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Номер 287, страница 85 - 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 3. Многогранники. Параграф 3. Правильные многогранники - номер 287, страница 85.
App\Models\Task {#1024 // 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" => 744702 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "85" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/00-287" "field_display_title" => "287" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#331 #items: array:1 [ 0 => App\Models\Term {#1029 #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 {#1026 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#342 #items: array:1 [ 0 => App\Models\Branch {#1023 #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" => 744380 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Многогранники" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333 #items: array:1 [ 0 => App\Models\Term {#1033 #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" => "63" "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 {#1030 #items: array:1 [ 0 => App\Models\Book {#1032 #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 {#1034 #items: array:1 [ 0 => App\Models\Term {#1035 #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 {#1036 #items: array:2 [ 0 => App\Models\Term {#1037 #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 {#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: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 {#1039 #items: array:1 [ 0 => App\Models\Term {#1040 #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 {#1041 #items: array:5 [ 0 => App\Models\Term {#1042 #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 {#1043 #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 {#1044 #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 {#1045 #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 {#1046 #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 {#1047 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1048 #items: array:1 [ 0 => App\Models\Term {#1049 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ …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 {#1050 #items: array:1 [ 0 => App\Models\Term {#1051 #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 {#1052 #items: array:1 [ 0 => App\Models\Term {#1053 #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 {#1054 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1055 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1056 #items: array:1 [ 0 => App\Models\Term {#1057 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array: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 {#1058 #items: array:1 [ 0 => App\Models\Term {#1059 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ …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 {#1060 #items: array:1 [ 0 => App\Models\Term {#1061 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array: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 {#1062 #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 {#1063 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1064 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1065 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1066 #items: array:1 [ 0 => App\Models\Term {#1067 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array: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 {#1034} "field_class" => Illuminate\Database\Eloquent\Collection {#1036} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1039} "field_author" => Illuminate\Database\Eloquent\Collection {#1041} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1047} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1048} "field_country" => Illuminate\Database\Eloquent\Collection {#1050} "field_city" => Illuminate\Database\Eloquent\Collection {#1052} "field_series" => Illuminate\Database\Eloquent\Collection {#1054} "field_umk" => Illuminate\Database\Eloquent\Collection {#1055} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1056} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1058} "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1060} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1062} "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 {#1063} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1064} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1065} "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1066} "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 {#1068 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 744380 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Многогранники" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333} "field_page_start" => "63" "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 {#1030} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1068} ] #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 {#1069 #items: array:2 [ 0 => App\Models\Branch {#1023} 1 => App\Models\Branch {#1070 #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" => 744383 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Правильные многогранники" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1071 #items: array:1 [ 0 => App\Models\Term {#1072 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array: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" => "78" "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 {#1073 #items: array:1 [ 0 => App\Models\Book {#1032} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1074 #items: array:1 [ 0 => App\Models\Branch {#1075 #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" => 744380 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Многогранники" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1076 …2} "field_page_start" => "63" "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 {#1077 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1078 …2} ] #original: array:24 [ "id" => 744380 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Многогранники" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1076 …2} "field_page_start" => "63" "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 {#1077 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1078 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 744383 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Правильные многогранники" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1071} "field_page_start" => "78" "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 {#1073} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1074} ] #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 {#1079 #items: array:6 [ 0 => App\Models\Element {#1080 #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" => 938936 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1081 #items: array:1 [ 0 => App\Models\Edition {#1082 #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 {#1083 …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 {#1085 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_process_formula" => "wiris" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1087 …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 {#1083 …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" …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 } "task" => array:2 [ "refs" => "744702" "type" => "task" ] "text" => "<p> <b>287.</b> Ребро правильного октаэдра равно <i>а</i>. Найдите расстояние между: а) двумя его противоположными вершинами; б) центрами двух смежных граней; в) противоположными гранями. </p>" "img" => array:1 [ 0 => array:5 [ "name" => "287-1.jpg" "alt" => null "width" => "1575" "height" => 233 "path" => "/media/geometrija_10/atanasjan-u23/0-00/287-1.webp?ts=1739436519" ] ] ] #original: array:7 [ "id" => 938936 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1081} "task" => array:2 [ "refs" => "744702" "type" => "task" ] "text" => "<p> <b>287.