}$ см.</p><p><strong>2)</strong> Пусть концы отрезка $AB$ расположены по разные стороны от плоскости $\alpha$</p><p>2) Пусть концы отрезка $AB$ расположены по разные стороны от плоскости $\alpha$ (см. рис. б) и пусть $AA_1$ и $BB_1$ — перпендикуляры к плоскости $\alpha$, $AA_1 = 1$ см, $BB_1 = 4$ см. Так как $AA_1 \perp \alpha$ и $BB_1 \perp \alpha$, то $AA_1 \parallel \text{—}$, и прямые $AA_1$, $BB_1$, $A_1B_1$ лежат в одной —. Проведем через точку $P$ — середину отрезка $AB$ — прямую, параллельную $B_1B$. Тогда по — точки $P_1$ и $F$ пересечения этой прямой с прямыми $A_1B_1$ и $A_1B$ будут серединами отрезков — и —, а отрезки $P_1F$ и $PF$ — средними — . $P_1P = P_1F = \frac{1}{2} \text{\_\_\_\_\_\_\_} = \text{\_\_\_\_\_\_\_}$ см.</p><p>Ответ. — см или — см.</p>" "img" => array:2 [ 0 => array:5 [ "name" => "50-1.jpg" "alt" => null "width" => "1646" "height" => 1077 "path" => "/media/geometrija_10/glazkov-rt/0-00/50-1.webp?ts=1739375720" ] 1 => array:5 [ "name" => "50-2.jpg" "alt" => null "width" => "1646" "height" => 1605 "path" => "/media/geometrija_10/glazkov-rt/0-00/50-2.webp?ts=1739375720" ] ] ] #original: array:7 [ "id" => 937129 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1100} "task" => array:2 [ "refs" => "738521" "type" => "task" ] "text" => "<p><strong>50</strong> Концы отрезка отстоят от плоскости $\alpha$ на расстояниях 1 см и 4 см. Найдите расстояние от середины отрезка до плоскости $\alpha$ (задача 142 учебника).</p><p>Решение. Рассмотрим два случая:</p><p>1) концы отрезка находятся по одну сторону от плоскости $\alpha$;</p><p>2) концы отрезка находятся по разные стороны от плоскости $\alpha$.</p><p><strong>1)</strong> Пусть отрезок $AB$ расположен по одну сторону от плоскости $\alpha$</p><p>1) Пусть отрезок $AB$ расположен по одну сторону от плоскости $\alpha$ (см. рис. а), $AA_1 \perp \alpha$, $AA_1 = 1$ см, $BB_1 \perp \alpha$, $BB_1 = 4$ см. Так как $AA_1 \perp \alpha$ и $BB_1 \perp \alpha$, то $AA_1 \parallel \text{—}$, и поэтому четырехугольник $A_1ABB_1$ — —. Проведем в ней среднюю линию $PP_1$, тогда $PP_1 \parallel \text{—}$, и так как $AA_1 \perp \alpha$, то и $PP_1 \perp \text{—}$. Следовательно, длина отрезка $PP_1$ и есть искомое расстояние от середины отрезка $AB$ до плоскости $\alpha$, $PP_1 = \frac{1}{2} \text{\_\_\_\_\_\_\_} = \text{\_\_\_\_\_\_\_}$ см.</p><p><strong>2)</strong> Пусть концы отрезка $AB$ расположены по разные стороны от плоскости $\alpha$</p><p>2) Пусть концы отрезка $AB$ расположены по разные стороны от плоскости $\alpha$ (см. рис. б) и пусть $AA_1$ и $BB_1$ — перпендикуляры к плоскости $\alpha$, $AA_1 = 1$ см, $BB_1 = 4$ см. Так как $AA_1 \perp \alpha$ и $BB_1 \perp \alpha$, то $AA_1 \parallel \text{—}$, и прямые $AA_1$, $BB_1$, $A_1B_1$ лежат в одной —. Проведем через точку $P$ — середину отрезка $AB$ — прямую, параллельную $B_1B$. Тогда по — точки $P_1$ и $F$ пересечения этой прямой с прямыми $A_1B_1$ и $A_1B$ будут серединами отрезков — и —, а отрезки $P_1F$ и $PF$ — средними — . $P_1P = P_1F = \frac{1}{2} \text{\_\_\_\_\_\_\_} = \text{\_\_\_\_\_\_\_}$ см.</p><p>Ответ. — см или — см.</p>" "img" => array:2 [ 0 => array:5 [ "name" => "50-1.jpg" "alt" => null "width" => "1646" "height" => 1077 "path" => "/media/geometrija_10/glazkov-rt/0-00/50-1.webp?ts=1739375720" ] 1 => array:5 [ "name" => "50-2.jpg" "alt" => null "width" => "1646" "height" => 1605 "path" => "/media/geometrija_10/glazkov-rt/0-00/50-2.webp?ts=1739375720" ] ] ] #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 {#1093 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 937239 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108 #items: array:1 [ 0 => App\Models\Edition {#1099 #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 {#1102 #items: array:1 [ 0 => App\Models\Term {#1101 #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 {#1103 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1104 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1105 #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 {#1102} "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 {#1103} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1104} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1105} "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" => "738521" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "50-1.png" "alt" => null "width" => "694" "height" => 2234 "path" => "/media/geometrija_10/glazkov-rt/1-00/50-1.webp?ts=1739375853" ] ] ] #original: array:6 [ "id" => 937239 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108} "task" => array:2 [ "refs" => "738521" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "50-1.png" "alt" => null "width" => "694" "height" => 2234 "path" => "/media/geometrija_10/glazkov-rt/1-00/50-1.webp?ts=1739375853" ] ] ] #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 {#1106 #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" => 1434313 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116 #items: array:1 [ 0 => App\Models\Edition {#1107 #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 {#1110 #items: array:1 [ 0 => App\Models\Term {#1109 #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 {#1111 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1112 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1113 #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 {#1110} "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 {#1111} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1112} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1113} "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" => "738521" "type" => "task" ] "text" => "<p><strong>1)</strong> Пусть отрезок $AB$ расположен по одну сторону от плоскости $\alpha$ (см. рис. а), $AA_1 \perp \alpha$, $AA_1 = 1$ см, $BB_1 \perp \alpha$, $BB_1 = 4$ см. Так как $AA_1 \perp \alpha$ и $BB_1 \perp \alpha$, то $AA_1 \parallel BB_1$, и поэтому четырехугольник $A_1ABB_1$ — трапеция. Проведем в ней среднюю линию $PP_1$, где $P$ — середина $AB$, а $P_1$ — ее проекция на плоскость $\alpha$. Тогда $PP_1$ будет параллельна основаниям трапеции $AA_1$ и $BB_1$. Так как $AA_1 \perp \alpha$, то и $PP_1 \perp \alpha$. Следовательно, длина отрезка $PP_1$ и есть искомое расстояние от середины отрезка $AB$ до плоскости $\alpha$. Длина средней линии трапеции равна полусумме ее оснований:$PP_1 = \frac{1}{2}(AA_1 + BB_1) = \frac{1+4}{2} = 2,5$ см.Ответ: 2,5 см.</p><p><strong>2)</strong> Пусть концы отрезка $AB$ расположены по разные стороны от плоскости $\alpha$ (см. рис. б) и пусть $AA_1$ и $BB_1$ — перпендикуляры к плоскости $\alpha$, $AA_1 = 1$ см, $BB_1 = 4$ см. Так как $AA_1 \perp \alpha$ и $BB_1 \perp \alpha$, то $AA_1 \parallel BB_1$. Прямые $AA_1$ и $BB_1$ задают плоскость, в которой также лежат точки $A_1$ и $B_1$. Проведем через точку $P$ — середину отрезка $AB$ — прямую, перпендикулярную плоскости $\alpha$. Пусть $P_1$ — точка ее пересечения с плоскостью $\alpha$, а $F$ — точка ее пересечения с прямой $A_1B$. Так как прямая $PP_1F$ параллельна $AA_1$ и $BB_1$, то по теореме Фалеса точка $F$ является серединой отрезка $A_1B$, а точка $P_1$ является серединой отрезка $A_1B_1$. Таким образом, отрезок $PF$ является средней линией в треугольнике $A_1AB$, а отрезок $P_1F$ — средней линией в треугольнике $A_1BB_1$. Их длины равны:$PF = \frac{1}{2}AA_1 = \frac{1}{2} \cdot 1 = 0,5$ см.$P_1F = \frac{1}{2}BB_1 = \frac{1}{2} \cdot 4 = 2$ см.Искомое расстояние $PP_1$ равно разности длин отрезков $P_1F$ и $PF$:$PP_1 = P_1F - PF = 2 - 0,5 = 1,5$ см.Ответ: 1,5 см.</p><p>Таким образом, в зависимости от расположения отрезка относительно плоскости, возможно два ответа.