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Номер 304, страница 87 - 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, дополнительные задачи - номер 304, страница 87.
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" => 744719 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "87" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/00-304" "field_display_title" => "304" "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" => 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 {#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" => "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 {#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" => 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 {#1039} "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 {#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" => 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 {#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" => "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 {#1168 #items: array:1 [ 0 => App\Models\Book {#1131 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 36 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1133 #items: array:1 [ 0 => App\Models\Term {#1130 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1134 #items: array:2 [ 0 => App\Models\Term {#1135 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Term {#1136 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1137 #items: array:1 [ 0 => App\Models\Term {#1138 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1139 #items: array:5 [ 0 => App\Models\Term {#1140 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Term {#1141 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 2 => App\Models\Term {#1142 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 3 => App\Models\Term {#1143 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 4 => App\Models\Term {#1144 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1145 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1146 #items: array:1 [ 0 => App\Models\Term {#1147 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1148 #items: array:1 [ 0 => App\Models\Term {#1149 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1150 #items: array:1 [ 0 => App\Models\Term {#1151 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1152 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1153 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1154 #items: array:1 [ 0 => App\Models\Term {#1155 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1156 #items: array:1 [ 0 => App\Models\Term {#1157 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1158 #items: array:1 [ 0 => App\Models\Term {#1159 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1160 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => "с ромбами" "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для общеобразовательных организаций" "field_allowed" => "Допущено Министерством просвещения Российской Федерации" "field_reserve_field" => "Математика: алгебра и начала математического анализа, геометрия" "field_link_to_source" => null "field_tasks_count" => "1147" "field_priority" => "12" "field_default_folder" => "/geometrija_10/atanasjan-u23/" "field_isbn" => "978-5-09-103606-0 (2023)" "field_cover" => array:1 [ 0 => "/media/geometrija_10/atanasjan-u23/covers/cover1.webp?ts=1739363242" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_10/atanasjan-u23/covers/cover1.webp?ts=1739363242" "alt" => "" "width" => "1200" "height" => "1661" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1161 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1162 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1163 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1164 #items: array:1 [ 0 => App\Models\Term {#1165 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 543 "class_subject" => 392 ] ] #original: array:50 [ "id" => 36 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1133} "field_class" => Illuminate\Database\Eloquent\Collection {#1134} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1137} "field_author" => Illuminate\Database\Eloquent\Collection {#1139} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1145} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1146} "field_country" => Illuminate\Database\Eloquent\Collection {#1148} "field_city" => Illuminate\Database\Eloquent\Collection {#1150} "field_series" => Illuminate\Database\Eloquent\Collection {#1152} "field_umk" => Illuminate\Database\Eloquent\Collection {#1153} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1154} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1156} "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1158} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1160} "field_under_the_edition_degree" => null "field_cover_description" => "с ромбами" "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для общеобразовательных организаций" "field_allowed" => "Допущено Министерством просвещения Российской Федерации" "field_reserve_field" => "Математика: алгебра и начала математического анализа, геометрия" "field_link_to_source" => null "field_tasks_count" => "1147" "field_priority" => "12" "field_default_folder" => "/geometrija_10/atanasjan-u23/" "field_isbn" => "978-5-09-103606-0 (2023)" "field_cover" => array:1 [ 0 => "/media/geometrija_10/atanasjan-u23/covers/cover1.webp?ts=1739363242" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_10/atanasjan-u23/covers/cover1.webp?ts=1739363242" "alt" => "" "width" => "1200" "height" => "1661" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1161} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1162} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1163} "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1164} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 543 "class_subject" => 392 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1166 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 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 {#1132} "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 {#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" => 744385 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Правильные многогранники, 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" => "86" "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" => 744385 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Правильные многогранники, дополнительные задачи" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1172} "field_page_start" => "86" "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 {#1112 #items: array:5 [ 0 => App\Models\Element {#1099 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 938953 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1092 #items: array:1 [ 0 => App\Models\Edition {#1100 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 2315 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1098 …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 {#1096 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1090 …2} "field_process_formula" => "wiris" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1095 …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 {#1098 …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 {#1096 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1090 …2} "field_process_formula" => "wiris" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1095 …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" => "744719" "type" => "task" ] "text" => "<p> <b>304.</b> В правильной четырёхугольной пирамиде плоский угол при вершине равен 60°. Докажите, что двугранный угол между боковой гранью и основанием пирамиды вдвое меньше двугранного угла при боковом ребре. </p>" "img" => array:1 [ 0 => array:5 [ "name" => "304-1.jpg" "alt" => null "width" => "1532" "height" => 271 "path" => "/media/geometrija_10/atanasjan-u23/0-00/304-1.webp?ts=1739436538" ] ] ] #original: array:7 [ "id" => 938953 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1092} "task" => array:2 [ "refs" => "744719" "type" => "task" ] "text" => "<p> <b>304.</b> В правильной четырёхугольной пирамиде плоский угол при вершине равен 60°. Докажите, что двугранный угол между боковой гранью и основанием пирамиды вдвое меньше двугранного угла при боковом ребре. </p>" "img" => array:1 [ 0 => array:5 [ "name" => "304-1.jpg" "alt" => null "width" => "1532" "height" => 271 "path" => "/media/geometrija_10/atanasjan-u23/0-00/304-1.