gdz.top
Номер 254, страница 77 - 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 2. Пирамида - номер 254, страница 77.
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" => 744669 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "77" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/00-254" "field_display_title" => "254" "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 {#1046 #items: array:1 [ 0 => App\Models\Branch {#1045 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 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 {#1050 #items: array:1 [ 0 => App\Models\Term {#1047 #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 {#1086 #items: array:1 [ 0 => App\Models\Book {#1049 #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 {#1051 #items: array:1 [ 0 => App\Models\Term {#1048 #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 {#1052 #items: array:2 [ 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] } 1 => App\Models\Term {#1054 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …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 {#1055 #items: array:1 [ 0 => App\Models\Term {#1056 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array: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 {#1057 #items: array:5 [ 0 => App\Models\Term {#1058 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …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 {#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: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 {#1060 #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 {#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: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 {#1062 #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 {#1063 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1064 #items: array:1 [ 0 => App\Models\Term {#1065 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array: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 {#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_city" => Illuminate\Database\Eloquent\Collection {#1068 #items: array:1 [ 0 => App\Models\Term {#1069 #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 {#1070 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1071 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1072 #items: array:1 [ 0 => App\Models\Term {#1073 #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 {#1074 #items: array:1 [ 0 => App\Models\Term {#1075 #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 {#1076 #items: array:1 [ 0 => App\Models\Term {#1077 #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 {#1078 #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 {#1079 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1080 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1081 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1082 #items: array:1 [ 0 => App\Models\Term {#1083 #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 {#1051} "field_class" => Illuminate\Database\Eloquent\Collection {#1052} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1055} "field_author" => Illuminate\Database\Eloquent\Collection {#1057} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1063} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1064} "field_country" => Illuminate\Database\Eloquent\Collection {#1066} "field_city" => Illuminate\Database\Eloquent\Collection {#1068} "field_series" => Illuminate\Database\Eloquent\Collection {#1070} "field_umk" => Illuminate\Database\Eloquent\Collection {#1071} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1072} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1074} "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1076} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1078} "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 {#1079} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1080} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1081} "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1082} "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 {#1084 #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 {#1050} "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 {#1086} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1084} ] #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 {#1121 #items: array:2 [ 0 => App\Models\Branch {#1129 #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 {#1130 #items: array:1 [ 0 => App\Models\Term {#1047} ] #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 {#1131 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1132 #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 {#1130} "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 {#1131} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1132} ] #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 {#1133 #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" => 744382 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Пирамида" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1137 #items: array:1 [ 0 => App\Models\Term {#1134 #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" => "72" "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 {#1136 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1135 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 744382 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Пирамида" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1137} "field_page_start" => "72" "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 {#1136} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1135} ] #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 {#1038 #items: array:6 [ 0 => App\Models\Element {#1109 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 938903 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1098 #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 [ "id" => 2315 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1105 …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 {#1104 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1102 …2} "field_process_formula" => "wiris" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1103 …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 {#1105 …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 {#1104 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1102 …2} "field_process_formula" => "wiris" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1103 …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" => "744669" "type" => "task" ] "text" => "<p> <b>254.</b> В правильной треугольной пирамиде сторона основания равна <i>а</i>, высота равна <i>Н</i>. Найдите: а) боковое ребро пирамиды; б) плоский угол при вершине пирамиды; в) угол между боковым ребром и плоскостью основания пирамиды; г) угол между боковой гранью и основанием пирамиды; д) двугранный угол при боковом ребре пирамиды . </p>" "img" => array:1 [ 0 => array:5 [ "name" => "254-1.jpg" "alt" => null "width" => "1545" "height" => 335 "path" => "/media/geometrija_10/atanasjan-u23/0-00/254-1.webp?ts=1739436493" ] ] ] #original: array:7 [ "id" => 938903 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1098} "task" => array:2 [ "refs" => "744669" "type" => "task" ] "text" => "<p> <b>254.</b> В правильной треугольной пирамиде сторона основания равна <i>а</i>, высота равна <i>Н</i>. Найдите: а) боковое ребро пирамиды; б) плоский угол при вершине пирамиды; в) угол между боковым ребром и плоскостью основания пирамиды; г) угол между боковой гранью и основанием пирамиды; д) двугранный угол при боковом ребре пирамиды . </p>" "img" => array:1 [ 0 => array:5 [ "name" => "254-1.jpg" "alt" => null "width" => "1545" "height" => 335 "path" => "/media/geometrija_10/atanasjan-u23/0-00/254-1.webp?ts=1739436493" ] ] ] #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 {#1122 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 940825 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1145 #items: array:1 [ 0 => App\Models\Edition {#1125 #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 {#1139 …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 {#1140 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1141 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1142 …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 {#1139 …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 {#1140 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1141 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1142 …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" => "744669" "type" => "task" ] "img" => array:5 [ 0 => array:5 [ "name" => "254-1.png" "alt" => null "width" => "694" "height" => 2105 "path" => "/media/geometrija_10/atanasjan-u23/2-00/254-1.webp?ts=1739441107" ] 1 => array:5 [ "name" => "254-2.png" "alt" => null "width" => "694" "height" => 3505 "path" => "/media/geometrija_10/atanasjan-u23/2-00/254-2.webp?ts=1739441107" ] 2 => array:5 [ "name" => "254-3.png" "alt" => null "width" => "694" "height" => 2366 "path" => "/media/geometrija_10/atanasjan-u23/2-00/254-3.webp?ts=1739441107" ] 3 => array:5 [ "name" => "254-4.png" "alt" => null "width" => "694" "height" => 2409 "path" => "/media/geometrija_10/atanasjan-u23/2-00/254-4.webp?ts=1739441107" ] 4 => array:5 [ "name" => "254-5.png" "alt" => null "width" => "694" "height" => 5979 "path" => "/media/geometrija_10/atanasjan-u23/2-00/254-5.webp?ts=1739441107" ] ] ] #original: array:6 [ "id" => 940825 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1145} "task" => array:2 [ "refs" => "744669" "type" => "task" ] "img" => array:5 [ 0 => array:5 [ "name" => "254-1.png" "alt" => null "width" => "694" "height" => 2105 "path" => "/media/geometrija_10/atanasjan-u23/2-00/254-1.webp?ts=1739441107" ] 1 => array:5 [ "name" => "254-2.png" "alt" => null "width" => "694" "height" => 3505 "path" => "/media/geometrija_10/atanasjan-u23/2-00/254-2.webp?ts=1739441107" ] 2 => array:5 [ "name" => "254-3.png" "alt" => null "width" => "694" "height" => 2366 "path" => "/media/geometrija_10/atanasjan-u23/2-00/254-3.webp?ts=1739441107" ] 3 => array:5 [ "name" => "254-4.png" "alt" => null "width" => "694" "height" => 2409 "path" => "/media/geometrija_10/atanasjan-u23/2-00/254-4.webp?ts=1739441107" ] 4 => array:5 [ "name" => "254-5.png" "alt" => null "width" => "694" "height" => 5979 "path" => "/media/geometrija_10/atanasjan-u23/2-00/254-5.webp?ts=1739441107" ] ] ] #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 {#1100 #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" => 940569 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1092 #items: array:1 [ 0 => App\Models\Edition {#1101 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 2318 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 3" "field_order" => "4" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1097 …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 {#1091 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1096 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1095 …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 {#1097 …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 {#1091 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1096 …2} "field_process_formula" => "" "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" => "744669" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "254-1.jpg" "alt" => null "width" => "1300" "height" => 1778 "path" => "/media/geometrija_10/atanasjan-u23/3-00/254-1.webp?ts=1739436698" ] 1 => array:5 [ "name" => "254-2.jpg" "alt" => null "width" => "1300" "height" => 2037 "path" => "/media/geometrija_10/atanasjan-u23/3-00/254-2.webp?ts=1739436698" ] 2 => array:5 [ "name" => "254-3.jpg" "alt" => null "width" => "1300" "height" => 1324 "path" => "/media/geometrija_10/atanasjan-u23/3-00/254-3.webp?ts=1739436698" ] ] ] #original: array:6 [ "id" => 940569 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1092} "task" => array:2 [ "refs" => "744669" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "254-1.jpg" "alt" => null "width" => "1300" "height" => 1778 "path" => "/media/geometrija_10/atanasjan-u23/3-00/254-1.webp?ts=1739436698" ] 1 => array:5 [ "name" => "254-2.jpg" "alt" => null "width" => "1300" "height" => 2037 "path" => "/media/geometrija_10/atanasjan-u23/3-00/254-2.webp?ts=1739436698" ] 2 => array:5 [ "name" => "254-3.jpg" "alt" => null "width" => "1300" "height" => 1324 "path" => "/media/geometrija_10/atanasjan-u23/3-00/254-3.webp?ts=1739436698" ] ] ] #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 {#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" => 940625 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1034 #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" => 2319 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 4" "field_order" => "5" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1090 …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" => "4-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_content_source" => null ] #original: array:21 [ "id" => 2319 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 4" "field_order" => "5" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1090 …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" => "4-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1088 …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" => "744669" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "254-1.jpg" "alt" => null "width" => "1148" "height" => 1146 "path" => "/media/geometrija_10/atanasjan-u23/4-00/254-1.webp?ts=1739436477" ] 1 => array:5 [ "name" => "254-2.jpg" "alt" => null "width" => "1148" "height" => 1502 "path" => "/media/geometrija_10/atanasjan-u23/4-00/254-2.webp?ts=1739436477" ] ] ] #original: array:6 [ "id" => 940625 