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Номер 78, страница 32 - гдз по геометрии 10-11 класс учебник Атанасян, Бутузов
Авторы: Атанасян Л. С., Бутузов В. Ф., Кадомцев С. Б., Позняк Э. Г., Киселёва Л. С.
Тип: Учебник
Издательство: Просвещение
Год издания: 2019 - 2026
Уровень обучения: базовый и углублённый
Цвет обложки: коричневый с ромбами
ISBN: 978-5-09-103606-0 (2023)
Допущено Министерством просвещения Российской Федерации
Математика: алгебра и начала математического анализа, геометрия
Популярные ГДЗ в 10 классе
Глава 1. Параллельность прямых и плоскостей. Параграф 4. Тетраэдр и параллелепипед - номер 78, страница 32.
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" => 744493 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "32" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/00-78" "field_display_title" => "78" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1079 #items: array:1 [ 0 => App\Models\Branch {#1034 #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" => 744367 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Параллельность прямых и плоскостей" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039 #items: array:1 [ 0 => App\Models\Term {#1038 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "9" "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 {#1040 #items: array:1 [ 0 => App\Models\Book {#1041 #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" => 36 "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 {#1042 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1047 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1049 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1070 …2} "field_under_the_edition_degree" => null "field_cover_description" => "с ромбами" "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для общеобразовательных организаций" "field_allowed" => "Допущено Министерством просвещения Российской Федерации" "field_reserve_field" => "Математика: алгебра и начала математического анализа, геометрия" "field_link_to_source" => null "field_tasks_count" => "1147" "field_priority" => "12" "field_default_folder" => "/geometrija_10/atanasjan-u23/" "field_isbn" => "978-5-09-103606-0 (2023)" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1071 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1072 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1073 …2} "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1074 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 36 "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 {#1042 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1047 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1049 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1070 …2} "field_under_the_edition_degree" => null "field_cover_description" => "с ромбами" "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для общеобразовательных организаций" "field_allowed" => "Допущено Министерством просвещения Российской Федерации" "field_reserve_field" => "Математика: алгебра и начала математического анализа, геометрия" "field_link_to_source" => null "field_tasks_count" => "1147" "field_priority" => "12" "field_default_folder" => "/geometrija_10/atanasjan-u23/" "field_isbn" => "978-5-09-103606-0 (2023)" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1071 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1072 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1073 …2} "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1074 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 {#1076 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 744367 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Параллельность прямых и плоскостей" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039} "field_page_start" => "9" "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 {#1040} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1076} ] #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 {#1120 #items: array:2 [ 0 => App\Models\Branch {#1128 #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" => 744367 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Параллельность прямых и плоскостей" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1132 #items: array:1 [ 0 => App\Models\Term {#1129 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "9" "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 {#1168 #items: array:1 [ 0 => App\Models\Book {#1131 #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" => 36 "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 {#1133 #items: array:1 [ 0 => App\Models\Term {#1130 #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 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1134 #items: array:2 [ 0 => App\Models\Term {#1135 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Term {#1136 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1137 #items: array:1 [ 0 => App\Models\Term {#1138 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1139 #items: array:5 [ 0 => App\Models\Term {#1140 #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 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Term {#1141 #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 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 2 => App\Models\Term {#1142 #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 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 3 => App\Models\Term {#1143 #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 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 4 => App\Models\Term {#1144 #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 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1145 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1146 #items: array:1 [ 0 => App\Models\Term {#1147 #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 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1148 #items: array:1 [ 0 => App\Models\Term {#1149 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1150 #items: array:1 [ 0 => App\Models\Term {#1151 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1152 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1153 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1154 #items: array:1 [ 0 => App\Models\Term {#1155 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1156 #items: array:1 [ 0 => App\Models\Term {#1157 #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 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1158 #items: array:1 [ 0 => App\Models\Term {#1159 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1160 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => "с ромбами" "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для общеобразовательных организаций" "field_allowed" => "Допущено Министерством просвещения Российской Федерации" "field_reserve_field" => "Математика: алгебра и начала математического анализа, геометрия" "field_link_to_source" => null "field_tasks_count" => "1147" "field_priority" => "12" "field_default_folder" => "/geometrija_10/atanasjan-u23/" "field_isbn" => "978-5-09-103606-0 (2023)" "field_cover" => array:1 [ 0 => "/media/geometrija_10/atanasjan-u23/covers/cover1.webp?ts=1739363242" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_10/atanasjan-u23/covers/cover1.webp?ts=1739363242" "alt" => "" "width" => "1200" "height" => "1661" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1161 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1162 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1163 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1164 #items: array:1 [ 0 => App\Models\Term {#1165 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #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" => 36 "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 {#1133} "field_class" => Illuminate\Database\Eloquent\Collection {#1134} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1137} "field_author" => Illuminate\Database\Eloquent\Collection {#1139} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1145} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1146} "field_country" => Illuminate\Database\Eloquent\Collection {#1148} "field_city" => Illuminate\Database\Eloquent\Collection {#1150} "field_series" => Illuminate\Database\Eloquent\Collection {#1152} "field_umk" => Illuminate\Database\Eloquent\Collection {#1153} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1154} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1156} "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1158} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1160} "field_under_the_edition_degree" => null "field_cover_description" => "с ромбами" "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для общеобразовательных организаций" "field_allowed" => "Допущено Министерством просвещения Российской Федерации" "field_reserve_field" => "Математика: алгебра и начала математического анализа, геометрия" "field_link_to_source" => null "field_tasks_count" => "1147" "field_priority" => "12" "field_default_folder" => "/geometrija_10/atanasjan-u23/" "field_isbn" => "978-5-09-103606-0 (2023)" "field_cover" => array:1 [ 0 => "/media/geometrija_10/atanasjan-u23/covers/cover1.webp?ts=1739363242" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_10/atanasjan-u23/covers/cover1.webp?ts=1739363242" "alt" => "" "width" => "1200" "height" => "1661" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1161} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1162} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1163} "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1164} "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 {#1166 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 744367 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Параллельность прямых и плоскостей" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1132} "field_page_start" => "9" "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 {#1168} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1166} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Branch {#1167 #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" => 744371 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Тетраэдр и параллелепипед" "field_branch_order" => "4" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1172 #items: array:1 [ 0 => App\Models\Term {#1169 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => "§" ] #original: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => "§" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "25" "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 {#1171 #items: array:1 [ 0 => App\Models\Book {#1131} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1170 #items: array:1 [ 0 => App\Models\Branch {#1034} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 744371 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Тетраэдр и параллелепипед" "field_branch_order" => "4" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1172} "field_page_start" => "25" "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 {#1171} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1170} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1077 #items: array:6 [ 0 => App\Models\Element {#1108 #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" => 938727 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1097 #items: array:1 [ 0 => App\Models\Edition {#1105 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 2315 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1104 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1103 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1101 …2} "field_process_formula" => "wiris" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1102 …2} "field_content_source" => null ] #original: array:21 [ "id" => 2315 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1104 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1103 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1101 …2} "field_process_formula" => "wiris" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1102 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "744493" "type" => "task" ] "text" => """ <p><strong>78.</strong> На рисунке 42 изображён параллелепипед ABCDA₁B₁C₁D₁, на рёбрах которого отмечены точки М, N, М₁ и N₁ так, что <span class="long">AM = CN = A₁M₁ = C₁N₁.</span> Докажите, что MBNDM₁B₁N₁D₁ — параллелепипед.</p><figure><figure class="align-center">\n <img src="/media/geometrija_10/atanasjan-u23/0-00/78.webp?ts=1745530887" alt="Рисунок 42 параллелепипед" loading="lazy" width="495" height="516">\n </figure>\n <figcaption> </figcaption></figure> """ "img" => array:2 [ 0 => array:5 [ "name" => "78-1.jpg" "alt" => null "width" => "1044" "height" => 285 "path" => "/media/geometrija_10/atanasjan-u23/0-00/78-1.webp?ts=1739436353" ] 1 => array:5 [ "name" => "78-2.jpg" "alt" => null "width" => "479" "height" => 569 "path" => "/media/geometrija_10/atanasjan-u23/0-00/78-2.webp?ts=1739436353" ] ] ] #original: array:7 [ "id" => 938727 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1097} "task" => array:2 [ "refs" => "744493" "type" => "task" ] "text" => """ <p><strong>78.</strong> На рисунке 42 изображён параллелепипед ABCDA₁B₁C₁D₁, на рёбрах которого отмечены точки М, N, М₁ и N₁ так, что <span class="long">AM = CN = A₁M₁ = C₁N₁.</span> Докажите, что MBNDM₁B₁N₁D₁ — параллелепипед.</p><figure><figure class="align-center">\n <img src="/media/geometrija_10/atanasjan-u23/0-00/78.webp?ts=1745530887" alt="Рисунок 42 параллелепипед" loading="lazy" width="495" height="516">\n </figure>\n <figcaption> </figcaption></figure> """ "img" => array:2 [ 0 => array:5 [ "name" => "78-1.jpg" "alt" => null "width" => "1044" "height" => 285 "path" => "/media/geometrija_10/atanasjan-u23/0-00/78-1.webp?ts=1739436353" ] 1 => array:5 [ "name" => "78-2.jpg" "alt" => null "width" => "479" "height" => 569 "path" => "/media/geometrija_10/atanasjan-u23/0-00/78-2.webp?ts=1739436353" ] ] ] #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" => 939942 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1080 #items: array:1 [ 0 => App\Models\Edition {#1092 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 