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 => "*" ] } 1 => App\Models\Element {#1123 #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" => 941123 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1179 #items: array:1 [ 0 => App\Models\Edition {#1121 #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 {#1173 …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 {#1174 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1175 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1176 …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 {#1173 …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 {#1174 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1175 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1176 …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" => "744719" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "304-1.png" "alt" => null "width" => "694" "height" => 4522 "path" => "/media/geometrija_10/atanasjan-u23/2-00/304-1.webp?ts=1739441169" ] ] ] #original: array:6 [ "id" => 941123 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1179} "task" => array:2 [ "refs" => "744719" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "304-1.png" "alt" => null "width" => "694" "height" => 4522 "path" => "/media/geometrija_10/atanasjan-u23/2-00/304-1.webp?ts=1739441169" ] ] ] #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 {#1094 #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" => 940940 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1084 #items: array:1 [ 0 => App\Models\Edition {#1093 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 2318 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 3" "field_order" => "4" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1088 …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" => "3-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_content_source" => null ] #original: array:21 [ "id" => 2318 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 3" "field_order" => "4" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1088 …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" => "3-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1087 …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" => "744719" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "304-1.jpg" "alt" => null "width" => "1328" "height" => 904 "path" => "/media/geometrija_10/atanasjan-u23/3-00/304-1.webp?ts=1739436757" ] 1 => array:5 [ "name" => "304-2.jpg" "alt" => null "width" => "1328" "height" => 1831 "path" => "/media/geometrija_10/atanasjan-u23/3-00/304-2.webp?ts=1739436757" ] ] ] #original: array:6 [ "id" => 940940 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1084} "task" => array:2 [ "refs" => "744719" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "304-1.jpg" "alt" => null "width" => "1328" "height" => 904 "path" => "/media/geometrija_10/atanasjan-u23/3-00/304-1.webp?ts=1739436757" ] 1 => array:5 [ "name" => "304-2.jpg" "alt" => null "width" => "1328" "height" => 1831 "path" => "/media/geometrija_10/atanasjan-u23/3-00/304-2.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 {#1086 #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" => 940949 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1124 #items: array:1 [ 0 => App\Models\Edition {#1083 #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" => 2320 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 5" "field_order" => "6" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1078 …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" => "5-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1118 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1127 …2} "field_content_source" => null ] #original: array:21 [ "id" => 2320 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 5" "field_order" => "6" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1078 …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" …8 ] #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" => "744719" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "304-1.jpg" "alt" => null "width" => "572" "height" => 678 "path" => "/media/geometrija_10/atanasjan-u23/5-00/304-1.webp?ts=1739436538" ] ] ] #original: array:6 [ "id" => 940949 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1124} "task" => array:2 [ "refs" => "744719" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "304-1.jpg" "alt" => null "width" => "572" "height" => 678 "path" => "/media/geometrija_10/atanasjan-u23/5-00/304-1.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 => "*" ] } 4 => App\Models\Element {#1177 #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" => 1352032 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1187 #items: array:1 [ 0 => App\Models\Edition {#1178 #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" => "744719" "type" => "task" ] "text" => "<p>Пусть дана правильная четырехугольная пирамида $SABCD$, где $S$ — вершина, а $ABCD$ — квадратное основание. По условию, плоский угол при вершине равен $60^\circ$. Это означает, что угол в каждой боковой грани при вершине $S$ равен $60^\circ$, например, $\angle ASB = 60^\circ$.