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Номер 776, страница 190 - 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 7. Метод координат в пространстве. Движения. Параграф 3. Движения, дополнительные задачи - номер 776, страница 190.
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" => 745191 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "190" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/00-776" "field_display_title" => "776" "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" => 744405 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Метод координат в пространстве. Движения" "field_branch_order" => "7" "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" => "160" "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" => 744405 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Метод координат в пространстве. Движения" "field_branch_order" => "7" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039} "field_page_start" => "160" "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" => 744405 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Метод координат в пространстве. Движения" "field_branch_order" => "7" "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" => "160" "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 [ "id" => 6477 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Геометрия" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "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 [ …6] "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 {#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 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ …6] "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 [ …6] "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 => "*" ] } 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 [ "id" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ …6] "field_translit" => "odinadcatyj" ] #original: array:6 [ "id" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ …6] "field_translit" => "odinadcatyj" ] #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 {#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 [ "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 {#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 [ "id" => 3753 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Атанасян" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Левон" "field_patronymic" => "Сергеевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 3753 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Атанасян" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Левон" "field_patronymic" => "Сергеевич" "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 {#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 [ "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 => "*" ] } 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 [ "id" => 4518 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кадомцев" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Сергей" "field_patronymic" => "Борисович" "field_surname_rp" => null ] #original: array:12 [ "id" => 4518 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кадомцев" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Сергей" "field_patronymic" => "Борисович" "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 => "*" ] } 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 [ "id" => 5494 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Позняк" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Эдуард" "field_patronymic" => "Генрихович" "field_surname_rp" => null ] #original: array:12 [ "id" => 5494 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Позняк" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Эдуард" "field_patronymic" => "Генрихович" "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 => "*" ] } 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 [ "id" => 6905 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Киселёва" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Людмила" "field_patronymic" => "Сергеевна" "field_surname_rp" => null ] #original: array:12 [ "id" => 6905 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Киселёва" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Людмила" "field_patronymic" => "Сергеевна" "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 {#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 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #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 {#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 [ "id" => 9 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Россия" "field_cases" => array:6 [ …6] "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 [ …6] "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 {#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 [ "id" => 15 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Москва" "field_cases" => array:6 [ …6] "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 [ …6] "field_translit" => "moskva" ] #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" => 744405 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Метод координат в пространстве. Движения" "field_branch_order" => "7" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1132} "field_page_start" => "160" "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" => 744410 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Движения, дополнительные задачи" "field_branch_order" => "3" "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" => "187" "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" => 744410 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Движения, дополнительные задачи" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1172} "field_page_start" => "187" "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 {#1117 #items: array:3 [ 0 => App\Models\Element {#1109 #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" => 939883 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1102 #items: array:1 [ 0 => App\Models\Edition {#1110 #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 {#1108 …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 {#1105 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1106 …2} "field_process_formula" => "wiris" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1104 …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 {#1108 …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 {#1105 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1106 …2} "field_process_formula" => "wiris" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1104 …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" => "745191" "type" => "task" ] "text" => "<p> <b>776.</b> Разбейте куб на шесть равных тетраэдров. </p>" "img" => array:1 [ 0 => array:5 [ "name" => "776-1.jpg" "alt" => null "width" => "1575" "height" => 126 "path" => "/media/geometrija_10/atanasjan-u23/0-00/776-1.webp?ts=1739436900" ] ] ] #original: array:7 [ "id" => 939883 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1102} "task" => array:2 [ "refs" => "745191" "type" => "task" ] "text" => "<p> <b>776.</b> Разбейте куб на шесть равных тетраэдров. </p>" "img" => array:1 [ 0 => array:5 [ "name" => "776-1.jpg" "alt" => null "width" => "1575" "height" => 126 "path" => "/media/geometrija_10/atanasjan-u23/0-00/776-1.webp?ts=1739436900" ] ] ] #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 {#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" => 942737 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1094 #items: array:1 [ 0 => App\Models\Edition {#1101 #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 {#1100 …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 {#1097 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1098 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1096 …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 {#1100 …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 {#1097 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1098 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1096 …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" => "745191" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "776-1.png" "alt" => null "width" => "694" "height" => 2793 "path" => "/media/geometrija_10/atanasjan-u23/2-00/776-1.webp?ts=1739441669" ] ] ] #original: array:6 [ "id" => 942737 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1094} "task" => array:2 [ "refs" => "745191" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "776-1.png" "alt" => null "width" => "694" "height" => 2793 "path" => "/media/geometrija_10/atanasjan-u23/2-00/776-1.webp?ts=1739441669" ] ] ] #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 {#1090 #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" => 1352504 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1086 #items: array:1 [ 0 => App\Models\Edition {#1095 #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" => 5120 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 6" "field_order" => "7" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1092 …2} "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" => "6-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5120 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 6" "field_order" => "7" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1092 …2} "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" => "6-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1089 …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" => "745191" "type" => "task" ] "text" => "<p>Для того чтобы разбить куб на шесть равных тетраэдров, можно использовать следующий метод, основанный на выборе одной из главных диагоналей куба.</p><p><strong>1. Определение геометрии</strong></p><p>Рассмотрим куб $ABCDA_1B_1C_1D_1$ с длиной ребра $a$. Введем систему координат так, чтобы его вершины имели следующие координаты:</p><ul><li>$A(0,0,0)$</li> <li>$B(a,0,0)$</li> <li>$C(a,a,0)$</li> <li>$D(0,a,0)$</li> <li>$A_1(0,0,a)$</li> <li>$B_1(a,0,a)$</li> <li>$C_1(a,a,a)$</li> <li>$D_1(0,a,a)$</li></ul><p>Объем куба равен $V_{куб} = a^3$. Следовательно, каждый из шести равных тетраэдров должен иметь объем $V_{тетр} = \frac{a^3}{6}$.</p><p><strong>2. Построение тетраэдров</strong></p><p>Выберем одну из четырех главных диагоналей куба. Пусть это будет диагональ $AC_1$. Эта диагональ будет служить общим ребром для всех шести тетраэдров.</p><p>Остальные вершины каждого тетраэдра будут определяться диагоналями граней куба. В кубе 12 диагоналей граней. Нам нужны те, которые не проходят через вершины $A$ или $C_1$. Таких диагоналей ровно шесть:</p><ol><li>$BD$ на грани $ABCD$</li> <li>$B_1D_1$ на грани $A_1B_1C_1D_1$</li> <li>$A_1B$ на грани $ABB_1A_1$</li> <li>$CD_1$ на грани $CDD_1C_1$</li> <li>$B_1C$ на грани $BCC_1B_1$</li> <li>$A_1D$ на грани $ADD_1A_1$</li></ol><p>Теперь построим шесть тетраэдров. Каждый тетраэдр образуется вершинами главной диагонали $A, C_1$ и двумя вершинами одной из перечисленных выше диагоналей граней.</p><ul><li>$T_1$: вершины $A, C_1, B, D$</li> <li>$T_2$: вершины $A, C_1, B_1, D_1$</li> <li>$T_3$: вершины $A, C_1, A_1, B$</li> <li>$T_4$: вершины $A, C_1, C, D_1$</li> <li>$T_5$: вершины $A, C_1, B_1, C$</li> <li>$T_6$: вершины $A, C_1, A_1, D$</li></ul><p><strong>3. Доказательство равенства тетраэдров</strong></p><p>Чтобы доказать, что все шесть тетраэдров равны (конгруэнтны), достаточно показать, что они имеют одинаковые наборы длин ребер. В силу симметрии куба, все эти тетраэдры будут конгруэнтны. Проверим это, рассчитав длины ребер для двух из них, например, для $T_1$ и $T_4$.</p><p>Для тетраэдра $T_1(A, C_1, B, D)$:</p><ul><li>$AC_1 = \sqrt{a^2+a^2+a^2} = a\sqrt{3}$ (главная диагональ)</li> <li>$AB = a$ (ребро куба)</li> <li>$AD = a$ (ребро куба)</li> <li>$C_1B = \sqrt{(a-a)^2+(0-a)^2+(0-a)^2} = a\sqrt{2}$ (диагональ грани)</li> <li>$C_1D = \sqrt{(0-a)^2+(a-a)^2+(0-a)^2} = a\sqrt{2}$ (диагональ грани)</li> <li>$BD = \sqrt{(0-a)^2+(a-0)^2+(0-0)^2} = a\sqrt{2}$ (диагональ грани)</li></ul><p>Набор длин ребер для $T_1$: $\{a, a, a\sqrt{2}, a\sqrt{2}, a\sqrt{2}, a\sqrt{3}\}$.</p><p>Для тетраэдра $T_4(A, C_1, C, D_1)$:</p><ul><li>$AC_1 = a\sqrt{3}$ (главная диагональ)</li> <li>$AC = \sqrt{a^2+a^2+0^2} = a\sqrt{2}$ (диагональ грани)</li> <li>$AD_1 = \sqrt{0^2+a^2+a^2} = a\sqrt{2}$ (диагональ грани)</li> <li>$C_1C = \sqrt{(a-a)^2+(a-a)^2+(a-0)^2} = a$ (ребро куба)</li> <li>$C_1D_1 = \sqrt{(a-0)^2+(a-a)^2+(a-a)^2} = a$ (ребро куба)</li> <li>$CD_1 = \sqrt{(0-a)^2+(a-a)^2+(a-0)^2} = a\sqrt{2}$ (диагональ грани)</li></ul><p>Набор длин ребер для $T_4$: $\{a, a, a\sqrt{2}, a\sqrt{2}, a\sqrt{2}, a\sqrt{3}\}$.</p><p>Наборы длин ребер совпадают, следовательно, тетраэдры $T_1$ и $T_4$ конгруэнтны. Аналогично можно показать конгруэнтность всех шести тетраэдров.