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Номер 24, страница 19 - 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
1.4. Тетраэдр и параллелепипед - номер 24, страница 19.
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" => 738496 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "19" "field_page_end" => "20" "field_url" => "/10-klass/geometrija/glazkov-tetrad/00-24" "field_display_title" => "24" "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" => 738456 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "1.4. Тетраэдр и параллелепипед" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1047 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "19" "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" => 738456 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "1.4. Тетраэдр и параллелепипед" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1047} "field_page_start" => "19" "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" => 937105 "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" => "738496" "type" => "task" ] "text" => "<p><strong>24</strong> В тетраэдре $MNPQ$ ребро $MN = 3\sqrt{2}$ см, $NP = NQ = 7$ см, $PQ = 8$ см, $\angle MNP = \angle MNQ = 45^\circ$. Найдите площадь грани $MPQ$.</p><p>Решение.</p><p><strong>1)</strong> $\triangle MNP = \triangle MNQ$, так как ___________, поэтому $MP = \_$.</p><p><strong>2)</strong> По теореме косинусов для треугольника $MNP$ имеем: $MP^2 = \_$.</p><p>= ___________, откуда $MP = \_$ см.</p><p><strong>3)</strong> $\triangle MPQ$ равнобедренный, так как ________________, а потому его высота $ME$ является ________________, т. е. $PE = \_$ см. Итак, в прямоугольном треугольнике $MEP$ гипотенуза ________________, катет ________________, следовательно, $ME = \_$ см.</p><p><strong>4)</strong> $S_{MPQ} = \frac{1}{2} \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ = \frac{1}{2} \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ см^2 = \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ см^2$.</p><p>Ответ.</p>" "img" => array:2 [ 0 => array:5 [ "name" => "24-1.jpg" "alt" => null "width" => "1646" "height" => 729 "path" => "/media/geometrija_10/glazkov-rt/0-00/24-1.webp?ts=1739375698" ] 1 => array:5 [ "name" => "24-2.jpg" "alt" => null "width" => "1646" "height" => 672 "path" => "/media/geometrija_10/glazkov-rt/0-00/24-2.webp?ts=1739375698" ] ] ] #original: array:7 [ "id" => 937105 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1097} "task" => array:2 [ "refs" => "738496" "type" => "task" ] "text" => "<p><strong>24</strong> В тетраэдре $MNPQ$ ребро $MN = 3\sqrt{2}$ см, $NP = NQ = 7$ см, $PQ = 8$ см, $\angle MNP = \angle MNQ = 45^\circ$. Найдите площадь грани $MPQ$.</p><p>Решение.</p><p><strong>1)</strong> $\triangle MNP = \triangle MNQ$, так как ___________, поэтому $MP = \_$.</p><p><strong>2)</strong> По теореме косинусов для треугольника $MNP$ имеем: $MP^2 = \_$.</p><p>= ___________, откуда $MP = \_$ см.</p><p><strong>3)</strong> $\triangle MPQ$ равнобедренный, так как ________________, а потому его высота $ME$ является ________________, т. е. $PE = \_$ см. Итак, в прямоугольном треугольнике $MEP$ гипотенуза ________________, катет ________________, следовательно, $ME = \_$ см.</p><p><strong>4)</strong> $S_{MPQ} = \frac{1}{2} \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ = \frac{1}{2} \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ см^2 = \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ см^2$.</p><p>Ответ.