</b> Ребро правильного октаэдра равно <i>а</i>. Найдите расстояние между: а) двумя его противоположными вершинами; б) центрами двух смежных граней; в) противоположными гранями. </p>" "img" => array:1 [ 0 => array:5 [ "name" => "287-1.jpg" "alt" => null "width" => "1575" "height" => 233 "path" => "/media/geometrija_10/atanasjan-u23/0-00/287-1.webp?ts=1739436519" ] ] ] #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 {#1088 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 941041 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1089 #items: array:1 [ 0 => App\Models\Edition {#1090 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ …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" => "744702" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "287-1.png" "alt" => null "width" => "694" "height" => 2204 "path" => "/media/geometrija_10/atanasjan-u23/2-00/287-1.webp?ts=1739441139" ] 1 => array:5 [ "name" => "287-2.png" "alt" => null "width" => "694" "height" => 3607 "path" => "/media/geometrija_10/atanasjan-u23/2-00/287-2.webp?ts=1739441139" ] 2 => array:5 [ "name" => "287-3.png" "alt" => null "width" => "694" "height" => 5197 "path" => "/media/geometrija_10/atanasjan-u23/2-00/287-3.webp?ts=1739441139" ] ] ] #original: array:6 [ "id" => 941041 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1089} "task" => array:2 [ "refs" => "744702" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "287-1.png" "alt" => null "width" => "694" "height" => 2204 "path" => "/media/geometrija_10/atanasjan-u23/2-00/287-1.webp?ts=1739441139" ] 1 => array:5 [ "name" => "287-2.png" "alt" => null "width" => "694" "height" => 3607 "path" => "/media/geometrija_10/atanasjan-u23/2-00/287-2.webp?ts=1739441139" ] 2 => array:5 [ "name" => "287-3.png" "alt" => null "width" => "694" "height" => 5197 "path" => "/media/geometrija_10/atanasjan-u23/2-00/287-3.webp?ts=1739441139" ] ] ] #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 {#1096 #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" => 940804 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1097 #items: array:1 [ 0 => App\Models\Edition {#1098 #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" => "744702" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "287-1.jpg" "alt" => null "width" => "1360" "height" => 931 "path" => "/media/geometrija_10/atanasjan-u23/3-00/287-1.webp?ts=1739436757" ] 1 => array:5 [ "name" => "287-2.jpg" "alt" => null "width" => "1360" "height" => 2111 "path" => "/media/geometrija_10/atanasjan-u23/3-00/287-2.webp?ts=1739436757" ] 2 => array:5 [ "name" => "287-3.jpg" "alt" => null "width" => "1360" "height" => 1250 "path" => "/media/geometrija_10/atanasjan-u23/3-00/287-3.webp?ts=1739436757" ] ] ] #original: array:6 [ "id" => 940804 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1097} "task" => array:2 [ "refs" => "744702" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "287-1.jpg" "alt" => null "width" => "1360" "height" => 931 "path" => "/media/geometrija_10/atanasjan-u23/3-00/287-1.webp?ts=1739436757" ] 1 => array:5 [ "name" => "287-2.jpg" "alt" => null "width" => "1360" "height" => 2111 "path" => "/media/geometrija_10/atanasjan-u23/3-00/287-2.webp?ts=1739436757" ] 2 => array:5 [ "name" => "287-3.jpg" "alt" => null "width" => "1360" "height" => 1250 "path" => "/media/geometrija_10/atanasjan-u23/3-00/287-3.webp?ts=1739436757" ] ] ] #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 {#1104 #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" => 940754 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1105 #items: array:1 [ 0 => App\Models\Edition {#1106 #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" => "744702" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "287-1.jpg" "alt" => null "width" => "1124" "height" => 1432 "path" => "/media/geometrija_10/atanasjan-u23/4-00/287-1.webp?ts=1739436506" ] ] ] #original: array:6 [ "id" => 940754 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1105} "task" => array:2 [ "refs" => "744702" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "287-1.jpg" "alt" => null "width" => "1124" "height" => 1432 "path" => "/media/geometrija_10/atanasjan-u23/4-00/287-1.webp?ts=1739436506" ] ] ] #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 {#1112 #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" => 940881 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1113 #items: array:1 [ 0 => App\Models\Edition {#1114 #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" => "744702" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "287-1.jpg" "alt" => null "width" => "582" "height" => 1602 "path" => "/media/geometrija_10/atanasjan-u23/5-00/287-1.webp?ts=1739436538" ] ] ] #original: array:6 [ "id" => 940881 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1113} "task" => array:2 [ "refs" => "744702" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "287-1.jpg" "alt" => null "width" => "582" "height" => 1602 "path" => "/media/geometrija_10/atanasjan-u23/5-00/287-1.webp?ts=1739436538" ] ] ] #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 {#1120 #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" => 1352015 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1121 #items: array:1 [ 0 => App\Models\Edition {#1122 #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" => "744702" "type" => "task" ] "text" => "<p>Правильный октаэдр — это многогранник, состоящий из восьми граней, каждая из которых является равносторонним треугольником. Все 12 ребер октаэдра имеют одинаковую длину $a$. Удобно представить октаэдр как две правильные четырехугольные пирамиды, соединенные общим основанием, которое является квадратом.</p><p><strong>а) двумя его противоположными вершинами;</strong></p><p>Пусть октаэдр состоит из двух пирамид с общим квадратным основанием $BCDE$ и вершинами-полюсами $A$ и $F$. Все ребра октаэдра равны $a$, значит, стороны квадрата $BCDE$ также равны $a$ (например, $BC=CD=a$).</p><p>Противоположные вершины — это, например, вершины $B$ и $D$, лежащие в плоскости основания, или вершины-полюсы $A$ и $F$.</p><p>Расстояние между вершинами $B$ и $D$ — это диагональ квадрата $BCDE$. Рассмотрим прямоугольный треугольник $\triangle BCD$. По теореме Пифагора:</p><p>$BD^2 = BC^2 + CD^2 = a^2 + a^2 = 2a^2$</p><p>Отсюда находим расстояние $BD$:</p><p>$BD = \sqrt{2a^2} = a\sqrt{2}$</p><p>Это расстояние одинаково для любой пары противоположных вершин правильного октаэдра.</p><p>Ответ: $a\sqrt{2}$</p><p><strong>б) центрами двух смежных граней;</strong></p><p>Смежные грани — это грани, имеющие общее ребро. Для решения этой задачи воспользуемся координатным методом. Разместим центр октаэдра в начале координат $O(0,0,0)$. Как было показано в пункте (а), расстояние между противоположными вершинами (диагональ октаэдра) равно $a\sqrt{2}$. Следовательно, расстояние от центра до любой вершины равно половине этой диагонали, то есть $\frac{a\sqrt{2}}{2}$.</p><p>Расположим вершины на осях координат: $A(\frac{a\sqrt{2}}{2}, 0, 0)$, $B(-\frac{a\sqrt{2}}{2}, 0, 0)$, $C(0, \frac{a\sqrt{2}}{2}, 0)$, $D(0, -\frac{a\sqrt{2}}{2}, 0)$, $E(0, 0, \frac{a\sqrt{2}}{2})$, $F(0, 0, -\frac{a\sqrt{2}}{2})$.</p><p>Возьмем две смежные грани, например, грань с вершинами $A, C, E$ и грань с вершинами $A, D, E$. Они имеют общее ребро $AE$.</p><p>Центр грани (правильного треугольника) — это ее центроид, координаты которого равны среднему арифметическому координат вершин.</p><p>Найдем центр $M_1$ грани $ACE$:</p><p>$M_1 = \left(\frac{a\sqrt{2}/2+0+0}{3}, \frac{0+a\sqrt{2}/2+0}{3}, \frac{0+0+a\sqrt{2}/2}{3}\right) = \left(\frac{a\sqrt{2}}{6}, \frac{a\sqrt{2}}{6}, \frac{a\sqrt{2}}{6}\right)$</p><p>Найдем центр $M_2$ грани $ADE$:</p><p>$M_2 = \left(\frac{a\sqrt{2}/2+0+0}{3}, \frac{0-a\sqrt{2}/2+0}{3}, \frac{0+0+a\sqrt{2}/2}{3}\right) = \left(\frac{a\sqrt{2}}{6}, -\frac{a\sqrt{2}}{6}, \frac{a\sqrt{2}}{6}\right)$</p><p>Теперь найдем расстояние $d$ между точками $M_1$ и $M_2$:</p><p>$d^2 = \left(\frac{a\sqrt{2}}{6}-\frac{a\sqrt{2}}{6}\right)^2 + \left(\frac{a\sqrt{2}}{6}-\left(-\frac{a\sqrt{2}}{6}\right)\right)^2 + \left(\frac{a\sqrt{2}}{6}-\frac{a\sqrt{2}}{6}\right)^2 = 0^2 + \left(\frac{2a\sqrt{2}}{6}\right)^2 + 0^2 = \left(\frac{a\sqrt{2}}{3}\right)^2 = \frac{2a^2}{9}$</p><p>$d = \sqrt{\frac{2a^2}{9}} = \frac{a\sqrt{2}}{3}$</p><p>Ответ: $\frac{a\sqrt{2}}{3}$</p><p><strong>в) противоположными гранями.