</p><p><strong>4. Проверка объема</strong></p><p>Объем тетраэдра, построенного на векторах $\vec{u}, \vec{v}, \vec{w}$, исходящих из одной вершины, равен $V = \frac{1}{6}|(\vec{u} \times \vec{v}) \cdot \vec{w}|$.</p><p>Для тетраэдра $T_1(A, C_1, B, D)$ можно выбрать вершину $A$ и векторы $\vec{AB}$, $\vec{AD}$ и $\vec{AC_1}$. Но эти векторы не образуют тетраэдр $T_1$. Правильнее выбрать векторы, исходящие из одной из четырех вершин тетраэдра, например, A: $\vec{AB}$, $\vec{AD}$, $\vec{AC_1}$. Но эти векторы определяют тетраэдр $ABDC_1$.Объем тетраэдра с вершинами $A(x_A, y_A, z_A)$, $B(x_B, y_B, z_B)$, $C(x_C, y_C, z_C)$, $D(x_D, y_D, z_D)$ вычисляется по формуле:$V = \frac{1}{6} \left| \det \begin{pmatrix} x_B-x_A & y_B-y_A & z_B-z_A \\ x_C-x_A & y_C-y_A & z_C-z_A \\ x_D-x_A & y_D-y_A & z_D-z_A \end{pmatrix} \right|$.Для $T_1(A, C_1, B, D)$ с вершиной $A(0,0,0)$:$\vec{AB} = (a,0,0)$, $\vec{AD} = (0,a,0)$, $\vec{AC_1} = (a,a,a)$.$V_{T_1} = \frac{1}{6} \left| \det \begin{pmatrix} a & 0 & 0 \\ 0 & a & 0 \\ a & a & a \end{pmatrix} \right| = \frac{1}{6} |a(a \cdot a - 0 \cdot a)| = \frac{1}{6} a^3$.</p><p>Таким образом, объем каждого тетраэдра равен $\frac{a^3}{6}$. Суммарный объем шести таких тетраэдров равен $6 \times \frac{a^3}{6} = a^3$, что совпадает с объемом куба. Геометрически эти шесть тетраэдров полностью заполняют пространство куба, располагаясь вокруг общей оси — главной диагонали $AC_1$.</p><p><strong>Ответ:</strong> Куб разбивается на шесть равных тетраэдров. Все тетраэдры имеют общее ребро, совпадающее с одной из главных диагоналей куба. Остальные две вершины каждого тетраэдра совпадают с концами одной из шести диагоналей граней, не пересекающих выбранную главную диагональ.</p>" ] #original: array:6 [ "id" => 1352504 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1086} "task" => array:2 [ "refs" => "745191" "type" => "task" ] "text" => "<p>Для того чтобы разбить куб на шесть равных тетраэдров, можно использовать следующий метод, основанный на выборе одной из главных диагоналей куба.</p><p><strong>1. Определение геометрии</strong></p><p>Рассмотрим куб $ABCDA_1B_1C_1D_1$ с длиной ребра $a$. Введем систему координат так, чтобы его вершины имели следующие координаты:</p><ul><li>$A(0,0,0)$</li> <li>$B(a,0,0)$</li> <li>$C(a,a,0)$</li> <li>$D(0,a,0)$</li> <li>$A_1(0,0,a)$</li> <li>$B_1(a,0,a)$</li> <li>$C_1(a,a,a)$</li> <li>$D_1(0,a,a)$</li></ul><p>Объем куба равен $V_{куб} = a^3$. Следовательно, каждый из шести равных тетраэдров должен иметь объем $V_{тетр} = \frac{a^3}{6}$.</p><p><strong>2. Построение тетраэдров</strong></p><p>Выберем одну из четырех главных диагоналей куба. Пусть это будет диагональ $AC_1$. Эта диагональ будет служить общим ребром для всех шести тетраэдров.</p><p>Остальные вершины каждого тетраэдра будут определяться диагоналями граней куба. В кубе 12 диагоналей граней. Нам нужны те, которые не проходят через вершины $A$ или $C_1$. Таких диагоналей ровно шесть:</p><ol><li>$BD$ на грани $ABCD$</li> <li>$B_1D_1$ на грани $A_1B_1C_1D_1$</li> <li>$A_1B$ на грани $ABB_1A_1$</li> <li>$CD_1$ на грани $CDD_1C_1$</li> <li>$B_1C$ на грани $BCC_1B_1$</li> <li>$A_1D$ на грани $ADD_1A_1$</li></ol><p>Теперь построим шесть тетраэдров. Каждый тетраэдр образуется вершинами главной диагонали $A, C_1$ и двумя вершинами одной из перечисленных выше диагоналей граней.</p><ul><li>$T_1$: вершины $A, C_1, B, D$</li> <li>$T_2$: вершины $A, C_1, B_1, D_1$</li> <li>$T_3$: вершины $A, C_1, A_1, B$</li> <li>$T_4$: вершины $A, C_1, C, D_1$</li> <li>$T_5$: вершины $A, C_1, B_1, C$</li> <li>$T_6$: вершины $A, C_1, A_1, D$</li></ul><p><strong>3. Доказательство равенства тетраэдров</strong></p><p>Чтобы доказать, что все шесть тетраэдров равны (конгруэнтны), достаточно показать, что они имеют одинаковые наборы длин ребер. В силу симметрии куба, все эти тетраэдры будут конгруэнтны. Проверим это, рассчитав длины ребер для двух из них, например, для $T_1$ и $T_4$.</p><p>Для тетраэдра $T_1(A, C_1, B, D)$:</p><ul><li>$AC_1 = \sqrt{a^2+a^2+a^2} = a\sqrt{3}$ (главная диагональ)</li> <li>$AB = a$ (ребро куба)</li> <li>$AD = a$ (ребро куба)</li> <li>$C_1B = \sqrt{(a-a)^2+(0-a)^2+(0-a)^2} = a\sqrt{2}$ (диагональ грани)</li> <li>$C_1D = \sqrt{(0-a)^2+(a-a)^2+(0-a)^2} = a\sqrt{2}$ (диагональ грани)</li> <li>$BD = \sqrt{(0-a)^2+(a-0)^2+(0-0)^2} = a\sqrt{2}$ (диагональ грани)</li></ul><p>Набор длин ребер для $T_1$: $\{a, a, a\sqrt{2}, a\sqrt{2}, a\sqrt{2}, a\sqrt{3}\}$.