</p>" "img" => array:2 [ 0 => array:5 [ "name" => "24-1.jpg" "alt" => null "width" => "1646" "height" => 729 "path" => "/media/geometrija_10/glazkov-rt/0-00/24-1.webp?ts=1739375698" ] 1 => array:5 [ "name" => "24-2.jpg" "alt" => null "width" => "1646" "height" => 672 "path" => "/media/geometrija_10/glazkov-rt/0-00/24-2.webp?ts=1739375698" ] ] ] #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" => 937215 "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" => "738496" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "24-1.png" "alt" => null "width" => "694" "height" => 2417 "path" => "/media/geometrija_10/glazkov-rt/1-00/24-1.webp?ts=1739375830" ] ] ] #original: array:6 [ "id" => 937215 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1105} "task" => array:2 [ "refs" => "738496" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "24-1.png" "alt" => null "width" => "694" "height" => 2417 "path" => "/media/geometrija_10/glazkov-rt/1-00/24-1.webp?ts=1739375830" ] ] ] #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" => 1434289 "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" => "738496" "type" => "task" ] "text" => "<p> <strong>1)</strong> Рассмотрим треугольники $MNP$ и $MNQ$. У них сторона $MN$ — общая, стороны $NP$ и $NQ$ равны по условию ($NP = NQ = 7$ см), и углы между этими сторонами также равны по условию ($\angle MNP = \angle MNQ = 45^\circ$). Следовательно, $\triangle MNP = \triangle MNQ$ по первому признаку равенства треугольников (по двум сторонам и углу между ними). Из равенства треугольников следует равенство их соответствующих сторон, поэтому $MP = MQ$.<br> Ответ: $MP = MQ$. </p> <p> <strong>2)</strong> Для нахождения длины стороны $MP$ применим теорему косинусов к треугольнику $MNP$: $MP^2 = MN^2 + NP^2 - 2 \cdot MN \cdot NP \cdot \cos(\angle MNP)$. Подставим известные значения: $MP^2 = (3\sqrt{2})^2 + 7^2 - 2 \cdot 3\sqrt{2} \cdot 7 \cdot \cos(45^\circ)$ $MP^2 = 18 + 49 - 42\sqrt{2} \cdot \frac{\sqrt{2}}{2} = 67 - \frac{42 \cdot 2}{2} = 67 - 42 = 25$. Отсюда $MP = \sqrt{25} = 5$ см.<br> Ответ: $MP = 5$ см. </p> <p> <strong>3)</strong> Рассмотрим грань $MPQ$. Так как $MP = MQ = 5$ см, треугольник $MPQ$ является равнобедренным с основанием $PQ$. Проведем в нем высоту $ME$ к основанию $PQ$. В равнобедренном треугольнике высота, проведенная к основанию, является также и медианой. Следовательно, точка $E$ — середина стороны $PQ$, и $PE = \frac{1}{2} PQ = \frac{8}{2} = 4$ см. Треугольник $MEP$ является прямоугольным, так как $ME$ — высота. По теореме Пифагора найдем катет $ME$: $ME^2 = MP^2 - PE^2 = 5^2 - 4^2 = 25 - 16 = 9$. $ME = \sqrt{9} = 3$ см.<br> Ответ: $ME = 3$ см. </p> <p> <strong>4)</strong> Площадь грани $MPQ$ найдем по формуле площади треугольника: $S_{MPQ} = \frac{1}{2} \cdot \text{основание} \cdot \text{высота} = \frac{1}{2} \cdot PQ \cdot ME$. $S_{MPQ} = \frac{1}{2} \cdot 8 \cdot 3 = 12 \text{ см}^2$.<br> Ответ: $12 \text{ см}^2$. </p>" ] #original: array:6 [ "id" => 1434289 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1113} "task" => array:2 [ "refs" => "738496" "type" => "task" ] "text" => "<p> <strong>1)</strong> Рассмотрим треугольники $MNP$ и $MNQ$. У них сторона $MN$ — общая, стороны $NP$ и $NQ$ равны по условию ($NP = NQ = 7$ см), и углы между этими сторонами также равны по условию ($\angle MNP = \angle MNQ = 45^\circ$). Следовательно, $\triangle MNP = \triangle MNQ$ по первому признаку равенства треугольников (по двум сторонам и углу между ними). Из равенства треугольников следует равенство их соответствующих сторон, поэтому $MP = MQ$.<br> Ответ: $MP = MQ$. </p> <p> <strong>2)</strong> Для нахождения длины стороны $MP$ применим теорему косинусов к треугольнику $MNP$: $MP^2 = MN^2 + NP^2 - 2 \cdot MN \cdot NP \cdot \cos(\angle MNP)$. Подставим известные значения: $MP^2 = (3\sqrt{2})^2 + 7^2 - 2 \cdot 3\sqrt{2} \cdot 7 \cdot \cos(45^\circ)$ $MP^2 = 18 + 49 - 42\sqrt{2} \cdot \frac{\sqrt{2}}{2} = 67 - \frac{42 \cdot 2}{2} = 67 - 42 = 25$. Отсюда $MP = \sqrt{25} = 5$ см.<br> Ответ: $MP = 5$ см. </p> <p> <strong>3)</strong> Рассмотрим грань $MPQ$. Так как $MP = MQ = 5$ см, треугольник $MPQ$ является равнобедренным с основанием $PQ$. Проведем в нем высоту $ME$ к основанию $PQ$. В равнобедренном треугольнике высота, проведенная к основанию, является также и медианой. Следовательно, точка $E$ — середина стороны $PQ$, и $PE = \frac{1}{2} PQ = \frac{8}{2} = 4$ см. Треугольник $MEP$ является прямоугольным, так как $ME$ — высота. По теореме Пифагора найдем катет $ME$: $ME^2 = MP^2 - PE^2 = 5^2 - 4^2 = 25 - 16 = 9$. $ME = \sqrt{9} = 3$ см.