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1034} "task" => array:2 [ "refs" => "744669" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "254-1.jpg" "alt" => null "width" => "1148" "height" => 1146 "path" => "/media/geometrija_10/atanasjan-u23/4-00/254-1.webp?ts=1739436477" ] 1 => array:5 [ "name" => "254-2.jpg" "alt" => null "width" => "1148" "height" => 1502 "path" => "/media/geometrija_10/atanasjan-u23/4-00/254-2.webp?ts=1739436477" ] ] ] #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 {#1043 #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" => 940792 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1123 #items: array:1 [ 0 => App\Models\Edition {#1041 #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 {#1120 …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 {#1128 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1124 …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 {#1120 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "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" => "744669" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "254-1.jpg" "alt" => null "width" => "582" "height" => 906 "path" => "/media/geometrija_10/atanasjan-u23/5-00/254-1.webp?ts=1739436519" ] ] ] #original: array:6 [ "id" => 940792 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1123} "task" => array:2 [ "refs" => "744669" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "254-1.jpg" "alt" => null "width" => "582" "height" => 906 "path" => "/media/geometrija_10/atanasjan-u23/5-00/254-1.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 => "*" ] } 5 => App\Models\Element {#1143 #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" => 1351982 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1153 #items: array:1 [ 0 => App\Models\Edition {#1144 #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" => "744669" "type" => "task" ] "text" => "<p>Пусть дана правильная треугольная пирамида <span>SABC</span>, где <span>S</span> — вершина, а <span>ABC</span> — равносторонний треугольник в основании со стороной <span>a</span>. <span>SO = H</span> — высота пирамиды, где <span>O</span> — центр основания (точка пересечения медиан, высот и биссектрис).</p><p>Для решения задачи нам понадобятся некоторые размеры в основании. <span>O</span> — центр описанной и вписанной окружностей для треугольника <span>ABC</span>.</p><p>Радиус описанной окружности <span>R</span> (отрезок <span>OA</span>) для равностороннего треугольника со стороной <span>a</span> равен:$R = OA = \frac{a}{\sqrt{3}}$</p><p>Радиус вписанной окружности <span>r</span> (длина перпендикуляра из <span>O</span> к стороне, например, <span>OM</span>, где <span>M</span> — середина <span>BC</span>) равен:$r = OM = \frac{a}{2\sqrt{3}}$</p><p><strong>а) боковое ребро пирамиды</strong></p><p>Боковое ребро, обозначим его <span>l</span> (например, <span>SA</span>), является гипотенузой в прямоугольном треугольнике <span>SOA</span>. Катеты этого треугольника — высота пирамиды <span>SO = H</span> и радиус описанной окружности основания <span>OA = R</span>.</p><p>По теореме Пифагора:$l^2 = SA^2 = SO^2 + OA^2 = H^2 + R^2$</p><p>Подставим значение <span>R</span>:$l^2 = H^2 + \left(\frac{a}{\sqrt{3}}\right)^2 = H^2 + \frac{a^2}{3}$</p><p>Отсюда длина бокового ребра:$l = \sqrt{H^2 + \frac{a^2}{3}}$</p><p>Ответ: $l = \sqrt{H^2 + \frac{a^2}{3}}$</p><p><strong>б) плоский угол при вершине пирамиды</strong></p><p>Плоский угол при вершине — это угол в боковой грани, например, угол <span>? = ?BSC</span>. Боковая грань (например, треугольник <span>SBC</span>) — равнобедренный треугольник с боковыми сторонами <span>SB = SC = l</span> и основанием <span>BC = a</span>.</p><p>Проведем в треугольнике <span>SBC</span> высоту (она же медиана и биссектриса) <span>SM</span> к основанию <span>BC</span>. <span>M</span> — середина <span>BC</span>, поэтому <span>BM = a/2</span>.</p><p>В прямоугольном треугольнике <span>SMB</span> угол <span>?BSM = ?/2</span>. Из определения синуса:$\sin\left(\frac{\alpha}{2}\right) = \frac{BM}{SB} = \frac{a/2}{l} = \frac{a}{2l}$</p><p>Подставим найденное ранее выражение для <span>l</span>:$\sin\left(\frac{\alpha}{2}\right) = \frac{a}{2\sqrt{H^2 + a^2/3}}$</p><p>Тогда искомый угол <span>?</span>:$\alpha = 2\arcsin\left(\frac{a}{2\sqrt{H^2 + a^2/3}}\right)$</p><p>Ответ: $\alpha = 2\arcsin\left(\frac{a}{2\sqrt{H^2 + a^2/3}}\right)$</p><p><strong>в) угол между боковым ребром и плоскостью основания пирамиды</strong></p><p>Угол между боковым ребром (например, <span>SA</span>) и плоскостью основания <span>ABC</span> — это угол между ребром <span>SA</span> и его проекцией на плоскость основания, которой является отрезок <span>OA</span>. Обозначим этот угол <span>? = ?SAO</span>.