2317 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_content_source" => null ] #original: array:21 [ "id" => 2317 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "744493" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "78-1.png" "alt" => null "width" => "694" "height" => 3453 "path" => "/media/geometrija_10/atanasjan-u23/2-00/78-1.webp?ts=1739440893" ] ] ] #original: array:6 [ "id" => 939942 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1080} "task" => array:2 [ "refs" => "744493" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "78-1.png" "alt" => null "width" => "694" "height" => 3453 "path" => "/media/geometrija_10/atanasjan-u23/2-00/78-1.webp?ts=1739440893" ] ] ] #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 {#1099 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 939548 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1091 #items: array:1 [ 0 => App\Models\Edition {#1100 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 2318 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 3" "field_order" => "4" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1096 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" …7 ] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "744493" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "78-1.jpg" "alt" => null "width" => "1387" "height" => 1042 "path" => "/media/geometrija_10/atanasjan-u23/3-00/78-1.webp?ts=1739436540" ] ] ] #original: array:6 [ "id" => 939548 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1091} "task" => array:2 [ "refs" => "744493" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "78-1.jpg" "alt" => null "width" => "1387" "height" => 1042 "path" => "/media/geometrija_10/atanasjan-u23/3-00/78-1.webp?ts=1739436540" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Element {#1083 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 939959 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1122 #items: array:1 [ 0 => App\Models\Edition {#1084 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "744493" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "78-1.jpg" "alt" => null "width" => "1192" "height" => 563 "path" => "/media/geometrija_10/atanasjan-u23/4-00/78-1.webp?ts=1739436373" ] ] ] #original: array:6 [ "id" => 939959 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1122} "task" => array:2 [ "refs" => "744493" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "78-1.jpg" "alt" => null "width" => "1192" "height" => 563 "path" => "/media/geometrija_10/atanasjan-u23/4-00/78-1.webp?ts=1739436373" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Element {#1121 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 940045 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1180 #items: array:1 [ 0 => App\Models\Edition {#1124 #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 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "744493" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "78-1.jpg" "alt" => null "width" => "1329" "height" => 907 "path" => "/media/geometrija_10/atanasjan-u23/5-00/78-1.webp?ts=1739436380" ] ] ] #original: array:6 [ "id" => 940045 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1180} "task" => array:2 [ "refs" => "744493" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "78-1.jpg" "alt" => null "width" => "1329" "height" => 907 "path" => "/media/geometrija_10/atanasjan-u23/5-00/78-1.webp?ts=1739436380" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 5 => App\Models\Element {#1178 #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" => 1351806 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1188 #items: array:1 [ 0 => App\Models\Edition {#1179 #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 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "744493" "type" => "task" ] "text" => "<p>Для доказательства того, что многогранник $MBNDM_1B_1N_1D_1$ является параллелепипедом, необходимо установить, что все его шесть граней являются параллелограммами.</p><p><strong>1. Докажем, что основания $MBND$ и $M_1B_1N_1D_1$ являются параллелограммами.</strong></p><p>Рассмотрим четырехугольник $MBND$, лежащий в плоскости основания $ABCD$.Поскольку $ABCDA_1B_1C_1D_1$ — параллелепипед, его основание $ABCD$ является параллелограммом. Следовательно, сторона $AB$ параллельна стороне $DC$ ($AB \parallel DC$) и их длины равны ($AB = DC$).Точки $M$ и $N$ лежат на ребрах $AB$ и $DC$ соответственно. Это означает, что отрезок $MB$ лежит на прямой $AB$, а отрезок $DN$ — на прямой $DC$. Так как прямые $AB$ и $DC$ параллельны, то и отрезки $MB$ и $DN$ параллельны ($MB \parallel DN$).По условию задачи, $AM = CN$.Длины отрезков $MB$ и $DN$ можно выразить как:$MB = AB - AM$$DN = DC - CN$Так как $AB = DC$ и $AM = CN$, то $MB = DN$.