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Номер 667, страница 170 - гдз по геометрии 10-11 класс учебник Атанасян, Бутузов
Авторы: Атанасян Л. С., Бутузов В. Ф., Кадомцев С. Б., Позняк Э. Г., Киселёва Л. С.
Тип: Учебник
Издательство: Просвещение
Год издания: 2019 - 2026
Уровень обучения: базовый и углублённый
Цвет обложки: коричневый с ромбами
ISBN: 978-5-09-103606-0 (2023)
Допущено Министерством просвещения Российской Федерации
Математика: алгебра и начала математического анализа, геометрия
Популярные ГДЗ в 10 классе
Глава 7. Метод координат в пространстве. Движения. Параграф 1. Координаты точки и координаты вектора - номер 667, страница 170.
App\Models\Task {#1024 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 745082 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "170" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/00-667" "field_display_title" => "667" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#335 #items: array:1 [ 0 => App\Models\Term {#1029 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1026 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#341 #items: array:1 [ 0 => App\Models\Branch {#1023 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 744405 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Метод координат в пространстве. Движения" "field_branch_order" => "7" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#331 #items: array:1 [ 0 => App\Models\Term {#1033 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "160" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1030 #items: array:1 [ 0 => App\Models\Book {#1032 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 36 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1034 #items: array:1 [ 0 => App\Models\Term {#1035 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1036 #items: array:2 [ 0 => App\Models\Term {#1037 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Term {#1038 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1039 #items: array:1 [ 0 => App\Models\Term {#1040 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1041 #items: array:5 [ 0 => App\Models\Term {#1042 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Term {#1043 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 2 => App\Models\Term {#1044 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 3 => App\Models\Term {#1045 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 4 => App\Models\Term {#1046 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1047 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1048 #items: array:1 [ 0 => App\Models\Term {#1049 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1050 #items: array:1 [ 0 => App\Models\Term {#1051 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1052 #items: array:1 [ 0 => App\Models\Term {#1053 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1054 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1055 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1056 #items: array:1 [ 0 => App\Models\Term {#1057 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1058 #items: array:1 [ 0 => App\Models\Term {#1059 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1060 #items: array:1 [ 0 => App\Models\Term {#1061 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1062 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => "с ромбами" "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для общеобразовательных организаций" "field_allowed" => "Допущено Министерством просвещения Российской Федерации" "field_reserve_field" => "Математика: алгебра и начала математического анализа, геометрия" "field_link_to_source" => null "field_tasks_count" => "1147" "field_priority" => "12" "field_default_folder" => "/geometrija_10/atanasjan-u23/" "field_isbn" => "978-5-09-103606-0 (2023)" "field_cover" => array:1 [ 0 => "/media/geometrija_10/atanasjan-u23/covers/cover1.webp?ts=1739363242" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_10/atanasjan-u23/covers/cover1.webp?ts=1739363242" "alt" => "" "width" => "1200" "height" => "1661" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1063 