</p><p>Ответ: 2,5 см или 1,5 см.</p>" ] #original: array:6 [ "id" => 1434313 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116} "task" => array:2 [ "refs" => "738521" "type" => "task" ] "text" => "<p><strong>1)</strong> Пусть отрезок $AB$ расположен по одну сторону от плоскости $\alpha$ (см. рис. а), $AA_1 \perp \alpha$, $AA_1 = 1$ см, $BB_1 \perp \alpha$, $BB_1 = 4$ см. Так как $AA_1 \perp \alpha$ и $BB_1 \perp \alpha$, то $AA_1 \parallel BB_1$, и поэтому четырехугольник $A_1ABB_1$ — трапеция. Проведем в ней среднюю линию $PP_1$, где $P$ — середина $AB$, а $P_1$ — ее проекция на плоскость $\alpha$. Тогда $PP_1$ будет параллельна основаниям трапеции $AA_1$ и $BB_1$. Так как $AA_1 \perp \alpha$, то и $PP_1 \perp \alpha$. Следовательно, длина отрезка $PP_1$ и есть искомое расстояние от середины отрезка $AB$ до плоскости $\alpha$. Длина средней линии трапеции равна полусумме ее оснований:$PP_1 = \frac{1}{2}(AA_1 + BB_1) = \frac{1+4}{2} = 2,5$ см.Ответ: 2,5 см.</p><p><strong>2)</strong> Пусть концы отрезка $AB$ расположены по разные стороны от плоскости $\alpha$ (см. рис. б) и пусть $AA_1$ и $BB_1$ — перпендикуляры к плоскости $\alpha$, $AA_1 = 1$ см, $BB_1 = 4$ см. Так как $AA_1 \perp \alpha$ и $BB_1 \perp \alpha$, то $AA_1 \parallel BB_1$. Прямые $AA_1$ и $BB_1$ задают плоскость, в которой также лежат точки $A_1$ и $B_1$. Проведем через точку $P$ — середину отрезка $AB$ — прямую, перпендикулярную плоскости $\alpha$. Пусть $P_1$ — точка ее пересечения с плоскостью $\alpha$, а $F$ — точка ее пересечения с прямой $A_1B$. Так как прямая $PP_1F$ параллельна $AA_1$ и $BB_1$, то по теореме Фалеса точка $F$ является серединой отрезка $A_1B$, а точка $P_1$ является серединой отрезка $A_1B_1$. Таким образом, отрезок $PF$ является средней линией в треугольнике $A_1AB$, а отрезок $P_1F$ — средней линией в треугольнике $A_1BB_1$. Их длины равны:$PF = \frac{1}{2}AA_1 = \frac{1}{2} \cdot 1 = 0,5$ см.$P_1F = \frac{1}{2}BB_1 = \frac{1}{2} \cdot 4 = 2$ см.Искомое расстояние $PP_1$ равно разности длин отрезков $P_1F$ и $PF$:$PP_1 = P_1F - PF = 2 - 0,5 = 1,5$ см.Ответ: 1,5 см.</p><p>Таким образом, в зависимости от расположения отрезка относительно плоскости, возможно два ответа.</p><p>Ответ: 2,5 см или 1,5 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" => "738522" "type" => "task" ] "previous" => array:2 [ "refs" => "738520" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1053 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1050 #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" => 971689 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "42" "field_url" => "/10-klass/geometrija/glazkov-tetrad/page-42" "field_display_title" => "42" "field_folder" => "1" "field_image_name" => "42" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1043 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1114 #items: array:1 [ 0 => App\Models\Book {#1056} ] #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 {#1115 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1126 #items: array:1 [ 0 => App\Models\Element {#1125 #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" => 1147805 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1127 #items: array:1 [ 0 => App\Models\Edition {#1090} ] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "971689" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "42-1.jpg" "alt" => null "width" => "1945" "height" => 2794 "path" => "/media/geometrija_10/glazkov-rt/0-1/42-1.webp?ts=1745219749" ] ] ] #original: array:6 [ "id" => 1147805 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1127} "book_page" => array:2 [ "refs" => "971689" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "42-1.jpg" "alt" => null "width" => "1945" "height" => 2794 "path" => "/media/geometrija_10/glazkov-rt/0-1/42-1.