</p><p>Так как пирамида правильная, ее боковые грани — равные равнобедренные треугольники. В нашем случае, $\triangle ASB$ является равнобедренным с $SA = SB$. Поскольку угол при вершине $\angle ASB = 60^\circ$, то углы при основании этого треугольника также равны $(180^\circ - 60^\circ) / 2 = 60^\circ$. Следовательно, все боковые грани пирамиды являются равносторонними треугольниками. Это означает, что все ребра пирамиды равны. Обозначим длину ребра за $a$: $AB = BC = CD = DA = SA = SB = SC = SD = a$.</p><p><strong>Нахождение двугранного угла между боковой гранью и основанием</strong></p><p>Пусть $\alpha$ — двугранный угол между боковой гранью, например, $(SBC)$, и основанием $(ABCD)$. Этот угол измеряется линейным углом, образованным двумя перпендикулярами, проведенными к линии их пересечения, то есть к ребру $BC$.</p><p>Проведем апофему $SM$ в грани $SBC$. Так как $\triangle SBC$ — равносторонний, $SM$ является его высотой и медианой, поэтому $M$ — середина $BC$ и $SM \perp BC$. Высота равностороннего треугольника со стороной $a$ равна $SM = \frac{a\sqrt{3}}{2}$.</p><p>Пусть $O$ — центр основания (точка пересечения диагоналей квадрата). Проведем отрезок $OM$. Так как $ABCD$ — квадрат, $OM$ является средней линией $\triangle ABC$ или $\triangle DBC$ (в зависимости от расположения), и $OM \perp BC$. Длина $OM$ равна половине стороны квадрата: $OM = \frac{1}{2}AB = \frac{a}{2}$.</p><p>Поскольку $SM \perp BC$ и $OM \perp BC$, то $\angle SMO$ — это линейный угол двугранного угла $\alpha$. Рассмотрим прямоугольный треугольник $\triangle SOM$ (где $\angle SOM = 90^\circ$, так как $SO$ — высота пирамиды).</p><p>Найдем косинус угла $\alpha = \angle SMO$:</p><p>$\cos \alpha = \frac{OM}{SM} = \frac{a/2}{a\sqrt{3}/2} = \frac{1}{\sqrt{3}}$.</p><p><strong>Нахождение двугранного угла при боковом ребре</strong></p><p>Пусть $\beta$ — двугранный угол при боковом ребре, например, при ребре $SB$. Этот угол образован двумя смежными боковыми гранями $(SAB)$ и $(SBC)$.</p><p>Для нахождения этого угла построим его линейный угол. Проведем в гранях $(SAB)$ и $(SBC)$ перпендикуляры к общему ребру $SB$.</p><p>В равностороннем треугольнике $\triangle SAB$ проведем высоту $AH$ к стороне $SB$. Длина этой высоты равна $AH = \frac{a\sqrt{3}}{2}$.</p><p>В равностороннем треугольнике $\triangle SBC$ проведем высоту $CH$ к стороне $SB$. Длина этой высоты равна $CH = \frac{a\sqrt{3}}{2}$.</p><p>Тогда $\angle AHC$ является линейным углом двугранного угла $\beta$. Рассмотрим треугольник $\triangle AHC$. Мы знаем длины его сторон: $AH = CH = \frac{a\sqrt{3}}{2}$. Третья сторона, $AC$, является диагональю квадрата $ABCD$, ее длина $AC = a\sqrt{2}$.</p><p>Применим к треугольнику $\triangle AHC$ теорему косинусов для нахождения угла $\beta = \angle AHC$:</p><p>$AC^2 = AH^2 + CH^2 - 2 \cdot AH \cdot CH \cdot \cos\beta$</p><p>$(a\sqrt{2})^2 = (\frac{a\sqrt{3}}{2})^2 + (\frac{a\sqrt{3}}{2})^2 - 2 \cdot \frac{a\sqrt{3}}{2} \cdot \frac{a\sqrt{3}}{2} \cdot \cos\beta$</p><p>$2a^2 = \frac{3a^2}{4} + \frac{3a^2}{4} - 2 \cdot \frac{3a^2}{4} \cdot \cos\beta$</p><p>$2a^2 = \frac{6a^2}{4} - \frac{6a^2}{4} \cdot \cos\beta$</p><p>$2a^2 = \frac{3a^2}{2} - \frac{3a^2}{2} \cdot \cos\beta$</p><p>Разделим обе части уравнения на $a^2$ (так как $a \neq 0$):</p><p>$2 = \frac{3}{2} - \frac{3}{2} \cos\beta$</p><p>$2 - \frac{3}{2} = - \frac{3}{2} \cos\beta$</p><p>$\frac{1}{2} = - \frac{3}{2} \cos\beta$</p><p>$\cos\beta = -\frac{1}{3}$.</p><p><strong>Доказательство</strong></p><p>Мы получили значения косинусов для обоих углов: $\cos \alpha = \frac{1}{\sqrt{3}}$ и $\cos\beta = -\frac{1}{3}$. Нам нужно доказать, что $\alpha = \frac{\beta}{2}$, что эквивалентно $\beta = 2\alpha$.</p><p>Воспользуемся формулой косинуса двойного угла: $\cos(2\alpha) = 2\cos^2\alpha - 1$.</p><p>Подставим найденное значение $\cos\alpha$:</p><p>$\cos(2\alpha) = 2 \cdot (\frac{1}{\sqrt{3}})^2 - 1 = 2 \cdot \frac{1}{3} - 1 = \frac{2}{3} - 1 = -\frac{1}{3}$.</p><p>Таким образом, мы получили, что $\cos(2\alpha) = -\frac{1}{3}$, и мы также знаем, что $\cos\beta = -\frac{1}{3}$. Следовательно, $\cos(2\alpha) = \cos\beta$.</p><p>Поскольку $\alpha$ — угол между плоскостью боковой грани и основанием, он острый ($0 < \alpha < 90^\circ$). Значит, $0 < 2\alpha < 180^\circ$. Угол $\beta$ — двугранный угол в пирамиде, он также находится в диапазоне $0 < \beta < 180^\circ$. Из равенства $\cos(2\alpha) = \cos\beta$ на этом интервале следует, что $2\alpha = \beta$.</p><p>Это доказывает, что двугранный угол между боковой гранью и основанием пирамиды ($\alpha$) вдвое меньше двугранного угла при боковом ребре ($\beta$).