<br> Ответ: $ME = 3$ см. </p> <p> <strong>4)</strong> Площадь грани $MPQ$ найдем по формуле площади треугольника: $S_{MPQ} = \frac{1}{2} \cdot \text{основание} \cdot \text{высота} = \frac{1}{2} \cdot PQ \cdot ME$. $S_{MPQ} = \frac{1}{2} \cdot 8 \cdot 3 = 12 \text{ см}^2$.<br> Ответ: $12 \text{ см}^2$. </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" => "738497" "type" => "task" ] "previous" => array:2 [ "refs" => "738495" "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" => 971666 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "19" "field_url" => "/10-klass/geometrija/glazkov-tetrad/page-19" "field_display_title" => "19" "field_folder" => "1" "field_image_name" => "19" "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" => 1147782 "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" => "971666" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "19-1.jpg" "alt" => null "width" => "1953" "height" => 2800 "path" => "/media/geometrija_10/glazkov-rt/0-1/19-1.webp?ts=1745219733" ] ] ] #original: array:6 [ "id" => 1147782 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1124} "book_page" => array:2 [ "refs" => "971666" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "19-1.jpg" "alt" => null "width" => "1953" "height" => 2800 "path" => "/media/geometrija_10/glazkov-rt/0-1/19-1.webp?ts=1745219733" ] ] ] #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" => "971667" "type" => "book_page" ] "previous" => array:2 [ "refs" => "971665" "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" => 738495 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "19" "field_page_end" => null "field_url" => "/10-klass/geometrija/glazkov-tetrad/00-23" "field_display_title" => "23" "field_outside_task" => null "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 {#1209 #items: array:1 [ 0 => App\Models\Branch {#1208 #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" => 738455 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "1.3. Параллельность плоскостей" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1210 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "17" "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 {#1211 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1212 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 738455 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "1.3. Параллельность плоскостей" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1210} "field_page_start" => "17" "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 {#1211} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1212} ] #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 {#1201 #items: array:1 [ 0 => App\Models\Branch {#1208} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1205 #items: array:3 [ 0 => 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:7 [ "id" => 937104 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1219 #items: array:1 [ 0 => App\Models\Edition {#1089} ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "738495" "type" => "task" ] "text" => "<p><strong>23</strong> На рисунке параллельные плоскости $\alpha$ и $\beta$ пересечены прямыми $MN$ и $MF$, $P_1$, $P_2$ и $Q_1$, $Q_2$ — точки пересечения прямых с плоскостями $\alpha$ и $\beta$. Найдите $P_1P_2$, если $MP_1 : MQ_1 = 3 : 4$ и $Q_1Q_2 = 72$ см.</p><p>Р е ш е н и е.</p><p><strong>1)</strong> Пересекающиеся прямые $MN$ и $MF$ задают некоторую ______ $\gamma$. $P_1$ и $P_2$ — общие точки плоскостей $\alpha$ и $\gamma$, поэтому прямая $P_1P_2$ ______ , аналогично $Q_1$ и $Q_2$ ______ поэтому прямая $Q_1Q_2$ ______ .</p><p><strong>Итак, параллельные плоскости $\alpha$ и $\beta$ пересечены плоскостью $\gamma$, поэтому, согласно ______ , линии их пересечения ______ , т. е. $P_1P_2 \parallel$ ______ .