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" => "971690" "type" => "book_page" ] "previous" => array:2 [ "refs" => "971688" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1192 #items: array:2 [ 0 => App\Models\Task {#1200 #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" => 738520 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "42" "field_page_end" => null "field_url" => "/10-klass/geometrija/glazkov-tetrad/00-49" "field_display_title" => "49" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1201 #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 {#1202 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1203 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1204 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1207 #items: array:3 [ 0 => App\Models\Element {#1216 #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" => 937128 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1217 #items: array:1 [ 0 => App\Models\Edition {#1090} ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "738520" "type" => "task" ] "text" => "<p><strong>49</strong> Из точки M к плоскости $\alpha$ проведены перпендикуляр MO и две наклонные MA и MB, которые образуют со своими проекциями на эту плоскость $\angle MAO = 45^\circ$, $\angle MBO = 30^\circ$, угол между наклонными равен $90^\circ$.</p><p>Найдите расстояние между основаниями наклонных, если проекция наклонной MA равна $\sqrt{3}$ см.</p><p><strong>Решение.</strong></p><p>$MO \perp \alpha$, поэтому</p><p>$MO \perp$ _______ и $MO \perp$ _______. $\triangle AMO$ прямоугольный и равнобедренный:</p><p>$\angle O =$ _______, $\angle A = \angle$ _______ = _______, $AO$ = _______, следовательно,</p><p>$MO$ = _______, $AM$ = _______. $\triangle BMO$ прямоугольный: $\angle O =$ _______,</p><p>$\angle B$ = _______, $MO$ = _______, поэтому $MB = 2$ _______ = _______ см.</p><p>$\triangle AMB$ прямоугольный: $\angle M$ = _______, $AM$ = _______, $BM$ = _______</p><p>= _______, поэтому $AB$ = _______ = _______ = _______ см.</p><p><strong>Ответ.</strong></p><p>_______ см.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "49-1.jpg" "alt" => null "width" => "1646" "height" => 1250 "path" => "/media/geometrija_10/glazkov-rt/0-00/49-1.webp?ts=1739375720" ] ] ] #original: array:7 [ "id" => 937128 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1217} "task" => array:2 [ "refs" => "738520" "type" => "task" ] "text" => "<p><strong>49</strong> Из точки M к плоскости $\alpha$ проведены перпендикуляр MO и две наклонные MA и MB, которые образуют со своими проекциями на эту плоскость $\angle MAO = 45^\circ$, $\angle MBO = 30^\circ$, угол между наклонными равен $90^\circ$.</p><p>Найдите расстояние между основаниями наклонных, если проекция наклонной MA равна $\sqrt{3}$ см.</p><p><strong>Решение.</strong></p><p>$MO \perp \alpha$, поэтому</p><p>$MO \perp$ _______ и $MO \perp$ _______. $\triangle AMO$ прямоугольный и равнобедренный:</p><p>$\angle O =$ _______, $\angle A = \angle$ _______ = _______, $AO$ = _______, следовательно,</p><p>$MO$ = _______, $AM$ = _______. $\triangle BMO$ прямоугольный: $\angle O =$ _______,</p><p>$\angle B$ = _______, $MO$ = _______, поэтому $MB = 2$ _______ = _______ см.</p><p>$\triangle AMB$ прямоугольный: $\angle M$ = _______, $AM$ = _______, $BM$ = _______</p><p>= _______, поэтому $AB$ = _______ = _______ = _______ см.</p><p><strong>Ответ.</strong></p><p>_______ см.