</p><p><strong>Ответ:</strong> Утверждение доказано. Мы показали, что $\cos \alpha = \frac{1}{\sqrt{3}}$ и $\cos \beta = -\frac{1}{3}$, а из формулы двойного угла $\cos(2\alpha) = 2\cos^2\alpha - 1$ следует, что $\cos(2\alpha) = \cos\beta$, откуда $2\alpha = \beta$.</p>" ] #original: array:6 [ "id" => 1352032 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1187} "task" => array:2 [ "refs" => "744719" "type" => "task" ] "text" => "<p>Пусть дана правильная четырехугольная пирамида $SABCD$, где $S$ — вершина, а $ABCD$ — квадратное основание. По условию, плоский угол при вершине равен $60^\circ$. Это означает, что угол в каждой боковой грани при вершине $S$ равен $60^\circ$, например, $\angle ASB = 60^\circ$.</p><p>Так как пирамида правильная, ее боковые грани — равные равнобедренные треугольники. В нашем случае, $\triangle ASB$ является равнобедренным с $SA = SB$. Поскольку угол при вершине $\angle ASB = 60^\circ$, то углы при основании этого треугольника также равны $(180^\circ - 60^\circ) / 2 = 60^\circ$. Следовательно, все боковые грани пирамиды являются равносторонними треугольниками. Это означает, что все ребра пирамиды равны. Обозначим длину ребра за $a$: $AB = BC = CD = DA = SA = SB = SC = SD = a$.</p><p><strong>Нахождение двугранного угла между боковой гранью и основанием</strong></p><p>Пусть $\alpha$ — двугранный угол между боковой гранью, например, $(SBC)$, и основанием $(ABCD)$. Этот угол измеряется линейным углом, образованным двумя перпендикулярами, проведенными к линии их пересечения, то есть к ребру $BC$.</p><p>Проведем апофему $SM$ в грани $SBC$. Так как $\triangle SBC$ — равносторонний, $SM$ является его высотой и медианой, поэтому $M$ — середина $BC$ и $SM \perp BC$. Высота равностороннего треугольника со стороной $a$ равна $SM = \frac{a\sqrt{3}}{2}$.</p><p>Пусть $O$ — центр основания (точка пересечения диагоналей квадрата). Проведем отрезок $OM$. Так как $ABCD$ — квадрат, $OM$ является средней линией $\triangle ABC$ или $\triangle DBC$ (в зависимости от расположения), и $OM \perp BC$. Длина $OM$ равна половине стороны квадрата: $OM = \frac{1}{2}AB = \frac{a}{2}$.</p><p>Поскольку $SM \perp BC$ и $OM \perp BC$, то $\angle SMO$ — это линейный угол двугранного угла $\alpha$. Рассмотрим прямоугольный треугольник $\triangle SOM$ (где $\angle SOM = 90^\circ$, так как $SO$ — высота пирамиды).</p><p>Найдем косинус угла $\alpha = \angle SMO$:</p><p>$\cos \alpha = \frac{OM}{SM} = \frac{a/2}{a\sqrt{3}/2} = \frac{1}{\sqrt{3}}$.</p><p><strong>Нахождение двугранного угла при боковом ребре</strong></p><p>Пусть $\beta$ — двугранный угол при боковом ребре, например, при ребре $SB$. Этот угол образован двумя смежными боковыми гранями $(SAB)$ и $(SBC)$.</p><p>Для нахождения этого угла построим его линейный угол. Проведем в гранях $(SAB)$ и $(SBC)$ перпендикуляры к общему ребру $SB$.</p><p>В равностороннем треугольнике $\triangle SAB$ проведем высоту $AH$ к стороне $SB$. Длина этой высоты равна $AH = \frac{a\sqrt{3}}{2}$.</p><p>В равностороннем треугольнике $\triangle SBC$ проведем высоту $CH$ к стороне $SB$. Длина этой высоты равна $CH = \frac{a\sqrt{3}}{2}$.</p><p>Тогда $\angle AHC$ является линейным углом двугранного угла $\beta$. Рассмотрим треугольник $\triangle AHC$. Мы знаем длины его сторон: $AH = CH = \frac{a\sqrt{3}}{2}$. Третья сторона, $AC$, является диагональю квадрата $ABCD$, ее длина $AC = a\sqrt{2}$.</p><p>Применим к треугольнику $\triangle AHC$ теорему косинусов для нахождения угла $\beta = \angle AHC$:</p><p>$AC^2 = AH^2 + CH^2 - 2 \cdot AH \cdot CH \cdot \cos\beta$</p><p>$(a\sqrt{2})^2 = (\frac{a\sqrt{3}}{2})^2 + (\frac{a\sqrt{3}}{2})^2 - 2 \cdot \frac{a\sqrt{3}}{2} \cdot \frac{a\sqrt{3}}{2} \cdot \cos\beta$</p><p>$2a^2 = \frac{3a^2}{4} + \frac{3a^2}{4} - 2 \cdot \frac{3a^2}{4} \cdot \cos\beta$</p><p>$2a^2 = \frac{6a^2}{4} - \frac{6a^2}{4} \cdot \cos\beta$</p><p>$2a^2 = \frac{3a^2}{2} - \frac{3a^2}{2} \cdot \cos\beta$</p><p>Разделим обе части уравнения на $a^2$ (так как $a \neq 0$):</p><p>$2 = \frac{3}{2} - \frac{3}{2} \cos\beta$</p><p>$2 - \frac{3}{2} = - \frac{3}{2} \cos\beta$</p><p>$\frac{1}{2} = - \frac{3}{2} \cos\beta$</p><p>$\cos\beta = -\frac{1}{3}$.</p><p><strong>Доказательство</strong></p><p>Мы получили значения косинусов для обоих углов: $\cos \alpha = \frac{1}{\sqrt{3}}$ и $\cos\beta = -\frac{1}{3}$. Нам нужно доказать, что $\alpha = \frac{\beta}{2}$, что эквивалентно $\beta = 2\alpha$.</p><p>Воспользуемся формулой косинуса двойного угла: $\cos(2\alpha) = 2\cos^2\alpha - 1$.