</p><p>Для тетраэдра $T_4(A, C_1, C, D_1)$:</p><ul><li>$AC_1 = a\sqrt{3}$ (главная диагональ)</li> <li>$AC = \sqrt{a^2+a^2+0^2} = a\sqrt{2}$ (диагональ грани)</li> <li>$AD_1 = \sqrt{0^2+a^2+a^2} = a\sqrt{2}$ (диагональ грани)</li> <li>$C_1C = \sqrt{(a-a)^2+(a-a)^2+(a-0)^2} = a$ (ребро куба)</li> <li>$C_1D_1 = \sqrt{(a-0)^2+(a-a)^2+(a-a)^2} = a$ (ребро куба)</li> <li>$CD_1 = \sqrt{(0-a)^2+(a-a)^2+(a-0)^2} = a\sqrt{2}$ (диагональ грани)</li></ul><p>Набор длин ребер для $T_4$: $\{a, a, a\sqrt{2}, a\sqrt{2}, a\sqrt{2}, a\sqrt{3}\}$.</p><p>Наборы длин ребер совпадают, следовательно, тетраэдры $T_1$ и $T_4$ конгруэнтны. Аналогично можно показать конгруэнтность всех шести тетраэдров.</p><p><strong>4. Проверка объема</strong></p><p>Объем тетраэдра, построенного на векторах $\vec{u}, \vec{v}, \vec{w}$, исходящих из одной вершины, равен $V = \frac{1}{6}|(\vec{u} \times \vec{v}) \cdot \vec{w}|$.</p><p>Для тетраэдра $T_1(A, C_1, B, D)$ можно выбрать вершину $A$ и векторы $\vec{AB}$, $\vec{AD}$ и $\vec{AC_1}$. Но эти векторы не образуют тетраэдр $T_1$. Правильнее выбрать векторы, исходящие из одной из четырех вершин тетраэдра, например, A: $\vec{AB}$, $\vec{AD}$, $\vec{AC_1}$. Но эти векторы определяют тетраэдр $ABDC_1$.Объем тетраэдра с вершинами $A(x_A, y_A, z_A)$, $B(x_B, y_B, z_B)$, $C(x_C, y_C, z_C)$, $D(x_D, y_D, z_D)$ вычисляется по формуле:$V = \frac{1}{6} \left| \det \begin{pmatrix} x_B-x_A & y_B-y_A & z_B-z_A \\ x_C-x_A & y_C-y_A & z_C-z_A \\ x_D-x_A & y_D-y_A & z_D-z_A \end{pmatrix} \right|$.Для $T_1(A, C_1, B, D)$ с вершиной $A(0,0,0)$:$\vec{AB} = (a,0,0)$, $\vec{AD} = (0,a,0)$, $\vec{AC_1} = (a,a,a)$.$V_{T_1} = \frac{1}{6} \left| \det \begin{pmatrix} a & 0 & 0 \\ 0 & a & 0 \\ a & a & a \end{pmatrix} \right| = \frac{1}{6} |a(a \cdot a - 0 \cdot a)| = \frac{1}{6} a^3$.</p><p>Таким образом, объем каждого тетраэдра равен $\frac{a^3}{6}$. Суммарный объем шести таких тетраэдров равен $6 \times \frac{a^3}{6} = a^3$, что совпадает с объемом куба. Геометрически эти шесть тетраэдров полностью заполняют пространство куба, располагаясь вокруг общей оси — главной диагонали $AC_1$.</p><p><strong>Ответ:</strong> Куб разбивается на шесть равных тетраэдров. Все тетраэдры имеют общее ребро, совпадающее с одной из главных диагоналей куба. Остальные две вершины каждого тетраэдра совпадают с концами одной из шести диагоналей граней, не пересекающих выбранную главную диагональ.</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" => "745192" "type" => "task" ] "previous" => array:2 [ "refs" => "745190" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1113 #items: array:1 [ 0 => App\Models\Book {#1131} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1116 #items: array:1 [ 0 => App\Models\BookPage {#1081 #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" => 933972 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "190" "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/page-190" "field_display_title" => "190" "field_folder" => "1" "field_image_name" => "190" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1082 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1085 #items: array:1 [ 0 => App\Models\Book {#1131} ] #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 {#1087 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1124 #items: array:1 [ 0 => 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 [ …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 } "next" => array:2 [ "refs" => "933973" "type" => "book_page" ] "previous" => array:2 [ "refs" => "933971" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1284 #items: array:13 [ 0 => App\Models\Task {#1304 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false 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#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 {#1381 #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 {#1395 #connection: "mysql" 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+usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] }
№776 (с. 190)
Условие. №776 (с. 190)
скриншот условия
776. Разбейте куб на шесть равных тетраэдров.