</strong></p><p>Противоположные грани октаэдра параллельны. Для решения этой задачи используем ту же систему координат, что и в пункте (б), где вершины октаэдра расположены на осях. Обозначим $k = \frac{a\sqrt{2}}{2}$ — расстояние от центра до вершины. Координаты вершин: $(\pm k, 0, 0)$, $(0, \pm k, 0)$, $(0, 0, \pm k)$.</p><p>Рассмотрим две противоположные грани: грань $F_1$ с вершинами $(k,0,0), (0,k,0), (0,0,k)$ и грань $F_2$ с вершинами $(-k,0,0), (0,-k,0), (0,0,-k)$.</p><p>Найдем уравнение плоскости, содержащей грань $F_1$. Вектор нормали $\vec{n}$ к этой плоскости можно взять как $(1,1,1)$. Уравнение плоскости, проходящей через точку $(k,0,0)$: $1(x-k) + 1(y-0) + 1(z-0) = 0$, что дает $x+y+z-k=0$.</p><p>Плоскость $F_2$ параллельна $F_1$ и проходит через точку $(-k,0,0)$. Ее уравнение: $1(x-(-k)) + 1(y-0) + 1(z-0) = 0$, что дает $x+y+z+k=0$.</p><p>Расстояние между двумя параллельными плоскостями $Ax+By+Cz+D_1=0$ и $Ax+By+Cz+D_2=0$ находится по формуле $d = \frac{|D_1 - D_2|}{\sqrt{A^2+B^2+C^2}}$.</p><p>$d = \frac{|-k - k|}{\sqrt{1^2+1^2+1^2}} = \frac{|-2k|}{\sqrt{3}} = \frac{2k}{\sqrt{3}}$</p><p>Подставим значение $k = \frac{a\sqrt{2}}{2}$:</p><p>$d = \frac{2(a\sqrt{2}/2)}{\sqrt{3}} = \frac{a\sqrt{2}}{\sqrt{3}} = \frac{a\sqrt{2}\sqrt{3}}{3} = \frac{a\sqrt{6}}{3}$</p><p>Ответ: $\frac{a\sqrt{6}}{3}$</p>" ] #original: array:6 [ "id" => 1352015 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1121} "task" => array:2 [ "refs" => "744702" "type" => "task" ] "text" => "<p>Правильный октаэдр — это многогранник, состоящий из восьми граней, каждая из которых является равносторонним треугольником. Все 12 ребер октаэдра имеют одинаковую длину $a$. Удобно представить октаэдр как две правильные четырехугольные пирамиды, соединенные общим основанием, которое является квадратом.</p><p><strong>а) двумя его противоположными вершинами;</strong></p><p>Пусть октаэдр состоит из двух пирамид с общим квадратным основанием $BCDE$ и вершинами-полюсами $A$ и $F$. Все ребра октаэдра равны $a$, значит, стороны квадрата $BCDE$ также равны $a$ (например, $BC=CD=a$).</p><p>Противоположные вершины — это, например, вершины $B$ и $D$, лежащие в плоскости основания, или вершины-полюсы $A$ и $F$.</p><p>Расстояние между вершинами $B$ и $D$ — это диагональ квадрата $BCDE$. Рассмотрим прямоугольный треугольник $\triangle BCD$. По теореме Пифагора:</p><p>$BD^2 = BC^2 + CD^2 = a^2 + a^2 = 2a^2$</p><p>Отсюда находим расстояние $BD$:</p><p>$BD = \sqrt{2a^2} = a\sqrt{2}$</p><p>Это расстояние одинаково для любой пары противоположных вершин правильного октаэдра.</p><p>Ответ: $a\sqrt{2}$</p><p><strong>б) центрами двух смежных граней;</strong></p><p>Смежные грани — это грани, имеющие общее ребро. Для решения этой задачи воспользуемся координатным методом. Разместим центр октаэдра в начале координат $O(0,0,0)$. Как было показано в пункте (а), расстояние между противоположными вершинами (диагональ октаэдра) равно $a\sqrt{2}$. Следовательно, расстояние от центра до любой вершины равно половине этой диагонали, то есть $\frac{a\sqrt{2}}{2}$.</p><p>Расположим вершины на осях координат: $A(\frac{a\sqrt{2}}{2}, 0, 0)$, $B(-\frac{a\sqrt{2}}{2}, 0, 0)$, $C(0, \frac{a\sqrt{2}}{2}, 0)$, $D(0, -\frac{a\sqrt{2}}{2}, 0)$, $E(0, 0, \frac{a\sqrt{2}}{2})$, $F(0, 0, -\frac{a\sqrt{2}}{2})$.</p><p>Возьмем две смежные грани, например, грань с вершинами $A, C, E$ и грань с вершинами $A, D, E$. Они имеют общее ребро $AE$.</p><p>Центр грани (правильного треугольника) — это ее центроид, координаты которого равны среднему арифметическому координат вершин.