</strong></p><p><strong>2)</strong> $\Delta P_1MP_2 \sim$ ______ , так как ______ , следовательно, $MP_1 : MQ_1 = P_1P_2 :$ ______ , $P_1P_2 = $ ______ = ______ .</p><p>О т в е т.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "23-1.jpg" "alt" => null "width" => "1646" "height" => 1571 "path" => "/media/geometrija_10/glazkov-rt/0-00/23-1.webp?ts=1739375698" ] ] ] #original: array:7 [ "id" => 937104 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1219} "task" => array:2 [ "refs" => "738495" "type" => "task" ] "text" => "<p><strong>23</strong> На рисунке параллельные плоскости $\alpha$ и $\beta$ пересечены прямыми $MN$ и $MF$, $P_1$, $P_2$ и $Q_1$, $Q_2$ — точки пересечения прямых с плоскостями $\alpha$ и $\beta$. Найдите $P_1P_2$, если $MP_1 : MQ_1 = 3 : 4$ и $Q_1Q_2 = 72$ см.</p><p>Р е ш е н и е.</p><p><strong>1)</strong> Пересекающиеся прямые $MN$ и $MF$ задают некоторую ______ $\gamma$. $P_1$ и $P_2$ — общие точки плоскостей $\alpha$ и $\gamma$, поэтому прямая $P_1P_2$ ______ , аналогично $Q_1$ и $Q_2$ ______ поэтому прямая $Q_1Q_2$ ______ .</p><p><strong>Итак, параллельные плоскости $\alpha$ и $\beta$ пересечены плоскостью $\gamma$, поэтому, согласно ______ , линии их пересечения ______ , т. е. $P_1P_2 \parallel$ ______ .</strong></p><p><strong>2)</strong> $\Delta P_1MP_2 \sim$ ______ , так как ______ , следовательно, $MP_1 : MQ_1 = P_1P_2 :$ ______ , $P_1P_2 = $ ______ = ______ .</p><p>О т в е т.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "23-1.jpg" "alt" => null "width" => "1646" "height" => 1571 "path" => "/media/geometrija_10/glazkov-rt/0-00/23-1.webp?ts=1739375698" ] ] ] #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 {#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" => 937214 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1221 #items: array:1 [ 0 => App\Models\Edition {#1096} ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "738495" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "23-1.png" "alt" => null "width" => "694" "height" => 2034 "path" => "/media/geometrija_10/glazkov-rt/1-00/23-1.webp?ts=1739375830" ] ] ] #original: array:6 [ "id" => 937214 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1221} "task" => array:2 [ "refs" => "738495" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "23-1.png" "alt" => null "width" => "694" "height" => 2034 "path" => "/media/geometrija_10/glazkov-rt/1-00/23-1.webp?ts=1739375830" ] ] ] #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 {#1222 #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" => 1434288 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1223 #items: array:1 [ 0 => App\Models\Edition {#1104} ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "738495" "type" => "task" ] "text" => "<p><strong>1)</strong> Две пересекающиеся в точке $M$ прямые $MN$ и $MF$ задают единственную плоскость. Обозначим эту плоскость $\gamma$.</p> <p>Точки $P_1$ и $P_2$ лежат в плоскости $\alpha$ по условию и в плоскости $\gamma$ по построению. Следовательно, прямая $P_1P_2$ является линией пересечения плоскостей $\alpha$ и $\gamma$.</p> <p>Аналогично, точки $Q_1$ и $Q_2$ лежат в плоскости $\beta$ по условию и в плоскости $\gamma$ по построению. Следовательно, прямая $Q_1Q_2$ является линией пересечения плоскостей $\beta$ и $\gamma$.</p> <p>По свойству параллельных плоскостей: если две параллельные плоскости ($\alpha \parallel \beta$) пересекаются третьей плоскостью ($\gamma$), то линии их пересечения параллельны. Таким образом, $P_1P_2 \parallel Q_1Q_2$.</p> <p><strong>2)</strong> Рассмотрим треугольники $\triangle P_1MP_2$ и $\triangle Q_1MQ_2$. Они подобны.</p> <p>Докажем их подобие: Угол $\angle M$ (или $\angle P_1MP_2$) является общим для обоих треугольников. Углы $\angle MP_1P_2$ и $\angle MQ_1Q_2$ равны как соответственные углы при параллельных прямых $P_1P_2$ и $Q_1Q_2$ и секущей $MN$. Следовательно, $\triangle P_1MP_2 \sim \triangle Q_1MQ_2$ по двум углам.