</p><p>Рассмотрим прямоугольный треугольник <span>SOA</span>.$\tan(\beta) = \frac{SO}{OA} = \frac{H}{R} = \frac{H}{a/\sqrt{3}} = \frac{H\sqrt{3}}{a}$</p><p>Отсюда угол <span>?</span>:$\beta = \arctan\left(\frac{H\sqrt{3}}{a}\right)$</p><p>Ответ: $\beta = \arctan\left(\frac{H\sqrt{3}}{a}\right)$</p><p><strong>г) угол между боковой гранью и основанием пирамиды</strong></p><p>Угол между боковой гранью (например, <span>SBC</span>) и основанием <span>ABC</span> — это линейный угол двугранного угла при ребре <span>BC</span>.</p><p>Проведем апофему <span>SM</span> (<span>M</span> — середина <span>BC</span>). Так как <span>SBC</span> — равнобедренный треугольник, <span>SM ? BC</span>. В основании <span>AM</span> — медиана и высота, поэтому <span>AM ? BC</span>. Следовательно, искомый угол <span>?</span> — это угол <span>?SMA</span>. Так как точка <span>O</span> лежит на <span>AM</span>, это угол <span>?SMO</span>.</p><p>Рассмотрим прямоугольный треугольник <span>SOM</span>.$\tan(\gamma) = \frac{SO}{OM} = \frac{H}{r} = \frac{H}{a/(2\sqrt{3})} = \frac{2H\sqrt{3}}{a}$</p><p>Отсюда угол <span>?</span>:$\gamma = \arctan\left(\frac{2H\sqrt{3}}{a}\right)$</p><p>Ответ: $\gamma = \arctan\left(\frac{2H\sqrt{3}}{a}\right)$</p><p><strong>д) двугранный угол при боковом ребре пирамиды</strong></p><p>Двугранный угол при боковом ребре (например, <span>SA</span>) — это угол <span>?</span> между плоскостями боковых граней <span>SAB</span> и <span>SAC</span>.</p><p>Для нахождения этого угла построим его линейный угол. Проведем в грани <span>SAB</span> перпендикуляр <span>BK</span> к ребру <span>SA</span>. Так как боковые грани <span>SAB</span> и <span>SAC</span> равны, то перпендикуляр из точки <span>C</span> к ребру <span>SA</span> также придет в точку <span>K</span>, то есть <span>CK ? SA</span>.</p><p>Тогда искомый угол <span>? = ?BKC</span>. Треугольник <span>BKC</span> — равнобедренный с основанием <span>BC = a</span> и равными сторонами <span>BK = CK</span>.</p><p>По теореме косинусов для треугольника <span>BKC</span>:$BC^2 = BK^2 + CK^2 - 2 \cdot BK \cdot CK \cdot \cos(\delta) \implies a^2 = 2BK^2(1 - \cos\delta)$$\cos(\delta) = 1 - \frac{a^2}{2BK^2}$</p><p>Найдем длину <span>BK</span>. Площадь треугольника <span>SAB</span> можно выразить двумя способами:$S_{\triangle SAB} = \frac{1}{2} AB \cdot SM = \frac{1}{2} SA \cdot BK \implies a \cdot SM = l \cdot BK \implies BK = \frac{a \cdot SM}{l}$</p><p>Найдем квадраты длин <span>SM</span> и <span>l</span>.Из прямоугольного треугольника <span>SOM</span>: $SM^2 = SO^2 + OM^2 = H^2 + r^2 = H^2 + \left(\frac{a}{2\sqrt{3}}\right)^2 = H^2 + \frac{a^2}{12}$.Длина бокового ребра: $l^2 = H^2 + \frac{a^2}{3}$.</p><p>Теперь найдем $BK^2$:$BK^2 = \frac{a^2 \cdot SM^2}{l^2} = \frac{a^2 \left(H^2 + \frac{a^2}{12}\right)}{H^2 + \frac{a^2}{3}} = \frac{a^2 \frac{12H^2 + a^2}{12}}{\frac{3H^2 + a^2}{3}} = \frac{a^2 (12H^2 + a^2)}{4(3H^2 + a^2)}$</p><p>Подставим <span>BK?</span> в формулу для косинуса угла <span>?</span>:$\cos(\delta) = 1 - \frac{a^2}{2 \cdot \frac{a^2(12H^2+a^2)}{4(3H^2+a^2)}} = 1 - \frac{2(3H^2+a^2)}{12H^2+a^2}$</p><p>$\cos(\delta) = \frac{(12H^2+a^2) - (6H^2+2a^2)}{12H^2+a^2} = \frac{6H^2 - a^2}{12H^2 + a^2}$</p><p>Отсюда угол <span>?</span>:$\delta = \arccos\left(\frac{6H^2 - a^2}{12H^2 + a^2}\right)$</p><p>Ответ: $\delta = \arccos\left(\frac{6H^2 - a^2}{12H^2 + a^2}\right)$</p>" ] #original: array:6 [ "id" => 1351982 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1153} "task" => array:2 [ "refs" => "744669" "type" => "task" ] "text" => "<p>Пусть дана правильная треугольная пирамида <span>SABC</span>, где <span>S</span> — вершина, а <span>ABC</span> — равносторонний треугольник в основании со стороной <span>a</span>. <span>SO = H</span> — высота пирамиды, где <span>O</span> — центр основания (точка пересечения медиан, высот и биссектрис).</p><p>Для решения задачи нам понадобятся некоторые размеры в основании. <span>O</span> — центр описанной и вписанной окружностей для треугольника <span>ABC</span>.</p><p>Радиус описанной окружности <span>R</span> (отрезок <span>OA</span>) для равностороннего треугольника со стороной <span>a</span> равен:$R = OA = \frac{a}{\sqrt{3}}$</p><p>Радиус вписанной окружности <span>r</span> (длина перпендикуляра из <span>O</span> к стороне, например, <span>OM</span>, где <span>M</span> — середина <span>BC</span>) равен:$r = OM = \frac{a}{2\sqrt{3}}$</p><p><strong>а) боковое ребро пирамиды</strong></p><p>Боковое ребро, обозначим его <span>l</span> (например, <span>SA</span>), является гипотенузой в прямоугольном треугольнике <span>SOA</span>. Катеты этого треугольника — высота пирамиды <span>SO = H</span> и радиус описанной окружности основания <span>OA = R</span>.