</p><p>Найдем центр $M_1$ грани $ACE$:</p><p>$M_1 = \left(\frac{a\sqrt{2}/2+0+0}{3}, \frac{0+a\sqrt{2}/2+0}{3}, \frac{0+0+a\sqrt{2}/2}{3}\right) = \left(\frac{a\sqrt{2}}{6}, \frac{a\sqrt{2}}{6}, \frac{a\sqrt{2}}{6}\right)$</p><p>Найдем центр $M_2$ грани $ADE$:</p><p>$M_2 = \left(\frac{a\sqrt{2}/2+0+0}{3}, \frac{0-a\sqrt{2}/2+0}{3}, \frac{0+0+a\sqrt{2}/2}{3}\right) = \left(\frac{a\sqrt{2}}{6}, -\frac{a\sqrt{2}}{6}, \frac{a\sqrt{2}}{6}\right)$</p><p>Теперь найдем расстояние $d$ между точками $M_1$ и $M_2$:</p><p>$d^2 = \left(\frac{a\sqrt{2}}{6}-\frac{a\sqrt{2}}{6}\right)^2 + \left(\frac{a\sqrt{2}}{6}-\left(-\frac{a\sqrt{2}}{6}\right)\right)^2 + \left(\frac{a\sqrt{2}}{6}-\frac{a\sqrt{2}}{6}\right)^2 = 0^2 + \left(\frac{2a\sqrt{2}}{6}\right)^2 + 0^2 = \left(\frac{a\sqrt{2}}{3}\right)^2 = \frac{2a^2}{9}$</p><p>$d = \sqrt{\frac{2a^2}{9}} = \frac{a\sqrt{2}}{3}$</p><p>Ответ: $\frac{a\sqrt{2}}{3}$</p><p><strong>в) противоположными гранями.</strong></p><p>Противоположные грани октаэдра параллельны. Для решения этой задачи используем ту же систему координат, что и в пункте (б), где вершины октаэдра расположены на осях. Обозначим $k = \frac{a\sqrt{2}}{2}$ — расстояние от центра до вершины. Координаты вершин: $(\pm k, 0, 0)$, $(0, \pm k, 0)$, $(0, 0, \pm k)$.</p><p>Рассмотрим две противоположные грани: грань $F_1$ с вершинами $(k,0,0), (0,k,0), (0,0,k)$ и грань $F_2$ с вершинами $(-k,0,0), (0,-k,0), (0,0,-k)$.</p><p>Найдем уравнение плоскости, содержащей грань $F_1$. Вектор нормали $\vec{n}$ к этой плоскости можно взять как $(1,1,1)$. Уравнение плоскости, проходящей через точку $(k,0,0)$: $1(x-k) + 1(y-0) + 1(z-0) = 0$, что дает $x+y+z-k=0$.</p><p>Плоскость $F_2$ параллельна $F_1$ и проходит через точку $(-k,0,0)$. Ее уравнение: $1(x-(-k)) + 1(y-0) + 1(z-0) = 0$, что дает $x+y+z+k=0$.</p><p>Расстояние между двумя параллельными плоскостями $Ax+By+Cz+D_1=0$ и $Ax+By+Cz+D_2=0$ находится по формуле $d = \frac{|D_1 - D_2|}{\sqrt{A^2+B^2+C^2}}$.</p><p>$d = \frac{|-k - k|}{\sqrt{1^2+1^2+1^2}} = \frac{|-2k|}{\sqrt{3}} = \frac{2k}{\sqrt{3}}$</p><p>Подставим значение $k = \frac{a\sqrt{2}}{2}$:</p><p>$d = \frac{2(a\sqrt{2}/2)}{\sqrt{3}} = \frac{a\sqrt{2}}{\sqrt{3}} = \frac{a\sqrt{2}\sqrt{3}}{3} = \frac{a\sqrt{6}}{3}$</p><p>Ответ: $\frac{a\sqrt{6}}{3}$</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" => "745469" "type" => "task" ] "previous" => array:2 [ "refs" => "744701" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1128 #items: array:1 [ 0 => App\Models\Book {#1032} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1143 #items: array:1 [ 0 => App\Models\BookPage {#1142 #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" => 933867 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "85" "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/page-85" "field_display_title" => "85" "field_folder" => "1" "field_image_name" => "85" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1144 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1182 #items: array:1 [ 0 => App\Models\Book {#1145 #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 [ …50] #original: array:50 [ …50] #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 "edition_groups" => Illuminate\Database\Eloquent\Collection {#1180 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1191 #items: array:1 [ 0 => App\Models\Element {#1190 #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" => "933868" "type" => "book_page" ] "previous" => array:2 [ "refs" => "933866" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#2017 #items: array:18 [ 0 => App\Models\Task {#2041 #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 {#2147 #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 {#2202 #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 {#2257 #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 {#2312 #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 {#2367 #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 {#2422 #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 {#2477 #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 {#2600 #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 {#2608 #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 {#2612 #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 {#2646 #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] } 12 => App\Models\Task {#2662 #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] } 13 => App\Models\Task {#2678 #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] } 14 => App\Models\Task {#2694 #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] } 15 => App\Models\Task {#2710 #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] } 16 => App\Models\Task {#2726 #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] } 17 => App\Models\Task {#2742 #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" => 933867 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "85" "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/page-85" "field_display_title" => "85" "field_folder" => "1" "field_image_name" => "85" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1144} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1182} "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 {#1180} "content" => Illuminate\Database\Eloquent\Collection {#1191} "next" => array:2 [ "refs" => "933868" "type" => "book_page" ] "previous" => array:2 [ "refs" => "933866" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#2017} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 744702 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "85" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/00-287" "field_display_title" => "287" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#331} "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 {#1026} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#342} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1069} "content" => Illuminate\Database\Eloquent\Collection {#1079} "next" => array:2 [ "refs" => "745469" "type" => "task" ] "previous" => array:2 [ "refs" => "744701" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1128} "page" => array:2 [ "refs" => "933867" "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 => "*" ] }
№287 (с. 85)
Условие. №287 (с. 85)
скриншот условия
287. Ребро правильного октаэдра равно а. Найдите расстояние между: а) двумя его противоположными вершинами; б) центрами двух смежных граней; в) противоположными гранями.
Решение 2. №287 (с. 85)
Решение 3. №287 (с. 85)
Решение 4. №287 (с. 85)
Решение 5. №287 (с. 85)
Решение 6. №287 (с. 85)
Правильный октаэдр — это многогранник, состоящий из восьми граней, каждая из которых является равносторонним треугольником. Все 12 ребер октаэдра имеют одинаковую длину $a$. Удобно представить октаэдр как две правильные четырехугольные пирамиды, соединенные общим основанием, которое является квадратом.
а) двумя его противоположными вершинами;
Пусть октаэдр состоит из двух пирамид с общим квадратным основанием $BCDE$ и вершинами-полюсами $A$ и $F$. Все ребра октаэдра равны $a$, значит, стороны квадрата $BCDE$ также равны $a$ (например, $BC=CD=a$).
Противоположные вершины — это, например, вершины $B$ и $D$, лежащие в плоскости основания, или вершины-полюсы $A$ и $F$.
Расстояние между вершинами $B$ и $D$ — это диагональ квадрата $BCDE$. Рассмотрим прямоугольный треугольник $\triangle BCD$. По теореме Пифагора:
$BD^2 = BC^2 + CD^2 = a^2 + a^2 = 2a^2$
Отсюда находим расстояние $BD$:
$BD = \sqrt{2a^2} = a\sqrt{2}$
Это расстояние одинаково для любой пары противоположных вершин правильного октаэдра.
Ответ: $a\sqrt{2}$
б) центрами двух смежных граней;
Смежные грани — это грани, имеющие общее ребро. Для решения этой задачи воспользуемся координатным методом. Разместим центр октаэдра в начале координат $O(0,0,0)$. Как было показано в пункте (а), расстояние между противоположными вершинами (диагональ октаэдра) равно $a\sqrt{2}$. Следовательно, расстояние от центра до любой вершины равно половине этой диагонали, то есть $\frac{a\sqrt{2}}{2}$.
Расположим вершины на осях координат: $A(\frac{a\sqrt{2}}{2}, 0, 0)$, $B(-\frac{a\sqrt{2}}{2}, 0, 0)$, $C(0, \frac{a\sqrt{2}}{2}, 0)$, $D(0, -\frac{a\sqrt{2}}{2}, 0)$, $E(0, 0, \frac{a\sqrt{2}}{2})$, $F(0, 0, -\frac{a\sqrt{2}}{2})$.
Возьмем две смежные грани, например, грань с вершинами $A, C, E$ и грань с вершинами $A, D, E$. Они имеют общее ребро $AE$.