</p><p>Подставим найденное значение $\cos\alpha$:</p><p>$\cos(2\alpha) = 2 \cdot (\frac{1}{\sqrt{3}})^2 - 1 = 2 \cdot \frac{1}{3} - 1 = \frac{2}{3} - 1 = -\frac{1}{3}$.</p><p>Таким образом, мы получили, что $\cos(2\alpha) = -\frac{1}{3}$, и мы также знаем, что $\cos\beta = -\frac{1}{3}$. Следовательно, $\cos(2\alpha) = \cos\beta$.</p><p>Поскольку $\alpha$ — угол между плоскостью боковой грани и основанием, он острый ($0 < \alpha < 90^\circ$). Значит, $0 < 2\alpha < 180^\circ$. Угол $\beta$ — двугранный угол в пирамиде, он также находится в диапазоне $0 < \beta < 180^\circ$. Из равенства $\cos(2\alpha) = \cos\beta$ на этом интервале следует, что $2\alpha = \beta$.</p><p>Это доказывает, что двугранный угол между боковой гранью и основанием пирамиды ($\alpha$) вдвое меньше двугранного угла при боковом ребре ($\beta$).</p><p><strong>Ответ:</strong> Утверждение доказано. Мы показали, что $\cos \alpha = \frac{1}{\sqrt{3}}$ и $\cos \beta = -\frac{1}{3}$, а из формулы двойного угла $\cos(2\alpha) = 2\cos^2\alpha - 1$ следует, что $\cos(2\alpha) = \cos\beta$, откуда $2\alpha = \beta$.</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" => "744720" "type" => "task" ] "previous" => array:2 [ "refs" => "744718" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1101 #items: array:1 [ 0 => App\Models\Book {#1131} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1451 #items: array:1 [ 0 => App\Models\BookPage {#1032 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: 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#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" => "933870" "type" => "book_page" ] "previous" => array:2 [ "refs" => "933868" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1213 #items: array:12 [ 0 => App\Models\Task {#1214 #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 {#1270 #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 {#1287 #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: [] 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[ …1] } 6 => App\Models\Task {#1355 #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 {#1372 #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 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#dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] }
№304 (с. 87)
Условие. №304 (с. 87)
скриншот условия
304. В правильной четырёхугольной пирамиде плоский угол при вершине равен 60°. Докажите, что двугранный угол между боковой гранью и основанием пирамиды вдвое меньше двугранного угла при боковом ребре.
Решение 2. №304 (с. 87)
Решение 3. №304 (с. 87)
Решение 5. №304 (с. 87)
Решение 6. №304 (с. 87)
Пусть дана правильная четырехугольная пирамида $SABCD$, где $S$ — вершина, а $ABCD$ — квадратное основание. По условию, плоский угол при вершине равен $60^\circ$. Это означает, что угол в каждой боковой грани при вершине $S$ равен $60^\circ$, например, $\angle ASB = 60^\circ$.
Так как пирамида правильная, ее боковые грани — равные равнобедренные треугольники. В нашем случае, $\triangle ASB$ является равнобедренным с $SA = SB$. Поскольку угол при вершине $\angle ASB = 60^\circ$, то углы при основании этого треугольника также равны $(180^\circ - 60^\circ) / 2 = 60^\circ$. Следовательно, все боковые грани пирамиды являются равносторонними треугольниками. Это означает, что все ребра пирамиды равны. Обозначим длину ребра за $a$: $AB = BC = CD = DA = SA = SB = SC = SD = a$.
Нахождение двугранного угла между боковой гранью и основанием
Пусть $\alpha$ — двугранный угол между боковой гранью, например, $(SBC)$, и основанием $(ABCD)$. Этот угол измеряется линейным углом, образованным двумя перпендикулярами, проведенными к линии их пересечения, то есть к ребру $BC$.
Проведем апофему $SM$ в грани $SBC$. Так как $\triangle SBC$ — равносторонний, $SM$ является его высотой и медианой, поэтому $M$ — середина $BC$ и $SM \perp BC$. Высота равностороннего треугольника со стороной $a$ равна $SM = \frac{a\sqrt{3}}{2}$.
Пусть $O$ — центр основания (точка пересечения диагоналей квадрата). Проведем отрезок $OM$. Так как $ABCD$ — квадрат, $OM$ является средней линией $\triangle ABC$ или $\triangle DBC$ (в зависимости от расположения), и $OM \perp BC$. Длина $OM$ равна половине стороны квадрата: $OM = \frac{1}{2}AB = \frac{a}{2}$.
Поскольку $SM \perp BC$ и $OM \perp BC$, то $\angle SMO$ — это линейный угол двугранного угла $\alpha$. Рассмотрим прямоугольный треугольник $\triangle SOM$ (где $\angle SOM = 90^\circ$, так как $SO$ — высота пирамиды).
Найдем косинус угла $\alpha = \angle SMO$:
$\cos \alpha = \frac{OM}{SM} = \frac{a/2}{a\sqrt{3}/2} = \frac{1}{\sqrt{3}}$.