</p><p>По теореме Пифагора:$l^2 = SA^2 = SO^2 + OA^2 = H^2 + R^2$</p><p>Подставим значение <span>R</span>:$l^2 = H^2 + \left(\frac{a}{\sqrt{3}}\right)^2 = H^2 + \frac{a^2}{3}$</p><p>Отсюда длина бокового ребра:$l = \sqrt{H^2 + \frac{a^2}{3}}$</p><p>Ответ: $l = \sqrt{H^2 + \frac{a^2}{3}}$</p><p><strong>б) плоский угол при вершине пирамиды</strong></p><p>Плоский угол при вершине — это угол в боковой грани, например, угол <span>? = ?BSC</span>. Боковая грань (например, треугольник <span>SBC</span>) — равнобедренный треугольник с боковыми сторонами <span>SB = SC = l</span> и основанием <span>BC = a</span>.</p><p>Проведем в треугольнике <span>SBC</span> высоту (она же медиана и биссектриса) <span>SM</span> к основанию <span>BC</span>. <span>M</span> — середина <span>BC</span>, поэтому <span>BM = a/2</span>.</p><p>В прямоугольном треугольнике <span>SMB</span> угол <span>?BSM = ?/2</span>. Из определения синуса:$\sin\left(\frac{\alpha}{2}\right) = \frac{BM}{SB} = \frac{a/2}{l} = \frac{a}{2l}$</p><p>Подставим найденное ранее выражение для <span>l</span>:$\sin\left(\frac{\alpha}{2}\right) = \frac{a}{2\sqrt{H^2 + a^2/3}}$</p><p>Тогда искомый угол <span>?</span>:$\alpha = 2\arcsin\left(\frac{a}{2\sqrt{H^2 + a^2/3}}\right)$</p><p>Ответ: $\alpha = 2\arcsin\left(\frac{a}{2\sqrt{H^2 + a^2/3}}\right)$</p><p><strong>в) угол между боковым ребром и плоскостью основания пирамиды</strong></p><p>Угол между боковым ребром (например, <span>SA</span>) и плоскостью основания <span>ABC</span> — это угол между ребром <span>SA</span> и его проекцией на плоскость основания, которой является отрезок <span>OA</span>. Обозначим этот угол <span>? = ?SAO</span>.</p><p>Рассмотрим прямоугольный треугольник <span>SOA</span>.$\tan(\beta) = \frac{SO}{OA} = \frac{H}{R} = \frac{H}{a/\sqrt{3}} = \frac{H\sqrt{3}}{a}$</p><p>Отсюда угол <span>?</span>:$\beta = \arctan\left(\frac{H\sqrt{3}}{a}\right)$</p><p>Ответ: $\beta = \arctan\left(\frac{H\sqrt{3}}{a}\right)$</p><p><strong>г) угол между боковой гранью и основанием пирамиды</strong></p><p>Угол между боковой гранью (например, <span>SBC</span>) и основанием <span>ABC</span> — это линейный угол двугранного угла при ребре <span>BC</span>.</p><p>Проведем апофему <span>SM</span> (<span>M</span> — середина <span>BC</span>). Так как <span>SBC</span> — равнобедренный треугольник, <span>SM ? BC</span>. В основании <span>AM</span> — медиана и высота, поэтому <span>AM ? BC</span>. Следовательно, искомый угол <span>?</span> — это угол <span>?SMA</span>. Так как точка <span>O</span> лежит на <span>AM</span>, это угол <span>?SMO</span>.</p><p>Рассмотрим прямоугольный треугольник <span>SOM</span>.$\tan(\gamma) = \frac{SO}{OM} = \frac{H}{r} = \frac{H}{a/(2\sqrt{3})} = \frac{2H\sqrt{3}}{a}$</p><p>Отсюда угол <span>?</span>:$\gamma = \arctan\left(\frac{2H\sqrt{3}}{a}\right)$</p><p>Ответ: $\gamma = \arctan\left(\frac{2H\sqrt{3}}{a}\right)$</p><p><strong>д) двугранный угол при боковом ребре пирамиды</strong></p><p>Двугранный угол при боковом ребре (например, <span>SA</span>) — это угол <span>?</span> между плоскостями боковых граней <span>SAB</span> и <span>SAC</span>.</p><p>Для нахождения этого угла построим его линейный угол. Проведем в грани <span>SAB</span> перпендикуляр <span>BK</span> к ребру <span>SA</span>. Так как боковые грани <span>SAB</span> и <span>SAC</span> равны, то перпендикуляр из точки <span>C</span> к ребру <span>SA</span> также придет в точку <span>K</span>, то есть <span>CK ? SA</span>.</p><p>Тогда искомый угол <span>? = ?BKC</span>. Треугольник <span>BKC</span> — равнобедренный с основанием <span>BC = a</span> и равными сторонами <span>BK = CK</span>.</p><p>По теореме косинусов для треугольника <span>BKC</span>:$BC^2 = BK^2 + CK^2 - 2 \cdot BK \cdot CK \cdot \cos(\delta) \implies a^2 = 2BK^2(1 - \cos\delta)$$\cos(\delta) = 1 - \frac{a^2}{2BK^2}$</p><p>Найдем длину <span>BK</span>. Площадь треугольника <span>SAB</span> можно выразить двумя способами:$S_{\triangle SAB} = \frac{1}{2} AB \cdot SM = \frac{1}{2} SA \cdot BK \implies a \cdot SM = l \cdot BK \implies BK = \frac{a \cdot SM}{l}$</p><p>Найдем квадраты длин <span>SM</span> и <span>l</span>.Из прямоугольного треугольника <span>SOM</span>: $SM^2 = SO^2 + OM^2 = H^2 + r^2 = H^2 + \left(\frac{a}{2\sqrt{3}}\right)^2 = H^2 + \frac{a^2}{12}$.Длина бокового ребра: $l^2 = H^2 + \frac{a^2}{3}$.