Центр грани (правильного треугольника) — это ее центроид, координаты которого равны среднему арифметическому координат вершин.
Найдем центр $M_1$ грани $ACE$:
$M_1 = \left(\frac{a\sqrt{2}/2+0+0}{3}, \frac{0+a\sqrt{2}/2+0}{3}, \frac{0+0+a\sqrt{2}/2}{3}\right) = \left(\frac{a\sqrt{2}}{6}, \frac{a\sqrt{2}}{6}, \frac{a\sqrt{2}}{6}\right)$
Найдем центр $M_2$ грани $ADE$:
$M_2 = \left(\frac{a\sqrt{2}/2+0+0}{3}, \frac{0-a\sqrt{2}/2+0}{3}, \frac{0+0+a\sqrt{2}/2}{3}\right) = \left(\frac{a\sqrt{2}}{6}, -\frac{a\sqrt{2}}{6}, \frac{a\sqrt{2}}{6}\right)$
Теперь найдем расстояние $d$ между точками $M_1$ и $M_2$:
$d^2 = \left(\frac{a\sqrt{2}}{6}-\frac{a\sqrt{2}}{6}\right)^2 + \left(\frac{a\sqrt{2}}{6}-\left(-\frac{a\sqrt{2}}{6}\right)\right)^2 + \left(\frac{a\sqrt{2}}{6}-\frac{a\sqrt{2}}{6}\right)^2 = 0^2 + \left(\frac{2a\sqrt{2}}{6}\right)^2 + 0^2 = \left(\frac{a\sqrt{2}}{3}\right)^2 = \frac{2a^2}{9}$
$d = \sqrt{\frac{2a^2}{9}} = \frac{a\sqrt{2}}{3}$
Ответ: $\frac{a\sqrt{2}}{3}$
в) противоположными гранями.
Противоположные грани октаэдра параллельны. Для решения этой задачи используем ту же систему координат, что и в пункте (б), где вершины октаэдра расположены на осях. Обозначим $k = \frac{a\sqrt{2}}{2}$ — расстояние от центра до вершины. Координаты вершин: $(\pm k, 0, 0)$, $(0, \pm k, 0)$, $(0, 0, \pm k)$.
Рассмотрим две противоположные грани: грань $F_1$ с вершинами $(k,0,0), (0,k,0), (0,0,k)$ и грань $F_2$ с вершинами $(-k,0,0), (0,-k,0), (0,0,-k)$.
Найдем уравнение плоскости, содержащей грань $F_1$. Вектор нормали $\vec{n}$ к этой плоскости можно взять как $(1,1,1)$. Уравнение плоскости, проходящей через точку $(k,0,0)$: $1(x-k) + 1(y-0) + 1(z-0) = 0$, что дает $x+y+z-k=0$.
Плоскость $F_2$ параллельна $F_1$ и проходит через точку $(-k,0,0)$. Ее уравнение: $1(x-(-k)) + 1(y-0) + 1(z-0) = 0$, что дает $x+y+z+k=0$.
Расстояние между двумя параллельными плоскостями $Ax+By+Cz+D_1=0$ и $Ax+By+Cz+D_2=0$ находится по формуле $d = \frac{|D_1 - D_2|}{\sqrt{A^2+B^2+C^2}}$.
$d = \frac{|-k - k|}{\sqrt{1^2+1^2+1^2}} = \frac{|-2k|}{\sqrt{3}} = \frac{2k}{\sqrt{3}}$
Подставим значение $k = \frac{a\sqrt{2}}{2}$:
$d = \frac{2(a\sqrt{2}/2)}{\sqrt{3}} = \frac{a\sqrt{2}}{\sqrt{3}} = \frac{a\sqrt{2}\sqrt{3}}{3} = \frac{a\sqrt{6}}{3}$
Ответ: $\frac{a\sqrt{6}}{3}$
Другие задания:
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