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "49-1.jpg" "alt" => null "width" => "1646" "height" => 1250 "path" => "/media/geometrija_10/glazkov-rt/0-00/49-1.webp?ts=1739375720" ] ] ] #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 {#1218 #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" => 937238 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1219 #items: array:1 [ 0 => App\Models\Edition {#1099} ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "738520" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "49-1.png" "alt" => null "width" => "694" "height" => 2162 "path" => "/media/geometrija_10/glazkov-rt/1-00/49-1.webp?ts=1739375853" ] ] ] #original: array:6 [ "id" => 937238 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1219} "task" => array:2 [ "refs" => "738520" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "49-1.png" "alt" => null "width" => "694" "height" => 2162 "path" => "/media/geometrija_10/glazkov-rt/1-00/49-1.webp?ts=1739375853" ] ] ] #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 {#1220 #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" => 1434312 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1221 #items: array:1 [ 0 => App\Models\Edition {#1107} ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "738520" "type" => "task" ] "text" => "<p> <strong>Решение.</strong></p><p> По условию, $MO$ — перпендикуляр к плоскости $\alpha$, значит, $MO$ перпендикулярен любой прямой, лежащей в этой плоскости и проходящей через точку $O$. Следовательно, $MO \perp OA$ и $MO \perp OB$. Это означает, что треугольники $\triangle AMO$ и $\triangle BMO$ являются прямоугольными с прямым углом при вершине $O$.</p><p> 1. Рассмотрим прямоугольный $\triangle AMO$. Угол между наклонной $MA$ и её проекцией $OA$ равен $\angle MAO = 45^\circ$. Так как $\triangle AMO$ прямоугольный, сумма его острых углов равна $90^\circ$, поэтому $\angle AMO = 90^\circ - 45^\circ = 45^\circ$. Поскольку углы при основании $AM$ равны ($\angle MAO = \angle AMO$), треугольник $\triangle AMO$ является равнобедренным, и его катеты равны: $MO = AO$. По условию, длина проекции $AO = \sqrt{3}$ см. Значит, $MO = \sqrt{3}$ см. Найдем длину наклонной $AM$ по теореме Пифагора: $AM^2 = MO^2 + AO^2 = (\sqrt{3})^2 + (\sqrt{3})^2 = 3 + 3 = 6$. $AM = \sqrt{6}$ см.</p><p> 2. Рассмотрим прямоугольный $\triangle BMO$. Угол между наклонной $MB$ и её проекцией $OB$ равен $\angle MBO = 30^\circ$. Мы уже нашли, что катет $MO = \sqrt{3}$ см. Найдем длину наклонной $MB$ (гипотенузы) через синус угла $\angle MBO$: $\sin(\angle MBO) = \frac{MO}{MB}$ $\sin(30^\circ) = \frac{\sqrt{3}}{MB}$ $\frac{1}{2} = \frac{\sqrt{3}}{MB}$ $MB = 2 \cdot \sqrt{3} = 2\sqrt{3}$ см.</p><p> 3. Рассмотрим $\triangle AMB$. По условию, угол между наклонными $\angle AMB = 90^\circ$, следовательно, $\triangle AMB$ — прямоугольный. Его катетами являются наклонные $AM$ и $MB$, а гипотенузой — отрезок $AB$, расстояние между основаниями наклонных. Применим теорему Пифагора для $\triangle AMB$: $AB^2 = AM^2 + MB^2 = (\sqrt{6})^2 + (2\sqrt{3})^2 = 6 + (4 \cdot 3) = 6 + 12 = 18$. $AB = \sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}$ см.</p><p> <strong>Ответ:</strong> $3\sqrt{2}$ см.</p>" ] #original: array:6 [ "id" => 1434312 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1221} "task" => array:2 [ "refs" => "738520" "type" => "task" ] "text" => "<p> <strong>Решение.</strong></p><p> По условию, $MO$ — перпендикуляр к плоскости $\alpha$, значит, $MO$ перпендикулярен любой прямой, лежащей в этой плоскости и проходящей через точку $O$. Следовательно, $MO \perp OA$ и $MO \perp OB$. Это означает, что треугольники $\triangle AMO$ и $\triangle BMO$ являются прямоугольными с прямым углом при вершине $O$.</p><p> 1. Рассмотрим прямоугольный $\triangle AMO$. Угол между наклонной $MA$ и её проекцией $OA$ равен $\angle MAO = 45^\circ$. Так как $\triangle AMO$ прямоугольный, сумма его острых углов равна $90^\circ$, поэтому $\angle AMO = 90^\circ - 45^\circ = 45^\circ$. Поскольку углы при основании $AM$ равны ($\angle MAO = \angle AMO$), треугольник $\triangle AMO$ является равнобедренным, и его катеты равны: $MO = AO$. По условию, длина проекции $AO = \sqrt{3}$ см. Значит, $MO = \sqrt{3}$ см. Найдем длину наклонной $AM$ по теореме Пифагора: $AM^2 = MO^2 + AO^2 = (\sqrt{3})^2 + (\sqrt{3})^2 = 3 + 3 = 6$. $AM = \sqrt{6}$ см.