В четырехугольнике $MBND$ противоположные стороны $MB$ и $DN$ параллельны и равны. По признаку параллелограмма, четырехугольник $MBND$ является параллелограммом.</p><p>Аналогично рассмотрим четырехугольник $M_1B_1N_1D_1$, лежащий в плоскости верхнего основания $A_1B_1C_1D_1$.Грань $A_1B_1C_1D_1$ — параллелограмм, поэтому $A_1B_1 \parallel D_1C_1$ и $A_1B_1 = D_1C_1$.Точки $M_1$ и $N_1$ лежат на ребрах $A_1B_1$ и $D_1C_1$ соответственно. Следовательно, $M_1B_1 \parallel D_1N_1$.По условию, $A_1M_1 = C_1N_1$.Длины отрезков $M_1B_1$ и $D_1N_1$ равны:$M_1B_1 = A_1B_1 - A_1M_1$$D_1N_1 = D_1C_1 - C_1N_1$Так как $A_1B_1 = D_1C_1$ и $A_1M_1 = C_1N_1$, то $M_1B_1 = D_1N_1$.В четырехугольнике $M_1B_1N_1D_1$ противоположные стороны $M_1B_1$ и $D_1N_1$ параллельны и равны. Следовательно, $M_1B_1N_1D_1$ является параллелограммом.</p><p><strong>2. Докажем, что боковые грани $MBB_1M_1$, $BNN_1B_1$, $NDD_1N_1$ и $DMM_1D_1$ являются параллелограммами.</strong></p><p>Сначала докажем, что отрезки $MM_1$ и $NN_1$ параллельны и равны боковым ребрам исходного параллелепипеда.Рассмотрим четырехугольник $AMM_1A_1$ в плоскости грани $ABB_1A_1$. Грань $ABB_1A_1$ — параллелограмм, поэтому $AB \parallel A_1B_1$ и $AA_1 \parallel BB_1$. Из $AB \parallel A_1B_1$ следует, что $AM \parallel A_1M_1$. По условию $AM = A_1M_1$. Четырехугольник $AMM_1A_1$, у которого две противоположные стороны ($AM$ и $A_1M_1$) параллельны и равны, является параллелограммом. Значит, $MM_1 \parallel AA_1$ и $MM_1 = AA_1$.</p><p>Теперь рассмотрим четырехугольник $CNN_1C_1$ в плоскости грани $DCC_1D_1$. Грань $DCC_1D_1$ — параллелограмм, поэтому $DC \parallel D_1C_1$ и $CC_1 \parallel DD_1$. Из $DC \parallel D_1C_1$ следует, что прямые, содержащие отрезки $CN$ и $C_1N_1$, параллельны. По условию $CN=C_1N_1$ (так как $CN=AM=A_1M_1=C_1N_1$). Рассмотрим четырехугольник $DNC_1N_1$. Он лежит в плоскости сечения, проходящего через параллельные прямые $DC$ и $D_1C_1$.Более строго, в параллелограмме $DCC_1D_1$ имеем $DD_1 \parallel CC_1$ и $DD_1=CC_1$. Векторно $\vec{DD_1} = \vec{CC_1}$. Также $\vec{DN} = \vec{MB}$ и $\vec{D_1N_1} = \vec{M_1B_1}$. Вектор $\vec{NN_1} = \vec{ND} + \vec{DD_1} + \vec{D_1N_1}$. Это сложно.Вернемся к $CNN_1C_1$. Так как $DCC_1D_1$ параллелограмм, то $\vec{DC} = \vec{D_1C_1}$ и $\vec{DD_1} = \vec{CC_1}$. Пусть $\vec{CN} = k \cdot \vec{CD}$ и $\vec{C_1N_1} = k \cdot \vec{C_1D_1}$ (поскольку $CN = C_1N_1$ и $CD = C_1D_1$, коэффициент $k$ одинаков). Тогда $\vec{NN_1} = \vec{NC} + \vec{CC_1} + \vec{C_1N_1} = -k\vec{CD} + \vec{CC_1} + k\vec{C_1D_1}$. Так как $\vec{CD} = \vec{C_1D_1}$, то $\vec{NN_1} = \vec{CC_1}$. Отсюда следует, что $NN_1 \parallel CC_1$ и $NN_1 = CC_1$.</p><p>В исходном параллелепипеде все боковые ребра параллельны и равны: $AA_1 \parallel BB_1 \parallel CC_1 \parallel DD_1$ и $AA_1 = BB_1 = CC_1 = DD_1$.Из доказанного выше ($MM_1 \parallel AA_1$, $MM_1 = AA_1$ и $NN_1 \parallel CC_1$, $NN_1 = CC_1$) следует, что все отрезки $MM_1, BB_1, NN_1, DD_1$ параллельны друг другу и равны по длине.</p><p>Теперь мы можем доказать, что боковые грани являются параллелограммами:</p><ul><li><b>Грань $MBB_1M_1$:</b> стороны $MM_1$ и $BB_1$ параллельны и равны. Следовательно, $MBB_1M_1$ — параллелограмм.</li><li><b>Грань $BNN_1B_1$:</b> стороны $BB_1$ и $NN_1$ параллельны и равны. Следовательно, $BNN_1B_1$ — параллелограмм.</li><li><b>Грань $NDD_1N_1$:</b> стороны $NN_1$ и $DD_1$ параллельны и равны. Следовательно, $NDD_1N_1$ — параллелограмм.</li><li><b>Грань $DMM_1D_1$:</b> стороны $DD_1$ и $MM_1$ параллельны и равны. Следовательно, $DMM_1D_1$ — параллелограмм.</li></ul><p><strong>Заключение</strong></p><p>Поскольку все шесть граней многогранника $MBNDM_1B_1N_1D_1$ (основания $MBND$, $M_1B_1N_1D_1$ и боковые грани $MBB_1M_1$, $BNN_1B_1$, $NDD_1N_1$, $DMM_1D_1$) являются параллелограммами, данный многогранник является параллелепипедом по определению.Что и требовалось доказать.</p><p><strong>Ответ:</strong> Утверждение, что $MBNDM_1B_1N_1D_1$ является параллелепипедом, доказано.</p>" ] #original: array:6 [ "id" => 1351806 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1188} "task" => array:2 [ "refs" => "744493" "type" => "task" ] "text" => "<p>Для доказательства того, что многогранник $MBNDM_1B_1N_1D_1$ является параллелепипедом, необходимо установить, что все его шесть граней являются параллелограммами.</p><p><strong>1. Докажем, что основания $MBND$ и $M_1B_1N_1D_1$ являются параллелограммами.</strong></p><p>Рассмотрим четырехугольник $MBND$, лежащий в плоскости основания $ABCD$.Поскольку $ABCDA_1B_1C_1D_1$ — параллелепипед, его основание $ABCD$ является параллелограммом. Следовательно, сторона $AB$ параллельна стороне $DC$ ($AB \parallel DC$) и их длины равны ($AB = DC$).