Нахождение двугранного угла при боковом ребре
Пусть $\beta$ — двугранный угол при боковом ребре, например, при ребре $SB$. Этот угол образован двумя смежными боковыми гранями $(SAB)$ и $(SBC)$.
Для нахождения этого угла построим его линейный угол. Проведем в гранях $(SAB)$ и $(SBC)$ перпендикуляры к общему ребру $SB$.
В равностороннем треугольнике $\triangle SAB$ проведем высоту $AH$ к стороне $SB$. Длина этой высоты равна $AH = \frac{a\sqrt{3}}{2}$.
В равностороннем треугольнике $\triangle SBC$ проведем высоту $CH$ к стороне $SB$. Длина этой высоты равна $CH = \frac{a\sqrt{3}}{2}$.
Тогда $\angle AHC$ является линейным углом двугранного угла $\beta$. Рассмотрим треугольник $\triangle AHC$. Мы знаем длины его сторон: $AH = CH = \frac{a\sqrt{3}}{2}$. Третья сторона, $AC$, является диагональю квадрата $ABCD$, ее длина $AC = a\sqrt{2}$.
Применим к треугольнику $\triangle AHC$ теорему косинусов для нахождения угла $\beta = \angle AHC$:
$AC^2 = AH^2 + CH^2 - 2 \cdot AH \cdot CH \cdot \cos\beta$
$(a\sqrt{2})^2 = (\frac{a\sqrt{3}}{2})^2 + (\frac{a\sqrt{3}}{2})^2 - 2 \cdot \frac{a\sqrt{3}}{2} \cdot \frac{a\sqrt{3}}{2} \cdot \cos\beta$
$2a^2 = \frac{3a^2}{4} + \frac{3a^2}{4} - 2 \cdot \frac{3a^2}{4} \cdot \cos\beta$
$2a^2 = \frac{6a^2}{4} - \frac{6a^2}{4} \cdot \cos\beta$
$2a^2 = \frac{3a^2}{2} - \frac{3a^2}{2} \cdot \cos\beta$
Разделим обе части уравнения на $a^2$ (так как $a \neq 0$):
$2 = \frac{3}{2} - \frac{3}{2} \cos\beta$
$2 - \frac{3}{2} = - \frac{3}{2} \cos\beta$
$\frac{1}{2} = - \frac{3}{2} \cos\beta$
$\cos\beta = -\frac{1}{3}$.
Доказательство
Мы получили значения косинусов для обоих углов: $\cos \alpha = \frac{1}{\sqrt{3}}$ и $\cos\beta = -\frac{1}{3}$. Нам нужно доказать, что $\alpha = \frac{\beta}{2}$, что эквивалентно $\beta = 2\alpha$.
Воспользуемся формулой косинуса двойного угла: $\cos(2\alpha) = 2\cos^2\alpha - 1$.
Подставим найденное значение $\cos\alpha$:
$\cos(2\alpha) = 2 \cdot (\frac{1}{\sqrt{3}})^2 - 1 = 2 \cdot \frac{1}{3} - 1 = \frac{2}{3} - 1 = -\frac{1}{3}$.
Таким образом, мы получили, что $\cos(2\alpha) = -\frac{1}{3}$, и мы также знаем, что $\cos\beta = -\frac{1}{3}$. Следовательно, $\cos(2\alpha) = \cos\beta$.
Поскольку $\alpha$ — угол между плоскостью боковой грани и основанием, он острый ($0 < \alpha < 90^\circ$). Значит, $0 < 2\alpha < 180^\circ$. Угол $\beta$ — двугранный угол в пирамиде, он также находится в диапазоне $0 < \beta < 180^\circ$. Из равенства $\cos(2\alpha) = \cos\beta$ на этом интервале следует, что $2\alpha = \beta$.
Это доказывает, что двугранный угол между боковой гранью и основанием пирамиды ($\alpha$) вдвое меньше двугранного угла при боковом ребре ($\beta$).
Ответ: Утверждение доказано. Мы показали, что $\cos \alpha = \frac{1}{\sqrt{3}}$ и $\cos \beta = -\frac{1}{3}$, а из формулы двойного угла $\cos(2\alpha) = 2\cos^2\alpha - 1$ следует, что $\cos(2\alpha) = \cos\beta$, откуда $2\alpha = \beta$.
Другие задания:
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