</p><p>Теперь найдем $BK^2$:$BK^2 = \frac{a^2 \cdot SM^2}{l^2} = \frac{a^2 \left(H^2 + \frac{a^2}{12}\right)}{H^2 + \frac{a^2}{3}} = \frac{a^2 \frac{12H^2 + a^2}{12}}{\frac{3H^2 + a^2}{3}} = \frac{a^2 (12H^2 + a^2)}{4(3H^2 + a^2)}$</p><p>Подставим <span>BK?</span> в формулу для косинуса угла <span>?</span>:$\cos(\delta) = 1 - \frac{a^2}{2 \cdot \frac{a^2(12H^2+a^2)}{4(3H^2+a^2)}} = 1 - \frac{2(3H^2+a^2)}{12H^2+a^2}$</p><p>$\cos(\delta) = \frac{(12H^2+a^2) - (6H^2+2a^2)}{12H^2+a^2} = \frac{6H^2 - a^2}{12H^2 + a^2}$</p><p>Отсюда угол <span>?</span>:$\delta = \arccos\left(\frac{6H^2 - a^2}{12H^2 + a^2}\right)$</p><p>Ответ: $\delta = \arccos\left(\frac{6H^2 - a^2}{12H^2 + a^2}\right)$</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" => "744670" "type" => "task" ] "previous" => array:2 [ "refs" => "744668" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1111 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1113 #items: array:1 [ 0 => App\Models\BookPage {#1115 #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" => 933859 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "77" "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/page-77" "field_display_title" => "77" "field_folder" => "1" "field_image_name" => "77" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1116 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1042 #items: array:1 [ 0 => App\Models\Book {#1049} ] #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 {#1117 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1160 #items: array:1 [ 0 => App\Models\Element {#1159 #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" => "933860" "type" => "book_page" ] "previous" => array:2 [ "refs" => "933858" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1262 #items: array:12 [ 0 => App\Models\Task {#1280 #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 {#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: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 2 => App\Models\Task {#1290 #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 {#1314 #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 {#1317 #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 {#1316 #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 {#1318 #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 {#1338 #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 {#1358 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 9 => App\Models\Task {#1378 #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 {#1398 #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 {#1418 #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" => 933859 "created_at" => 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№254 (с. 77)
Условие. №254 (с. 77)
скриншот условия
254. В правильной треугольной пирамиде сторона основания равна а, высота равна Н. Найдите: а) боковое ребро пирамиды; б) плоский угол при вершине пирамиды; в) угол между боковым ребром и плоскостью основания пирамиды; г) угол между боковой гранью и основанием пирамиды; д) двугранный угол при боковом ребре пирамиды .
Решение 2. №254 (с. 77)
Решение 3. №254 (с. 77)
Решение 4. №254 (с. 77)
Решение 5. №254 (с. 77)
Решение 6. №254 (с. 77)
Пусть дана правильная треугольная пирамида SABC, где S — вершина, а ABC — равносторонний треугольник в основании со стороной a. SO = H — высота пирамиды, где O — центр основания (точка пересечения медиан, высот и биссектрис).
Для решения задачи нам понадобятся некоторые размеры в основании. O — центр описанной и вписанной окружностей для треугольника ABC.
Радиус описанной окружности R (отрезок OA) для равностороннего треугольника со стороной a равен:$R = OA = \frac{a}{\sqrt{3}}$
Радиус вписанной окружности r (длина перпендикуляра из O к стороне, например, OM, где M — середина BC) равен:$r = OM = \frac{a}{2\sqrt{3}}$
а) боковое ребро пирамиды
Боковое ребро, обозначим его l (например, SA), является гипотенузой в прямоугольном треугольнике SOA. Катеты этого треугольника — высота пирамиды SO = H и радиус описанной окружности основания OA = R.
По теореме Пифагора:$l^2 = SA^2 = SO^2 + OA^2 = H^2 + R^2$
Подставим значение R:$l^2 = H^2 + \left(\frac{a}{\sqrt{3}}\right)^2 = H^2 + \frac{a^2}{3}$
Отсюда длина бокового ребра:$l = \sqrt{H^2 + \frac{a^2}{3}}$
Ответ: $l = \sqrt{H^2 + \frac{a^2}{3}}$
б) плоский угол при вершине пирамиды
Плоский угол при вершине — это угол в боковой грани, например, угол ? = ?BSC. Боковая грань (например, треугольник SBC) — равнобедренный треугольник с боковыми сторонами SB = SC = l и основанием BC = a.
Проведем в треугольнике SBC высоту (она же медиана и биссектриса) SM к основанию BC. M — середина BC, поэтому BM = a/2.