</p><p> 2. Рассмотрим прямоугольный $\triangle BMO$. Угол между наклонной $MB$ и её проекцией $OB$ равен $\angle MBO = 30^\circ$. Мы уже нашли, что катет $MO = \sqrt{3}$ см. Найдем длину наклонной $MB$ (гипотенузы) через синус угла $\angle MBO$: $\sin(\angle MBO) = \frac{MO}{MB}$ $\sin(30^\circ) = \frac{\sqrt{3}}{MB}$ $\frac{1}{2} = \frac{\sqrt{3}}{MB}$ $MB = 2 \cdot \sqrt{3} = 2\sqrt{3}$ см.</p><p> 3. Рассмотрим $\triangle AMB$. По условию, угол между наклонными $\angle AMB = 90^\circ$, следовательно, $\triangle AMB$ — прямоугольный. Его катетами являются наклонные $AM$ и $MB$, а гипотенузой — отрезок $AB$, расстояние между основаниями наклонных. Применим теорему Пифагора для $\triangle AMB$: $AB^2 = AM^2 + MB^2 = (\sqrt{6})^2 + (2\sqrt{3})^2 = 6 + (4 \cdot 3) = 6 + 12 = 18$. $AB = \sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}$ см.</p><p> <strong>Ответ:</strong> $3\sqrt{2}$ см.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "738521" "type" => "task" ] "previous" => array:2 [ "refs" => "738519" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1213 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "971689" "type" => "book_page" ] ] #original: array:24 [ "id" => 738520 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "42" "field_page_end" => null "field_url" => "/10-klass/geometrija/glazkov-tetrad/00-49" "field_display_title" => "49" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1201} "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 {#1202} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1203} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1204} "content" => Illuminate\Database\Eloquent\Collection {#1207} "next" => array:2 [ "refs" => "738521" "type" => "task" ] "previous" => array:2 [ "refs" => "738519" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1213} "page" => array:2 [ "refs" => "971689" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] 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=> null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1212 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1214 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1210 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1208 #items: array:3 [ 0 => App\Models\Element {#1052} 1 => App\Models\Element {#1093} 2 => App\Models\Element {#1106} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "738522" "type" => "task" ] "previous" => array:2 [ "refs" => "738520" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1211 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "page" => 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№50 (с. 42)
Условие. №50 (с. 42)
скриншот условия
50 Концы отрезка отстоят от плоскости $\alpha$ на расстояниях 1 см и 4 см. Найдите расстояние от середины отрезка до плоскости $\alpha$ (задача 142 учебника).
Решение. Рассмотрим два случая:
1) концы отрезка находятся по одну сторону от плоскости $\alpha$;
2) концы отрезка находятся по разные стороны от плоскости $\alpha$.
1) Пусть отрезок $AB$ расположен по одну сторону от плоскости $\alpha$
1) Пусть отрезок $AB$ расположен по одну сторону от плоскости $\alpha$ (см. рис. а), $AA_1 \perp \alpha$, $AA_1 = 1$ см, $BB_1 \perp \alpha$, $BB_1 = 4$ см. Так как $AA_1 \perp \alpha$ и $BB_1 \perp \alpha$, то $AA_1 \parallel \text{—}$, и поэтому четырехугольник $A_1ABB_1$ — —. Проведем в ней среднюю линию $PP_1$, тогда $PP_1 \parallel \text{—}$, и так как $AA_1 \perp \alpha$, то и $PP_1 \perp \text{—}$. Следовательно, длина отрезка $PP_1$ и есть искомое расстояние от середины отрезка $AB$ до плоскости $\alpha$, $PP_1 = \frac{1}{2} \text{\_\_\_\_\_\_\_} = \text{\_\_\_\_\_\_\_}$ см.