Решение 2. №776 (с. 190)
Решение 6. №776 (с. 190)
Для того чтобы разбить куб на шесть равных тетраэдров, можно использовать следующий метод, основанный на выборе одной из главных диагоналей куба.
1. Определение геометрии
Рассмотрим куб $ABCDA_1B_1C_1D_1$ с длиной ребра $a$. Введем систему координат так, чтобы его вершины имели следующие координаты:
- $A(0,0,0)$
- $B(a,0,0)$
- $C(a,a,0)$
- $D(0,a,0)$
- $A_1(0,0,a)$
- $B_1(a,0,a)$
- $C_1(a,a,a)$
- $D_1(0,a,a)$
Объем куба равен $V_{куб} = a^3$. Следовательно, каждый из шести равных тетраэдров должен иметь объем $V_{тетр} = \frac{a^3}{6}$.
2. Построение тетраэдров
Выберем одну из четырех главных диагоналей куба. Пусть это будет диагональ $AC_1$. Эта диагональ будет служить общим ребром для всех шести тетраэдров.
Остальные вершины каждого тетраэдра будут определяться диагоналями граней куба. В кубе 12 диагоналей граней. Нам нужны те, которые не проходят через вершины $A$ или $C_1$. Таких диагоналей ровно шесть:
- $BD$ на грани $ABCD$
- $B_1D_1$ на грани $A_1B_1C_1D_1$
- $A_1B$ на грани $ABB_1A_1$
- $CD_1$ на грани $CDD_1C_1$
- $B_1C$ на грани $BCC_1B_1$
- $A_1D$ на грани $ADD_1A_1$
Теперь построим шесть тетраэдров. Каждый тетраэдр образуется вершинами главной диагонали $A, C_1$ и двумя вершинами одной из перечисленных выше диагоналей граней.
- $T_1$: вершины $A, C_1, B, D$
- $T_2$: вершины $A, C_1, B_1, D_1$
- $T_3$: вершины $A, C_1, A_1, B$
- $T_4$: вершины $A, C_1, C, D_1$
- $T_5$: вершины $A, C_1, B_1, C$
- $T_6$: вершины $A, C_1, A_1, D$
3. Доказательство равенства тетраэдров
Чтобы доказать, что все шесть тетраэдров равны (конгруэнтны), достаточно показать, что они имеют одинаковые наборы длин ребер. В силу симметрии куба, все эти тетраэдры будут конгруэнтны. Проверим это, рассчитав длины ребер для двух из них, например, для $T_1$ и $T_4$.
Для тетраэдра $T_1(A, C_1, B, D)$:
- $AC_1 = \sqrt{a^2+a^2+a^2} = a\sqrt{3}$ (главная диагональ)
- $AB = a$ (ребро куба)
- $AD = a$ (ребро куба)
- $C_1B = \sqrt{(a-a)^2+(0-a)^2+(0-a)^2} = a\sqrt{2}$ (диагональ грани)
- $C_1D = \sqrt{(0-a)^2+(a-a)^2+(0-a)^2} = a\sqrt{2}$ (диагональ грани)
- $BD = \sqrt{(0-a)^2+(a-0)^2+(0-0)^2} = a\sqrt{2}$ (диагональ грани)
Набор длин ребер для $T_1$: $\{a, a, a\sqrt{2}, a\sqrt{2}, a\sqrt{2}, a\sqrt{3}\}$.