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1064 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1065 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1066 #items: array:1 [ 0 => App\Models\Term {#1067 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 543 "class_subject" => 392 ] ] #original: array:50 [ "id" => 36 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1034} "field_class" => Illuminate\Database\Eloquent\Collection {#1036} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1039} "field_author" => Illuminate\Database\Eloquent\Collection {#1041} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1047} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1048} "field_country" => Illuminate\Database\Eloquent\Collection {#1050} "field_city" => Illuminate\Database\Eloquent\Collection {#1052} "field_series" => Illuminate\Database\Eloquent\Collection {#1054} "field_umk" => Illuminate\Database\Eloquent\Collection {#1055} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1056} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1058} "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1060} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1062} "field_under_the_edition_degree" => null "field_cover_description" => "с ромбами" "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для общеобразовательных организаций" "field_allowed" => "Допущено Министерством просвещения Российской Федерации" "field_reserve_field" => "Математика: алгебра и начала математического анализа, геометрия" "field_link_to_source" => null "field_tasks_count" => "1147" "field_priority" => "12" "field_default_folder" => "/geometrija_10/atanasjan-u23/" "field_isbn" => "978-5-09-103606-0 (2023)" "field_cover" => array:1 [ 0 => "/media/geometrija_10/atanasjan-u23/covers/cover1.webp?ts=1739363242" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_10/atanasjan-u23/covers/cover1.webp?ts=1739363242" "alt" => "" "width" => "1200" "height" => "1661" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1063} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1064} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1065} "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1066} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 543 "class_subject" => 392 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1068 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 744405 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Метод координат в пространстве. Движения" "field_branch_order" => "7" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#331} "field_page_start" => "160" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1030} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1068} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1069 #items: array:2 [ 0 => App\Models\Branch {#1023} 1 => App\Models\Branch {#1070 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 744406 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Координаты точки и координаты вектора" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1071 #items: array:1 [ 0 => App\Models\Term {#1072 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => "§" ] #original: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => "§" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "160" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1073 #items: array:1 [ 0 => App\Models\Book {#1032} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1074 #items: array:1 [ 0 => App\Models\Branch {#1075 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 744405 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Метод координат в пространстве. Движения" "field_branch_order" => "7" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1076 …2} "field_page_start" => "160" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1077 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1078 …2} ] #original: array:24 [ "id" => 744405 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Метод координат в пространстве. Движения" "field_branch_order" => "7" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1076 …2} "field_page_start" => "160" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1077 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1078 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 744406 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Координаты точки и координаты вектора" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1071} "field_page_start" => "160" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1073} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1074} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1079 #items: array:6 [ 0 => App\Models\Element {#1080 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 939476 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1081 #items: array:1 [ 0 => App\Models\Edition {#1082 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 2315 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1083 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_process_formula" => "wiris" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_content_source" => null ] #original: array:21 [ "id" => 2315 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1083 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_process_formula" => "wiris" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_content_source" => null ] #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" => "745082" "type" => "task" ] "text" => """ <p><strong>667.</strong> Даны точки</p><figure><figure>\n <img src="/media/geometrija_10/atanasjan-u23/0-00/667_0.webp?ts=1745773989" alt="Найти периметр и медианы треугольника ABC" loading="lazy" width="883" height="177">\n </figure>\n <figcaption> </figcaption></figure><p>Найдите: а) периметр треугольника ABC; б) медианы треугольника ABC.</p> """ "img" => array:1 [ 0 => array:5 [ "name" => "667-1.jpg" "alt" => null "width" => "1562" "height" => 213 "path" => "/media/geometrija_10/atanasjan-u23/0-00/667-1.webp?ts=1739436810" ] ] ] #original: array:7 [ "id" => 939476 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1081} "task" => array:2 [ "refs" => "745082" "type" => "task" ] "text" => """ <p><strong>667.</strong> Даны точки</p><figure><figure>\n <img src="/media/geometrija_10/atanasjan-u23/0-00/667_0.webp?ts=1745773989" alt="Найти периметр и медианы треугольника ABC" loading="lazy" width="883" height="177">\n </figure>\n <figcaption> </figcaption></figure><p>Найдите: а) периметр треугольника ABC; б) медианы треугольника ABC.</p> """ "img" => array:1 [ 0 => array:5 [ "name" => "667-1.jpg" "alt" => null "width" => "1562" "height" => 213 "path" => "/media/geometrija_10/atanasjan-u23/0-00/667-1.webp?ts=1739436810" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1088 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 942459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1089 #items: array:1 [ 0 => App\Models\Edition {#1090 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "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 {#1091 …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 {#1093 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1094 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1095 …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 {#1091 …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 {#1093 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1094 …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" => "745082" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "667-1.png" "alt" => null "width" => "694" "height" => 1629 "path" => "/media/geometrija_10/atanasjan-u23/2-00/667-1.webp?ts=1739441554" ] 1 => array:5 [ "name" => "667-2.png" "alt" => null "width" => "694" "height" => 2143 "path" => "/media/geometrija_10/atanasjan-u23/2-00/667-2.webp?ts=1739441554" ] ] ] #original: array:6 [ "id" => 942459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1089} "task" => array:2 [ "refs" => "745082" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "667-1.png" "alt" => null "width" => "694" "height" => 1629 "path" => "/media/geometrija_10/atanasjan-u23/2-00/667-1.webp?ts=1739441554" ] 1 => array:5 [ "name" => "667-2.png" "alt" => null "width" => "694" "height" => 2143 "path" => "/media/geometrija_10/atanasjan-u23/2-00/667-2.webp?ts=1739441554" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Element {#1096 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 942324 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1097 #items: array:1 [ 0 => App\Models\Edition {#1098 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "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 {#1099 …2} "field_content_type" => "free" "field_content_mode" => "image" …13 ] #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" => "745082" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "667-1.jpg" "alt" => null "width" => "1320" "height" => 391 "path" => "/media/geometrija_10/atanasjan-u23/3-00/667-1.webp?ts=1739437126" ] 1 => array:5 [ "name" => "667-2.jpg" "alt" => null "width" => "1320" "height" => 1907 "path" => "/media/geometrija_10/atanasjan-u23/3-00/667-2.webp?ts=1739437126" ] ] ] #original: array:6 [ "id" => 942324 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1097} "task" => array:2 [ "refs" => "745082" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "667-1.jpg" "alt" => null "width" => "1320" "height" => 391 "path" => "/media/geometrija_10/atanasjan-u23/3-00/667-1.webp?ts=1739437126" ] 1 => array:5 [ "name" => "667-2.jpg" "alt" => null "width" => "1320" "height" => 1907 "path" => "/media/geometrija_10/atanasjan-u23/3-00/667-2.webp?ts=1739437126" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Element {#1104 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 941731 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1105 #items: array:1 [ 0 => App\Models\Edition {#1106 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "745082" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "667-1.jpg" "alt" => null "width" => "1404" "height" => 537 "path" => "/media/geometrija_10/atanasjan-u23/4-00/667-1.webp?ts=1739436691" ] 1 => array:5 [ "name" => "667-2.jpg" "alt" => null "width" => "1404" "height" => 2148 "path" => "/media/geometrija_10/atanasjan-u23/4-00/667-2.webp?ts=1739436691" ] 2 => array:5 [ "name" => "667-3.jpg" "alt" => null "width" => "1404" "height" => 237 "path" => "/media/geometrija_10/atanasjan-u23/4-00/667-3.webp?ts=1739436691" ] ] ] #original: array:6 [ "id" => 941731 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1105} "task" => array:2 [ "refs" => "745082" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "667-1.jpg" "alt" => null "width" => "1404" "height" => 537 "path" => "/media/geometrija_10/atanasjan-u23/4-00/667-1.webp?ts=1739436691" ] 1 => array:5 [ "name" => "667-2.jpg" "alt" => null "width" => "1404" "height" => 2148 "path" => "/media/geometrija_10/atanasjan-u23/4-00/667-2.webp?ts=1739436691" ] 2 => array:5 [ "name" => "667-3.jpg" "alt" => null "width" => "1404" "height" => 237 "path" => "/media/geometrija_10/atanasjan-u23/4-00/667-3.webp?ts=1739436691" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Element {#1112 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 941884 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1113 #items: array:1 [ 0 => App\Models\Edition {#1114 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "745082" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "667-1.jpg" "alt" => null "width" => "510" "height" => 564 "path" => "/media/geometrija_10/atanasjan-u23/5-00/667-1.webp?ts=1739436709" ] ] ] #original: array:6 [ "id" => 941884 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1113} "task" => array:2 [ "refs" => "745082" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "667-1.jpg" "alt" => null "width" => "510" "height" => 564 "path" => "/media/geometrija_10/atanasjan-u23/5-00/667-1.webp?ts=1739436709" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 5 => App\Models\Element {#1120 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1352395 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1121 #items: array:1 [ 0 => App\Models\Edition {#1122 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "745082" "type" => "task" ] "text" => "<p><strong>а)</strong></p><p>Периметр треугольника $P_{ABC}$ равен сумме длин его сторон: $P_{ABC} = AB + BC + AC$.