Точки $M$ и $N$ лежат на ребрах $AB$ и $DC$ соответственно. Это означает, что отрезок $MB$ лежит на прямой $AB$, а отрезок $DN$ — на прямой $DC$. Так как прямые $AB$ и $DC$ параллельны, то и отрезки $MB$ и $DN$ параллельны ($MB \parallel DN$).По условию задачи, $AM = CN$.Длины отрезков $MB$ и $DN$ можно выразить как:$MB = AB - AM$$DN = DC - CN$Так как $AB = DC$ и $AM = CN$, то $MB = DN$.В четырехугольнике $MBND$ противоположные стороны $MB$ и $DN$ параллельны и равны. По признаку параллелограмма, четырехугольник $MBND$ является параллелограммом.</p><p>Аналогично рассмотрим четырехугольник $M_1B_1N_1D_1$, лежащий в плоскости верхнего основания $A_1B_1C_1D_1$.Грань $A_1B_1C_1D_1$ — параллелограмм, поэтому $A_1B_1 \parallel D_1C_1$ и $A_1B_1 = D_1C_1$.Точки $M_1$ и $N_1$ лежат на ребрах $A_1B_1$ и $D_1C_1$ соответственно. Следовательно, $M_1B_1 \parallel D_1N_1$.По условию, $A_1M_1 = C_1N_1$.Длины отрезков $M_1B_1$ и $D_1N_1$ равны:$M_1B_1 = A_1B_1 - A_1M_1$$D_1N_1 = D_1C_1 - C_1N_1$Так как $A_1B_1 = D_1C_1$ и $A_1M_1 = C_1N_1$, то $M_1B_1 = D_1N_1$.В четырехугольнике $M_1B_1N_1D_1$ противоположные стороны $M_1B_1$ и $D_1N_1$ параллельны и равны. Следовательно, $M_1B_1N_1D_1$ является параллелограммом.</p><p><strong>2. Докажем, что боковые грани $MBB_1M_1$, $BNN_1B_1$, $NDD_1N_1$ и $DMM_1D_1$ являются параллелограммами.</strong></p><p>Сначала докажем, что отрезки $MM_1$ и $NN_1$ параллельны и равны боковым ребрам исходного параллелепипеда.Рассмотрим четырехугольник $AMM_1A_1$ в плоскости грани $ABB_1A_1$. Грань $ABB_1A_1$ — параллелограмм, поэтому $AB \parallel A_1B_1$ и $AA_1 \parallel BB_1$. Из $AB \parallel A_1B_1$ следует, что $AM \parallel A_1M_1$. По условию $AM = A_1M_1$. Четырехугольник $AMM_1A_1$, у которого две противоположные стороны ($AM$ и $A_1M_1$) параллельны и равны, является параллелограммом. Значит, $MM_1 \parallel AA_1$ и $MM_1 = AA_1$.</p><p>Теперь рассмотрим четырехугольник $CNN_1C_1$ в плоскости грани $DCC_1D_1$. Грань $DCC_1D_1$ — параллелограмм, поэтому $DC \parallel D_1C_1$ и $CC_1 \parallel DD_1$. Из $DC \parallel D_1C_1$ следует, что прямые, содержащие отрезки $CN$ и $C_1N_1$, параллельны. По условию $CN=C_1N_1$ (так как $CN=AM=A_1M_1=C_1N_1$). Рассмотрим четырехугольник $DNC_1N_1$. Он лежит в плоскости сечения, проходящего через параллельные прямые $DC$ и $D_1C_1$.Более строго, в параллелограмме $DCC_1D_1$ имеем $DD_1 \parallel CC_1$ и $DD_1=CC_1$. Векторно $\vec{DD_1} = \vec{CC_1}$. Также $\vec{DN} = \vec{MB}$ и $\vec{D_1N_1} = \vec{M_1B_1}$. Вектор $\vec{NN_1} = \vec{ND} + \vec{DD_1} + \vec{D_1N_1}$. Это сложно.Вернемся к $CNN_1C_1$. Так как $DCC_1D_1$ параллелограмм, то $\vec{DC} = \vec{D_1C_1}$ и $\vec{DD_1} = \vec{CC_1}$. Пусть $\vec{CN} = k \cdot \vec{CD}$ и $\vec{C_1N_1} = k \cdot \vec{C_1D_1}$ (поскольку $CN = C_1N_1$ и $CD = C_1D_1$, коэффициент $k$ одинаков). Тогда $\vec{NN_1} = \vec{NC} + \vec{CC_1} + \vec{C_1N_1} = -k\vec{CD} + \vec{CC_1} + k\vec{C_1D_1}$. Так как $\vec{CD} = \vec{C_1D_1}$, то $\vec{NN_1} = \vec{CC_1}$. Отсюда следует, что $NN_1 \parallel CC_1$ и $NN_1 = CC_1$.</p><p>В исходном параллелепипеде все боковые ребра параллельны и равны: $AA_1 \parallel BB_1 \parallel CC_1 \parallel DD_1$ и $AA_1 = BB_1 = CC_1 = DD_1$.Из доказанного выше ($MM_1 \parallel AA_1$, $MM_1 = AA_1$ и $NN_1 \parallel CC_1$, $NN_1 = CC_1$) следует, что все отрезки $MM_1, BB_1, NN_1, DD_1$ параллельны друг другу и равны по длине.</p><p>Теперь мы можем доказать, что боковые грани являются параллелограммами:</p><ul><li><b>Грань $MBB_1M_1$:</b> стороны $MM_1$ и $BB_1$ параллельны и равны. Следовательно, $MBB_1M_1$ — параллелограмм.</li><li><b>Грань $BNN_1B_1$:</b> стороны $BB_1$ и $NN_1$ параллельны и равны. Следовательно, $BNN_1B_1$ — параллелограмм.</li><li><b>Грань $NDD_1N_1$:</b> стороны $NN_1$ и $DD_1$ параллельны и равны. Следовательно, $NDD_1N_1$ — параллелограмм.</li><li><b>Грань $DMM_1D_1$:</b> стороны $DD_1$ и $MM_1$ параллельны и равны. Следовательно, $DMM_1D_1$ — параллелограмм.</li></ul><p><strong>Заключение</strong></p><p>Поскольку все шесть граней многогранника $MBNDM_1B_1N_1D_1$ (основания $MBND$, $M_1B_1N_1D_1$ и боковые грани $MBB_1M_1$, $BNN_1B_1$, $NDD_1N_1$, $DMM_1D_1$) являются параллелограммами, данный многогранник является параллелепипедом по определению.Что и требовалось доказать.</p><p><strong>Ответ:</strong> Утверждение, что $MBNDM_1B_1N_1D_1$ является параллелепипедом, доказано.