Для тетраэдра $T_4(A, C_1, C, D_1)$:
- $AC_1 = a\sqrt{3}$ (главная диагональ)
- $AC = \sqrt{a^2+a^2+0^2} = a\sqrt{2}$ (диагональ грани)
- $AD_1 = \sqrt{0^2+a^2+a^2} = a\sqrt{2}$ (диагональ грани)
- $C_1C = \sqrt{(a-a)^2+(a-a)^2+(a-0)^2} = a$ (ребро куба)
- $C_1D_1 = \sqrt{(a-0)^2+(a-a)^2+(a-a)^2} = a$ (ребро куба)
- $CD_1 = \sqrt{(0-a)^2+(a-a)^2+(a-0)^2} = a\sqrt{2}$ (диагональ грани)
Набор длин ребер для $T_4$: $\{a, a, a\sqrt{2}, a\sqrt{2}, a\sqrt{2}, a\sqrt{3}\}$.
Наборы длин ребер совпадают, следовательно, тетраэдры $T_1$ и $T_4$ конгруэнтны. Аналогично можно показать конгруэнтность всех шести тетраэдров.
4. Проверка объема
Объем тетраэдра, построенного на векторах $\vec{u}, \vec{v}, \vec{w}$, исходящих из одной вершины, равен $V = \frac{1}{6}|(\vec{u} \times \vec{v}) \cdot \vec{w}|$.
Для тетраэдра $T_1(A, C_1, B, D)$ можно выбрать вершину $A$ и векторы $\vec{AB}$, $\vec{AD}$ и $\vec{AC_1}$. Но эти векторы не образуют тетраэдр $T_1$. Правильнее выбрать векторы, исходящие из одной из четырех вершин тетраэдра, например, A: $\vec{AB}$, $\vec{AD}$, $\vec{AC_1}$. Но эти векторы определяют тетраэдр $ABDC_1$.Объем тетраэдра с вершинами $A(x_A, y_A, z_A)$, $B(x_B, y_B, z_B)$, $C(x_C, y_C, z_C)$, $D(x_D, y_D, z_D)$ вычисляется по формуле:$V = \frac{1}{6} \left| \det \begin{pmatrix} x_B-x_A & y_B-y_A & z_B-z_A \\ x_C-x_A & y_C-y_A & z_C-z_A \\ x_D-x_A & y_D-y_A & z_D-z_A \end{pmatrix} \right|$.Для $T_1(A, C_1, B, D)$ с вершиной $A(0,0,0)$:$\vec{AB} = (a,0,0)$, $\vec{AD} = (0,a,0)$, $\vec{AC_1} = (a,a,a)$.$V_{T_1} = \frac{1}{6} \left| \det \begin{pmatrix} a & 0 & 0 \\ 0 & a & 0 \\ a & a & a \end{pmatrix} \right| = \frac{1}{6} |a(a \cdot a - 0 \cdot a)| = \frac{1}{6} a^3$.
Таким образом, объем каждого тетраэдра равен $\frac{a^3}{6}$. Суммарный объем шести таких тетраэдров равен $6 \times \frac{a^3}{6} = a^3$, что совпадает с объемом куба. Геометрически эти шесть тетраэдров полностью заполняют пространство куба, располагаясь вокруг общей оси — главной диагонали $AC_1$.
Ответ: Куб разбивается на шесть равных тетраэдров. Все тетраэдры имеют общее ребро, совпадающее с одной из главных диагоналей куба. Остальные две вершины каждого тетраэдра совпадают с концами одной из шести диагоналей граней, не пересекающих выбранную главную диагональ.
Другие задания:
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