</p><p>Для нахождения длин сторон воспользуемся формулой расстояния между двумя точками $M_1(x_1; y_1; z_1)$ и $M_2(x_2; y_2; z_2)$ в трехмерном пространстве:</p><p>$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$</p><p>Имеем координаты вершин треугольника: $A(\frac{3}{2}; 1; -2)$, $B(2; 2; -3)$ и $C(2; 0; -1)$.</p><p>1. Найдем длину стороны $AB$:</p><p>$AB = \sqrt{(2 - \frac{3}{2})^2 + (2-1)^2 + (-3 - (-2))^2} = \sqrt{(\frac{1}{2})^2 + 1^2 + (-1)^2} = \sqrt{\frac{1}{4} + 1 + 1} = \sqrt{\frac{1+4+4}{4}} = \sqrt{\frac{9}{4}} = \frac{3}{2}$.</p><p>2. Найдем длину стороны $BC$:</p><p>$BC = \sqrt{(2-2)^2 + (0-2)^2 + (-1 - (-3))^2} = \sqrt{0^2 + (-2)^2 + 2^2} = \sqrt{0 + 4 + 4} = \sqrt{8} = 2\sqrt{2}$.</p><p>3. Найдем длину стороны $AC$:</p><p>$AC = \sqrt{(2 - \frac{3}{2})^2 + (0-1)^2 + (-1 - (-2))^2} = \sqrt{(\frac{1}{2})^2 + (-1)^2 + 1^2} = \sqrt{\frac{1}{4} + 1 + 1} = \sqrt{\frac{1+4+4}{4}} = \sqrt{\frac{9}{4}} = \frac{3}{2}$.</p><p>4. Вычислим периметр треугольника $ABC$:</p><p>$P_{ABC} = AB + BC + AC = \frac{3}{2} + 2\sqrt{2} + \frac{3}{2} = 3 + 2\sqrt{2}$.</p><p><strong>Ответ:</strong> $3 + 2\sqrt{2}$.</p><p><strong>б)</strong></p><p>Медиана треугольника — это отрезок, соединяющий вершину треугольника с серединой противоположной стороны. Для нахождения длин медиан ($m_a$, $m_b$, $m_c$) сначала необходимо найти координаты середин сторон треугольника.</p><p>Координаты середины отрезка с концами в точках $(x_1; y_1; z_1)$ и $(x_2; y_2; z_2)$ находятся по формуле: $(\frac{x_1+x_2}{2}; \frac{y_1+y_2}{2}; \frac{z_1+z_2}{2})$.</p><p>1. Найдем медиану $m_a$, проведенную из вершины $A$ к стороне $BC$. Обозначим середину $BC$ как $M_a$.</p><p>Координаты точки $M_a$:</p><p>$M_a(\frac{2+2}{2}; \frac{2+0}{2}; \frac{-3+(-1)}{2}) = M_a(2; 1; -2)$.</p><p>Длина медианы $m_a$ равна расстоянию между точками $A(\frac{3}{2}; 1; -2)$ и $M_a(2; 1; -2)$:</p><p>$m_a = \sqrt{(2 - \frac{3}{2})^2 + (1-1)^2 + (-2 - (-2))^2} = \sqrt{(\frac{1}{2})^2 + 0^2 + 0^2} = \sqrt{\frac{1}{4}} = \frac{1}{2}$.</p><p>2. Найдем медиану $m_b$, проведенную из вершины $B$ к стороне $AC$. Обозначим середину $AC$ как $M_b$.</p><p>Координаты точки $M_b$:</p><p>$M_b(\frac{\frac{3}{2}+2}{2}; \frac{1+0}{2}; \frac{-2+(-1)}{2}) = M_b(\frac{7/2}{2}; \frac{1}{2}; \frac{-3}{2}) = M_b(\frac{7}{4}; \frac{1}{2}; -\frac{3}{2})$.</p><p>Длина медианы $m_b$ равна расстоянию между точками $B(2; 2; -3)$ и $M_b(\frac{7}{4}; \frac{1}{2}; -\frac{3}{2})$:</p><p>$m_b = \sqrt{(\frac{7}{4}-2)^2 + (\frac{1}{2}-2)^2 + (-\frac{3}{2} - (-3))^2} = \sqrt{(-\frac{1}{4})^2 + (-\frac{3}{2})^2 + (\frac{3}{2})^2} = \sqrt{\frac{1}{16} + \frac{9}{4} + \frac{9}{4}} = \sqrt{\frac{1+36+36}{16}} = \sqrt{\frac{73}{16}} = \frac{\sqrt{73}}{4}$.</p><p>3. Найдем медиану $m_c$, проведенную из вершины $C$ к стороне $AB$. Обозначим середину $AB$ как $M_c$.</p><p>Координаты точки $M_c$:</p><p>$M_c(\frac{\frac{3}{2}+2}{2}; \frac{1+2}{2}; \frac{-2+(-3)}{2}) = M_c(\frac{7/2}{2}; \frac{3}{2}; \frac{-5}{2}) = M_c(\frac{7}{4}; \frac{3}{2}; -\frac{5}{2})$.</p><p>Длина медианы $m_c$ равна расстоянию между точками $C(2; 0; -1)$ и $M_c(\frac{7}{4}; \frac{3}{2}; -\frac{5}{2})$:</p><p>$m_c = \sqrt{(\frac{7}{4}-2)^2 + (\frac{3}{2}-0)^2 + (-\frac{5}{2} - (-1))^2} = \sqrt{(-\frac{1}{4})^2 + (\frac{3}{2})^2 + (-\frac{3}{2})^2} = \sqrt{\frac{1}{16} + \frac{9}{4} + \frac{9}{4}} = \sqrt{\frac{1+36+36}{16}} = \sqrt{\frac{73}{16}} = \frac{\sqrt{73}}{4}$.</p><p><strong>Ответ:</strong> длины медиан равны $\frac{1}{2}$, $\frac{\sqrt{73}}{4}$, $\frac{\sqrt{73}}{4}$.</p>" ] #original: array:6 [ "id" => 1352395 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1121} "task" => array:2 [ "refs" => "745082" "type" => "task" ] "text" => "<p><strong>а)</strong></p><p>Периметр треугольника $P_{ABC}$ равен сумме длин его сторон: $P_{ABC} = AB + BC + AC$.</p><p>Для нахождения длин сторон воспользуемся формулой расстояния между двумя точками $M_1(x_1; y_1; z_1)$ и $M_2(x_2; y_2; z_2)$ в трехмерном пространстве:</p><p>$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$</p><p>Имеем координаты вершин треугольника: $A(\frac{3}{2}; 1; -2)$, $B(2; 2; -3)$ и $C(2; 0; -1)$.