</p> <p>Из подобия треугольников следует пропорциональность их соответствующих сторон:</p> <p>$\frac{P_1P_2}{Q_1Q_2} = \frac{MP_1}{MQ_1}$</p> <p>По условию задачи дано, что $MP_1:MQ_1 = 3:4$ и $Q_1Q_2 = 72$ см. Подставим известные значения в полученную пропорцию:</p> <p>$\frac{P_1P_2}{72} = \frac{3}{4}$</p> <p>Теперь выразим и найдем длину отрезка $P_1P_2$:</p> <p>$P_1P_2 = \frac{3 \cdot 72}{4} = 3 \cdot 18 = 54$ (см).</p> <p><strong>Ответ:</strong> $54$ см.</p>" ] #original: array:6 [ "id" => 1434288 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1223} "task" => array:2 [ "refs" => "738495" "type" => "task" ] "text" => "<p><strong>1)</strong> Две пересекающиеся в точке $M$ прямые $MN$ и $MF$ задают единственную плоскость. Обозначим эту плоскость $\gamma$.</p> <p>Точки $P_1$ и $P_2$ лежат в плоскости $\alpha$ по условию и в плоскости $\gamma$ по построению. Следовательно, прямая $P_1P_2$ является линией пересечения плоскостей $\alpha$ и $\gamma$.</p> <p>Аналогично, точки $Q_1$ и $Q_2$ лежат в плоскости $\beta$ по условию и в плоскости $\gamma$ по построению. Следовательно, прямая $Q_1Q_2$ является линией пересечения плоскостей $\beta$ и $\gamma$.</p> <p>По свойству параллельных плоскостей: если две параллельные плоскости ($\alpha \parallel \beta$) пересекаются третьей плоскостью ($\gamma$), то линии их пересечения параллельны. Таким образом, $P_1P_2 \parallel Q_1Q_2$.</p> <p><strong>2)</strong> Рассмотрим треугольники $\triangle P_1MP_2$ и $\triangle Q_1MQ_2$. Они подобны.</p> <p>Докажем их подобие: Угол $\angle M$ (или $\angle P_1MP_2$) является общим для обоих треугольников. Углы $\angle MP_1P_2$ и $\angle MQ_1Q_2$ равны как соответственные углы при параллельных прямых $P_1P_2$ и $Q_1Q_2$ и секущей $MN$. Следовательно, $\triangle P_1MP_2 \sim \triangle Q_1MQ_2$ по двум углам.</p> <p>Из подобия треугольников следует пропорциональность их соответствующих сторон:</p> <p>$\frac{P_1P_2}{Q_1Q_2} = \frac{MP_1}{MQ_1}$</p> <p>По условию задачи дано, что $MP_1:MQ_1 = 3:4$ и $Q_1Q_2 = 72$ см. Подставим известные значения в полученную пропорцию:</p> <p>$\frac{P_1P_2}{72} = \frac{3}{4}$</p> <p>Теперь выразим и найдем длину отрезка $P_1P_2$:</p> <p>$P_1P_2 = \frac{3 \cdot 72}{4} = 3 \cdot 18 = 54$ (см).</p> <p><strong>Ответ:</strong> $54$ 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" => "738496" "type" => "task" ] "previous" => array:2 [ "refs" => "738494" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1215 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "971666" "type" => "book_page" ] ] #original: array:24 [ "id" => 738495 "created_at" => "2026-04-10 13:58:26" "updated_at" => null 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#attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => 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" => 738496 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "19" "field_page_end" => "20" "field_url" => "/10-klass/geometrija/glazkov-tetrad/00-24" "field_display_title" => "24" "field_outside_task" => "0" "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 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false } "page" => array:2 [ "refs" => "971666" "type" => "book_page" ] ] #original: array:24 [ "id" => 738496 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "19" "field_page_end" => "20" "field_url" => "/10-klass/geometrija/glazkov-tetrad/00-24" "field_display_title" => "24" "field_outside_task" => "0" "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 {#1214} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1216} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1204} "content" => Illuminate\Database\Eloquent\Collection {#1203} "next" => array:2 [ "refs" => "738497" "type" => "task" ] "previous" => array:2 [ "refs" => "738495" "type" => "task" ] "book" 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№24 (с. 19)
Условие. №24 (с. 19)
скриншот условия
24 В тетраэдре $MNPQ$ ребро $MN = 3\sqrt{2}$ см, $NP = NQ = 7$ см, $PQ = 8$ см, $\angle MNP = \angle MNQ = 45^\circ$. Найдите площадь грани $MPQ$.