2) Пусть концы отрезка $AB$ расположены по разные стороны от плоскости $\alpha$
2) Пусть концы отрезка $AB$ расположены по разные стороны от плоскости $\alpha$ (см. рис. б) и пусть $AA_1$ и $BB_1$ — перпендикуляры к плоскости $\alpha$, $AA_1 = 1$ см, $BB_1 = 4$ см. Так как $AA_1 \perp \alpha$ и $BB_1 \perp \alpha$, то $AA_1 \parallel \text{—}$, и прямые $AA_1$, $BB_1$, $A_1B_1$ лежат в одной —. Проведем через точку $P$ — середину отрезка $AB$ — прямую, параллельную $B_1B$. Тогда по — точки $P_1$ и $F$ пересечения этой прямой с прямыми $A_1B_1$ и $A_1B$ будут серединами отрезков — и —, а отрезки $P_1F$ и $PF$ — средними — . $P_1P = P_1F = \frac{1}{2} \text{\_\_\_\_\_\_\_} = \text{\_\_\_\_\_\_\_}$ см.
Ответ. — см или — см.
Решение. №50 (с. 42)
Решение 2. №50 (с. 42)
1) Пусть отрезок $AB$ расположен по одну сторону от плоскости $\alpha$ (см. рис. а), $AA_1 \perp \alpha$, $AA_1 = 1$ см, $BB_1 \perp \alpha$, $BB_1 = 4$ см. Так как $AA_1 \perp \alpha$ и $BB_1 \perp \alpha$, то $AA_1 \parallel BB_1$, и поэтому четырехугольник $A_1ABB_1$ — трапеция. Проведем в ней среднюю линию $PP_1$, где $P$ — середина $AB$, а $P_1$ — ее проекция на плоскость $\alpha$. Тогда $PP_1$ будет параллельна основаниям трапеции $AA_1$ и $BB_1$. Так как $AA_1 \perp \alpha$, то и $PP_1 \perp \alpha$. Следовательно, длина отрезка $PP_1$ и есть искомое расстояние от середины отрезка $AB$ до плоскости $\alpha$. Длина средней линии трапеции равна полусумме ее оснований:$PP_1 = \frac{1}{2}(AA_1 + BB_1) = \frac{1+4}{2} = 2,5$ см.Ответ: 2,5 см.
2) Пусть концы отрезка $AB$ расположены по разные стороны от плоскости $\alpha$ (см. рис. б) и пусть $AA_1$ и $BB_1$ — перпендикуляры к плоскости $\alpha$, $AA_1 = 1$ см, $BB_1 = 4$ см. Так как $AA_1 \perp \alpha$ и $BB_1 \perp \alpha$, то $AA_1 \parallel BB_1$. Прямые $AA_1$ и $BB_1$ задают плоскость, в которой также лежат точки $A_1$ и $B_1$. Проведем через точку $P$ — середину отрезка $AB$ — прямую, перпендикулярную плоскости $\alpha$. Пусть $P_1$ — точка ее пересечения с плоскостью $\alpha$, а $F$ — точка ее пересечения с прямой $A_1B$. Так как прямая $PP_1F$ параллельна $AA_1$ и $BB_1$, то по теореме Фалеса точка $F$ является серединой отрезка $A_1B$, а точка $P_1$ является серединой отрезка $A_1B_1$. Таким образом, отрезок $PF$ является средней линией в треугольнике $A_1AB$, а отрезок $P_1F$ — средней линией в треугольнике $A_1BB_1$. Их длины равны:$PF = \frac{1}{2}AA_1 = \frac{1}{2} \cdot 1 = 0,5$ см.$P_1F = \frac{1}{2}BB_1 = \frac{1}{2} \cdot 4 = 2$ см.Искомое расстояние $PP_1$ равно разности длин отрезков $P_1F$ и $PF$:$PP_1 = P_1F - PF = 2 - 0,5 = 1,5$ см.Ответ: 1,5 см.
Таким образом, в зависимости от расположения отрезка относительно плоскости, возможно два ответа.
Ответ: 2,5 см или 1,5 см.
Другие задания:
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