</p><p>1. Найдем длину стороны $AB$:</p><p>$AB = \sqrt{(2 - \frac{3}{2})^2 + (2-1)^2 + (-3 - (-2))^2} = \sqrt{(\frac{1}{2})^2 + 1^2 + (-1)^2} = \sqrt{\frac{1}{4} + 1 + 1} = \sqrt{\frac{1+4+4}{4}} = \sqrt{\frac{9}{4}} = \frac{3}{2}$.</p><p>2. Найдем длину стороны $BC$:</p><p>$BC = \sqrt{(2-2)^2 + (0-2)^2 + (-1 - (-3))^2} = \sqrt{0^2 + (-2)^2 + 2^2} = \sqrt{0 + 4 + 4} = \sqrt{8} = 2\sqrt{2}$.</p><p>3. Найдем длину стороны $AC$:</p><p>$AC = \sqrt{(2 - \frac{3}{2})^2 + (0-1)^2 + (-1 - (-2))^2} = \sqrt{(\frac{1}{2})^2 + (-1)^2 + 1^2} = \sqrt{\frac{1}{4} + 1 + 1} = \sqrt{\frac{1+4+4}{4}} = \sqrt{\frac{9}{4}} = \frac{3}{2}$.</p><p>4. Вычислим периметр треугольника $ABC$:</p><p>$P_{ABC} = AB + BC + AC = \frac{3}{2} + 2\sqrt{2} + \frac{3}{2} = 3 + 2\sqrt{2}$.</p><p><strong>Ответ:</strong> $3 + 2\sqrt{2}$.</p><p><strong>б)</strong></p><p>Медиана треугольника — это отрезок, соединяющий вершину треугольника с серединой противоположной стороны. Для нахождения длин медиан ($m_a$, $m_b$, $m_c$) сначала необходимо найти координаты середин сторон треугольника.</p><p>Координаты середины отрезка с концами в точках $(x_1; y_1; z_1)$ и $(x_2; y_2; z_2)$ находятся по формуле: $(\frac{x_1+x_2}{2}; \frac{y_1+y_2}{2}; \frac{z_1+z_2}{2})$.</p><p>1. Найдем медиану $m_a$, проведенную из вершины $A$ к стороне $BC$. Обозначим середину $BC$ как $M_a$.</p><p>Координаты точки $M_a$:</p><p>$M_a(\frac{2+2}{2}; \frac{2+0}{2}; \frac{-3+(-1)}{2}) = M_a(2; 1; -2)$.</p><p>Длина медианы $m_a$ равна расстоянию между точками $A(\frac{3}{2}; 1; -2)$ и $M_a(2; 1; -2)$:</p><p>$m_a = \sqrt{(2 - \frac{3}{2})^2 + (1-1)^2 + (-2 - (-2))^2} = \sqrt{(\frac{1}{2})^2 + 0^2 + 0^2} = \sqrt{\frac{1}{4}} = \frac{1}{2}$.</p><p>2. Найдем медиану $m_b$, проведенную из вершины $B$ к стороне $AC$. Обозначим середину $AC$ как $M_b$.</p><p>Координаты точки $M_b$:</p><p>$M_b(\frac{\frac{3}{2}+2}{2}; \frac{1+0}{2}; \frac{-2+(-1)}{2}) = M_b(\frac{7/2}{2}; \frac{1}{2}; \frac{-3}{2}) = M_b(\frac{7}{4}; \frac{1}{2}; -\frac{3}{2})$.</p><p>Длина медианы $m_b$ равна расстоянию между точками $B(2; 2; -3)$ и $M_b(\frac{7}{4}; \frac{1}{2}; -\frac{3}{2})$:</p><p>$m_b = \sqrt{(\frac{7}{4}-2)^2 + (\frac{1}{2}-2)^2 + (-\frac{3}{2} - (-3))^2} = \sqrt{(-\frac{1}{4})^2 + (-\frac{3}{2})^2 + (\frac{3}{2})^2} = \sqrt{\frac{1}{16} + \frac{9}{4} + \frac{9}{4}} = \sqrt{\frac{1+36+36}{16}} = \sqrt{\frac{73}{16}} = \frac{\sqrt{73}}{4}$.</p><p>3. Найдем медиану $m_c$, проведенную из вершины $C$ к стороне $AB$. Обозначим середину $AB$ как $M_c$.</p><p>Координаты точки $M_c$:</p><p>$M_c(\frac{\frac{3}{2}+2}{2}; \frac{1+2}{2}; \frac{-2+(-3)}{2}) = M_c(\frac{7/2}{2}; \frac{3}{2}; \frac{-5}{2}) = M_c(\frac{7}{4}; \frac{3}{2}; -\frac{5}{2})$.</p><p>Длина медианы $m_c$ равна расстоянию между точками $C(2; 0; -1)$ и $M_c(\frac{7}{4}; \frac{3}{2}; -\frac{5}{2})$:</p><p>$m_c = \sqrt{(\frac{7}{4}-2)^2 + (\frac{3}{2}-0)^2 + (-\frac{5}{2} - (-1))^2} = \sqrt{(-\frac{1}{4})^2 + (\frac{3}{2})^2 + (-\frac{3}{2})^2} = \sqrt{\frac{1}{16} + \frac{9}{4} + \frac{9}{4}} = \sqrt{\frac{1+36+36}{16}} = \sqrt{\frac{73}{16}} = \frac{\sqrt{73}}{4}$.</p><p><strong>Ответ:</strong> длины медиан равны $\frac{1}{2}$, $\frac{\sqrt{73}}{4}$, $\frac{\sqrt{73}}{4}$.</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" => "745083" "type" => "task" ] "previous" => array:2 [ "refs" => "745081" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1128 #items: array:1 [ 0 => App\Models\Book {#1032} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1534 #items: array:1 [ 0 => App\Models\BookPage {#1132 #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" => 933952 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "170" "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/page-170" "field_display_title" => "170" "field_folder" => "1" "field_image_name" => "170" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1136 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1135 #items: array:1 [ 0 => App\Models\Book {#1137 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ …50] #original: array:50 [ …50] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1172 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1173 #items: array:1 [ 0 => App\Models\Element {#1174 #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" => "933953" "type" => "book_page" ] "previous" => array:2 [ "refs" => "933951" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1182 #items: array:16 [ 0 => App\Models\Task {#1183 #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 {#1247 #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 {#1266 #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 {#1285 #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 {#1304 #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 {#1323 #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 {#1342 #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 {#1361 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 8 => App\Models\Task {#1380 #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 {#1399 #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 {#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] } 11 => App\Models\Task {#1437 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 12 => App\Models\Task {#1456 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 13 => App\Models\Task {#1475 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 14 => App\Models\Task {#1494 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 15 => App\Models\Task {#1513 #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" => 933952 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "170" "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/page-170" "field_display_title" => "170" "field_folder" => "1" "field_image_name" => "170" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1136} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1135} "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 {#1172} "content" => Illuminate\Database\Eloquent\Collection {#1173} "next" => array:2 [ "refs" => "933953" "type" => "book_page" ] "previous" => array:2 [ "refs" => "933951" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1182} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 745082 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "170" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/00-667" "field_display_title" => "667" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#335} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1026} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#341} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1069} "content" => Illuminate\Database\Eloquent\Collection {#1079} "next" => array:2 [ "refs" => "745083" "type" => "task" ] "previous" => array:2 [ "refs" => "745081" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1128} "page" => array:2 [ "refs" => "933952" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] }
№667 (с. 170)
Условие. №667 (с. 170)
скриншот условия
667. Даны точки
Найдите: а) периметр треугольника ABC; б) медианы треугольника ABC.
Решение 2. №667 (с. 170)
Решение 3. №667 (с. 170)
Решение 4. №667 (с. 170)
Решение 5. №667 (с. 170)
Решение 6. №667 (с. 170)
а)
Периметр треугольника $P_{ABC}$ равен сумме длин его сторон: $P_{ABC} = AB + BC + AC$.
Для нахождения длин сторон воспользуемся формулой расстояния между двумя точками $M_1(x_1; y_1; z_1)$ и $M_2(x_2; y_2; z_2)$ в трехмерном пространстве:
$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$
Имеем координаты вершин треугольника: $A(\frac{3}{2}; 1; -2)$, $B(2; 2; -3)$ и $C(2; 0; -1)$.
1. Найдем длину стороны $AB$:
$AB = \sqrt{(2 - \frac{3}{2})^2 + (2-1)^2 + (-3 - (-2))^2} = \sqrt{(\frac{1}{2})^2 + 1^2 + (-1)^2} = \sqrt{\frac{1}{4} + 1 + 1} = \sqrt{\frac{1+4+4}{4}} = \sqrt{\frac{9}{4}} = \frac{3}{2}$.
2. Найдем длину стороны $BC$:
$BC = \sqrt{(2-2)^2 + (0-2)^2 + (-1 - (-3))^2} = \sqrt{0^2 + (-2)^2 + 2^2} = \sqrt{0 + 4 + 4} = \sqrt{8} = 2\sqrt{2}$.
3. Найдем длину стороны $AC$:
$AC = \sqrt{(2 - \frac{3}{2})^2 + (0-1)^2 + (-1 - (-2))^2} = \sqrt{(\frac{1}{2})^2 + (-1)^2 + 1^2} = \sqrt{\frac{1}{4} + 1 + 1} = \sqrt{\frac{1+4+4}{4}} = \sqrt{\frac{9}{4}} = \frac{3}{2}$.