Решение.
1) $\triangle MNP = \triangle MNQ$, так как ___________, поэтому $MP = \_$.
2) По теореме косинусов для треугольника $MNP$ имеем: $MP^2 = \_$.
= ___________, откуда $MP = \_$ см.
3) $\triangle MPQ$ равнобедренный, так как ________________, а потому его высота $ME$ является ________________, т. е. $PE = \_$ см. Итак, в прямоугольном треугольнике $MEP$ гипотенуза ________________, катет ________________, следовательно, $ME = \_$ см.
4) $S_{MPQ} = \frac{1}{2} \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ = \frac{1}{2} \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ см^2 = \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ см^2$.
Ответ.
Решение. №24 (с. 19)
Решение 2. №24 (с. 19)
1) Рассмотрим треугольники $MNP$ и $MNQ$. У них сторона $MN$ — общая, стороны $NP$ и $NQ$ равны по условию ($NP = NQ = 7$ см), и углы между этими сторонами также равны по условию ($\angle MNP = \angle MNQ = 45^\circ$). Следовательно, $\triangle MNP = \triangle MNQ$ по первому признаку равенства треугольников (по двум сторонам и углу между ними). Из равенства треугольников следует равенство их соответствующих сторон, поэтому $MP = MQ$.
Ответ: $MP = MQ$.
2) Для нахождения длины стороны $MP$ применим теорему косинусов к треугольнику $MNP$: $MP^2 = MN^2 + NP^2 - 2 \cdot MN \cdot NP \cdot \cos(\angle MNP)$. Подставим известные значения: $MP^2 = (3\sqrt{2})^2 + 7^2 - 2 \cdot 3\sqrt{2} \cdot 7 \cdot \cos(45^\circ)$ $MP^2 = 18 + 49 - 42\sqrt{2} \cdot \frac{\sqrt{2}}{2} = 67 - \frac{42 \cdot 2}{2} = 67 - 42 = 25$. Отсюда $MP = \sqrt{25} = 5$ см.
Ответ: $MP = 5$ см.
3) Рассмотрим грань $MPQ$. Так как $MP = MQ = 5$ см, треугольник $MPQ$ является равнобедренным с основанием $PQ$. Проведем в нем высоту $ME$ к основанию $PQ$. В равнобедренном треугольнике высота, проведенная к основанию, является также и медианой. Следовательно, точка $E$ — середина стороны $PQ$, и $PE = \frac{1}{2} PQ = \frac{8}{2} = 4$ см. Треугольник $MEP$ является прямоугольным, так как $ME$ — высота. По теореме Пифагора найдем катет $ME$: $ME^2 = MP^2 - PE^2 = 5^2 - 4^2 = 25 - 16 = 9$. $ME = \sqrt{9} = 3$ см.
Ответ: $ME = 3$ см.
4) Площадь грани $MPQ$ найдем по формуле площади треугольника: $S_{MPQ} = \frac{1}{2} \cdot \text{основание} \cdot \text{высота} = \frac{1}{2} \cdot PQ \cdot ME$. $S_{MPQ} = \frac{1}{2} \cdot 8 \cdot 3 = 12 \text{ см}^2$.
Ответ: $12 \text{ см}^2$.
Другие задания:
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