В прямоугольном треугольнике SMB угол ?BSM = ?/2. Из определения синуса:$\sin\left(\frac{\alpha}{2}\right) = \frac{BM}{SB} = \frac{a/2}{l} = \frac{a}{2l}$
Подставим найденное ранее выражение для l:$\sin\left(\frac{\alpha}{2}\right) = \frac{a}{2\sqrt{H^2 + a^2/3}}$
Тогда искомый угол ?:$\alpha = 2\arcsin\left(\frac{a}{2\sqrt{H^2 + a^2/3}}\right)$
Ответ: $\alpha = 2\arcsin\left(\frac{a}{2\sqrt{H^2 + a^2/3}}\right)$
в) угол между боковым ребром и плоскостью основания пирамиды
Угол между боковым ребром (например, SA) и плоскостью основания ABC — это угол между ребром SA и его проекцией на плоскость основания, которой является отрезок OA. Обозначим этот угол ? = ?SAO.
Рассмотрим прямоугольный треугольник SOA.$\tan(\beta) = \frac{SO}{OA} = \frac{H}{R} = \frac{H}{a/\sqrt{3}} = \frac{H\sqrt{3}}{a}$
Отсюда угол ?:$\beta = \arctan\left(\frac{H\sqrt{3}}{a}\right)$
Ответ: $\beta = \arctan\left(\frac{H\sqrt{3}}{a}\right)$
г) угол между боковой гранью и основанием пирамиды
Угол между боковой гранью (например, SBC) и основанием ABC — это линейный угол двугранного угла при ребре BC.
Проведем апофему SM (M — середина BC). Так как SBC — равнобедренный треугольник, SM ? BC. В основании AM — медиана и высота, поэтому AM ? BC. Следовательно, искомый угол ? — это угол ?SMA. Так как точка O лежит на AM, это угол ?SMO.
Рассмотрим прямоугольный треугольник SOM.$\tan(\gamma) = \frac{SO}{OM} = \frac{H}{r} = \frac{H}{a/(2\sqrt{3})} = \frac{2H\sqrt{3}}{a}$
Отсюда угол ?:$\gamma = \arctan\left(\frac{2H\sqrt{3}}{a}\right)$
Ответ: $\gamma = \arctan\left(\frac{2H\sqrt{3}}{a}\right)$
д) двугранный угол при боковом ребре пирамиды
Двугранный угол при боковом ребре (например, SA) — это угол ? между плоскостями боковых граней SAB и SAC.
Для нахождения этого угла построим его линейный угол. Проведем в грани SAB перпендикуляр BK к ребру SA. Так как боковые грани SAB и SAC равны, то перпендикуляр из точки C к ребру SA также придет в точку K, то есть CK ? SA.
Тогда искомый угол ? = ?BKC. Треугольник BKC — равнобедренный с основанием BC = a и равными сторонами BK = CK.
По теореме косинусов для треугольника BKC:$BC^2 = BK^2 + CK^2 - 2 \cdot BK \cdot CK \cdot \cos(\delta) \implies a^2 = 2BK^2(1 - \cos\delta)$$\cos(\delta) = 1 - \frac{a^2}{2BK^2}$
Найдем длину BK. Площадь треугольника SAB можно выразить двумя способами:$S_{\triangle SAB} = \frac{1}{2} AB \cdot SM = \frac{1}{2} SA \cdot BK \implies a \cdot SM = l \cdot BK \implies BK = \frac{a \cdot SM}{l}$
Найдем квадраты длин SM и l.Из прямоугольного треугольника SOM: $SM^2 = SO^2 + OM^2 = H^2 + r^2 = H^2 + \left(\frac{a}{2\sqrt{3}}\right)^2 = H^2 + \frac{a^2}{12}$.Длина бокового ребра: $l^2 = H^2 + \frac{a^2}{3}$.
Теперь найдем $BK^2$:$BK^2 = \frac{a^2 \cdot SM^2}{l^2} = \frac{a^2 \left(H^2 + \frac{a^2}{12}\right)}{H^2 + \frac{a^2}{3}} = \frac{a^2 \frac{12H^2 + a^2}{12}}{\frac{3H^2 + a^2}{3}} = \frac{a^2 (12H^2 + a^2)}{4(3H^2 + a^2)}$
Подставим BK? в формулу для косинуса угла ?:$\cos(\delta) = 1 - \frac{a^2}{2 \cdot \frac{a^2(12H^2+a^2)}{4(3H^2+a^2)}} = 1 - \frac{2(3H^2+a^2)}{12H^2+a^2}$
$\cos(\delta) = \frac{(12H^2+a^2) - (6H^2+2a^2)}{12H^2+a^2} = \frac{6H^2 - a^2}{12H^2 + a^2}$
Отсюда угол ?:$\delta = \arccos\left(\frac{6H^2 - a^2}{12H^2 + a^2}\right)$
Ответ: $\delta = \arccos\left(\frac{6H^2 - a^2}{12H^2 + a^2}\right)$
Другие задания:
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