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "744494" "type" => "task" ] "previous" => array:2 [ "refs" => "744492" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1110 #items: array:1 [ 0 => App\Models\Book {#1131} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1112 #items: array:1 [ 0 => App\Models\BookPage {#1114 #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" => 933814 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "32" "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/page-32" "field_display_title" => "32" "field_folder" => "1" "field_image_name" => "32" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1115 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1081 #items: array:1 [ 0 => App\Models\Book {#1131} ] #escapeWhenCastingToString: false } "field_metatags_title" => 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"book_page" ] "previous" => array:2 [ "refs" => "933813" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1333 #items: array:12 [ 0 => App\Models\Task {#1351 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Task {#1358 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 2 => App\Models\Task {#1361 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 3 => App\Models\Task {#1360 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 4 => App\Models\Task {#1362 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 5 => App\Models\Task {#1425 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 6 => App\Models\Task {#1428 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 7 => App\Models\Task {#1452 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 8 => App\Models\Task {#1426 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 9 => App\Models\Task {#1486 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 10 => App\Models\Task {#1449 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 11 => App\Models\Task {#1487 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 933814 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "32" "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/page-32" "field_display_title" => "32" "field_folder" => "1" "field_image_name" => "32" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1115} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1081} "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 {#1116} "content" => Illuminate\Database\Eloquent\Collection {#1195} "next" => 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"field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1079} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1120} "content" => Illuminate\Database\Eloquent\Collection {#1077} "next" => array:2 [ "refs" => "744494" "type" => "task" ] "previous" => array:2 [ "refs" => "744492" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1110} "page" => array:2 [ "refs" => "933814" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] }
№78 (с. 32)
Условие. №78 (с. 32)
скриншот условия
78. На рисунке 42 изображён параллелепипед ABCDA₁B₁C₁D₁, на рёбрах которого отмечены точки М, N, М₁ и N₁ так, что AM = CN = A₁M₁ = C₁N₁. Докажите, что MBNDM₁B₁N₁D₁ — параллелепипед.
Решение 2. №78 (с. 32)
Решение 3. №78 (с. 32)
Решение 4. №78 (с. 32)
Решение 5. №78 (с. 32)
Решение 6. №78 (с. 32)
Для доказательства того, что многогранник $MBNDM_1B_1N_1D_1$ является параллелепипедом, необходимо установить, что все его шесть граней являются параллелограммами.
1. Докажем, что основания $MBND$ и $M_1B_1N_1D_1$ являются параллелограммами.
Рассмотрим четырехугольник $MBND$, лежащий в плоскости основания $ABCD$.Поскольку $ABCDA_1B_1C_1D_1$ — параллелепипед, его основание $ABCD$ является параллелограммом. Следовательно, сторона $AB$ параллельна стороне $DC$ ($AB \parallel DC$) и их длины равны ($AB = DC$).Точки $M$ и $N$ лежат на ребрах $AB$ и $DC$ соответственно. Это означает, что отрезок $MB$ лежит на прямой $AB$, а отрезок $DN$ — на прямой $DC$. Так как прямые $AB$ и $DC$ параллельны, то и отрезки $MB$ и $DN$ параллельны ($MB \parallel DN$).По условию задачи, $AM = CN$.Длины отрезков $MB$ и $DN$ можно выразить как:$MB = AB - AM$$DN = DC - CN$Так как $AB = DC$ и $AM = CN$, то $MB = DN$.В четырехугольнике $MBND$ противоположные стороны $MB$ и $DN$ параллельны и равны. По признаку параллелограмма, четырехугольник $MBND$ является параллелограммом.
Аналогично рассмотрим четырехугольник $M_1B_1N_1D_1$, лежащий в плоскости верхнего основания $A_1B_1C_1D_1$.Грань $A_1B_1C_1D_1$ — параллелограмм, поэтому $A_1B_1 \parallel D_1C_1$ и $A_1B_1 = D_1C_1$.Точки $M_1$ и $N_1$ лежат на ребрах $A_1B_1$ и $D_1C_1$ соответственно. Следовательно, $M_1B_1 \parallel D_1N_1$.По условию, $A_1M_1 = C_1N_1$.Длины отрезков $M_1B_1$ и $D_1N_1$ равны:$M_1B_1 = A_1B_1 - A_1M_1$$D_1N_1 = D_1C_1 - C_1N_1$Так как $A_1B_1 = D_1C_1$ и $A_1M_1 = C_1N_1$, то $M_1B_1 = D_1N_1$.В четырехугольнике $M_1B_1N_1D_1$ противоположные стороны $M_1B_1$ и $D_1N_1$ параллельны и равны. Следовательно, $M_1B_1N_1D_1$ является параллелограммом.
2. Докажем, что боковые грани $MBB_1M_1$, $BNN_1B_1$, $NDD_1N_1$ и $DMM_1D_1$ являются параллелограммами.