4. Вычислим периметр треугольника $ABC$:
$P_{ABC} = AB + BC + AC = \frac{3}{2} + 2\sqrt{2} + \frac{3}{2} = 3 + 2\sqrt{2}$.
Ответ: $3 + 2\sqrt{2}$.
б)
Медиана треугольника — это отрезок, соединяющий вершину треугольника с серединой противоположной стороны. Для нахождения длин медиан ($m_a$, $m_b$, $m_c$) сначала необходимо найти координаты середин сторон треугольника.
Координаты середины отрезка с концами в точках $(x_1; y_1; z_1)$ и $(x_2; y_2; z_2)$ находятся по формуле: $(\frac{x_1+x_2}{2}; \frac{y_1+y_2}{2}; \frac{z_1+z_2}{2})$.
1. Найдем медиану $m_a$, проведенную из вершины $A$ к стороне $BC$. Обозначим середину $BC$ как $M_a$.
Координаты точки $M_a$:
$M_a(\frac{2+2}{2}; \frac{2+0}{2}; \frac{-3+(-1)}{2}) = M_a(2; 1; -2)$.
Длина медианы $m_a$ равна расстоянию между точками $A(\frac{3}{2}; 1; -2)$ и $M_a(2; 1; -2)$:
$m_a = \sqrt{(2 - \frac{3}{2})^2 + (1-1)^2 + (-2 - (-2))^2} = \sqrt{(\frac{1}{2})^2 + 0^2 + 0^2} = \sqrt{\frac{1}{4}} = \frac{1}{2}$.
2. Найдем медиану $m_b$, проведенную из вершины $B$ к стороне $AC$. Обозначим середину $AC$ как $M_b$.
Координаты точки $M_b$:
$M_b(\frac{\frac{3}{2}+2}{2}; \frac{1+0}{2}; \frac{-2+(-1)}{2}) = M_b(\frac{7/2}{2}; \frac{1}{2}; \frac{-3}{2}) = M_b(\frac{7}{4}; \frac{1}{2}; -\frac{3}{2})$.
Длина медианы $m_b$ равна расстоянию между точками $B(2; 2; -3)$ и $M_b(\frac{7}{4}; \frac{1}{2}; -\frac{3}{2})$:
$m_b = \sqrt{(\frac{7}{4}-2)^2 + (\frac{1}{2}-2)^2 + (-\frac{3}{2} - (-3))^2} = \sqrt{(-\frac{1}{4})^2 + (-\frac{3}{2})^2 + (\frac{3}{2})^2} = \sqrt{\frac{1}{16} + \frac{9}{4} + \frac{9}{4}} = \sqrt{\frac{1+36+36}{16}} = \sqrt{\frac{73}{16}} = \frac{\sqrt{73}}{4}$.
3. Найдем медиану $m_c$, проведенную из вершины $C$ к стороне $AB$. Обозначим середину $AB$ как $M_c$.
Координаты точки $M_c$:
$M_c(\frac{\frac{3}{2}+2}{2}; \frac{1+2}{2}; \frac{-2+(-3)}{2}) = M_c(\frac{7/2}{2}; \frac{3}{2}; \frac{-5}{2}) = M_c(\frac{7}{4}; \frac{3}{2}; -\frac{5}{2})$.
Длина медианы $m_c$ равна расстоянию между точками $C(2; 0; -1)$ и $M_c(\frac{7}{4}; \frac{3}{2}; -\frac{5}{2})$:
$m_c = \sqrt{(\frac{7}{4}-2)^2 + (\frac{3}{2}-0)^2 + (-\frac{5}{2} - (-1))^2} = \sqrt{(-\frac{1}{4})^2 + (\frac{3}{2})^2 + (-\frac{3}{2})^2} = \sqrt{\frac{1}{16} + \frac{9}{4} + \frac{9}{4}} = \sqrt{\frac{1+36+36}{16}} = \sqrt{\frac{73}{16}} = \frac{\sqrt{73}}{4}$.
Ответ: длины медиан равны $\frac{1}{2}$, $\frac{\sqrt{73}}{4}$, $\frac{\sqrt{73}}{4}$.
Другие задания:
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