Сначала докажем, что отрезки $MM_1$ и $NN_1$ параллельны и равны боковым ребрам исходного параллелепипеда.Рассмотрим четырехугольник $AMM_1A_1$ в плоскости грани $ABB_1A_1$. Грань $ABB_1A_1$ — параллелограмм, поэтому $AB \parallel A_1B_1$ и $AA_1 \parallel BB_1$. Из $AB \parallel A_1B_1$ следует, что $AM \parallel A_1M_1$. По условию $AM = A_1M_1$. Четырехугольник $AMM_1A_1$, у которого две противоположные стороны ($AM$ и $A_1M_1$) параллельны и равны, является параллелограммом. Значит, $MM_1 \parallel AA_1$ и $MM_1 = AA_1$.
Теперь рассмотрим четырехугольник $CNN_1C_1$ в плоскости грани $DCC_1D_1$. Грань $DCC_1D_1$ — параллелограмм, поэтому $DC \parallel D_1C_1$ и $CC_1 \parallel DD_1$. Из $DC \parallel D_1C_1$ следует, что прямые, содержащие отрезки $CN$ и $C_1N_1$, параллельны. По условию $CN=C_1N_1$ (так как $CN=AM=A_1M_1=C_1N_1$). Рассмотрим четырехугольник $DNC_1N_1$. Он лежит в плоскости сечения, проходящего через параллельные прямые $DC$ и $D_1C_1$.Более строго, в параллелограмме $DCC_1D_1$ имеем $DD_1 \parallel CC_1$ и $DD_1=CC_1$. Векторно $\vec{DD_1} = \vec{CC_1}$. Также $\vec{DN} = \vec{MB}$ и $\vec{D_1N_1} = \vec{M_1B_1}$. Вектор $\vec{NN_1} = \vec{ND} + \vec{DD_1} + \vec{D_1N_1}$. Это сложно.Вернемся к $CNN_1C_1$. Так как $DCC_1D_1$ параллелограмм, то $\vec{DC} = \vec{D_1C_1}$ и $\vec{DD_1} = \vec{CC_1}$. Пусть $\vec{CN} = k \cdot \vec{CD}$ и $\vec{C_1N_1} = k \cdot \vec{C_1D_1}$ (поскольку $CN = C_1N_1$ и $CD = C_1D_1$, коэффициент $k$ одинаков). Тогда $\vec{NN_1} = \vec{NC} + \vec{CC_1} + \vec{C_1N_1} = -k\vec{CD} + \vec{CC_1} + k\vec{C_1D_1}$. Так как $\vec{CD} = \vec{C_1D_1}$, то $\vec{NN_1} = \vec{CC_1}$. Отсюда следует, что $NN_1 \parallel CC_1$ и $NN_1 = CC_1$.
В исходном параллелепипеде все боковые ребра параллельны и равны: $AA_1 \parallel BB_1 \parallel CC_1 \parallel DD_1$ и $AA_1 = BB_1 = CC_1 = DD_1$.Из доказанного выше ($MM_1 \parallel AA_1$, $MM_1 = AA_1$ и $NN_1 \parallel CC_1$, $NN_1 = CC_1$) следует, что все отрезки $MM_1, BB_1, NN_1, DD_1$ параллельны друг другу и равны по длине.
Теперь мы можем доказать, что боковые грани являются параллелограммами:
- Грань $MBB_1M_1$: стороны $MM_1$ и $BB_1$ параллельны и равны. Следовательно, $MBB_1M_1$ — параллелограмм.
- Грань $BNN_1B_1$: стороны $BB_1$ и $NN_1$ параллельны и равны. Следовательно, $BNN_1B_1$ — параллелограмм.
- Грань $NDD_1N_1$: стороны $NN_1$ и $DD_1$ параллельны и равны. Следовательно, $NDD_1N_1$ — параллелограмм.
- Грань $DMM_1D_1$: стороны $DD_1$ и $MM_1$ параллельны и равны. Следовательно, $DMM_1D_1$ — параллелограмм.
Заключение
Поскольку все шесть граней многогранника $MBNDM_1B_1N_1D_1$ (основания $MBND$, $M_1B_1N_1D_1$ и боковые грани $MBB_1M_1$, $BNN_1B_1$, $NDD_1N_1$, $DMM_1D_1$) являются параллелограммами, данный многогранник является параллелепипедом по определению.Что и требовалось доказать.
Ответ: Утверждение, что $MBNDM_1B_1N_1D_1$ является параллелепипедом, доказано.
Другие задания:
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