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Номер 571, страница 149 - 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 6. Векторы в пространстве. Параграф 2. Сложение и вычитание векторов. Умножение вектора на число - номер 571, страница 149.
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" => 744986 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "149" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/00-571" "field_display_title" => "571" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1079 #items: array:1 [ 0 => App\Models\Branch {#1034 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 744399 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Векторы в пространстве" "field_branch_order" => "6" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039 #items: array:1 [ 0 => App\Models\Term {#1038 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "142" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1040 #items: array:1 [ 0 => App\Models\Book {#1041 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 36 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1042 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1047 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1049 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1070 …2} "field_under_the_edition_degree" => null "field_cover_description" => "с ромбами" "field_publication_year" => "2019" "field_publication_year_until" => null …26 ] #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 } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1076 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 744399 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Векторы в пространстве" "field_branch_order" => "6" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039} "field_page_start" => "142" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1040} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1076} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1163 #items: array:2 [ 0 => App\Models\Branch {#1077 #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" => 744399 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Векторы в пространстве" "field_branch_order" => "6" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1080 #items: array:1 [ 0 => App\Models\Term {#1078 #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 [ …7] #original: array:7 [ …7] #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_page_start" => "142" "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 {#1081 #items: array:1 [ 0 => App\Models\Book {#1082 #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 } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1117 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 744399 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Векторы в пространстве" "field_branch_order" => "6" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1080} "field_page_start" => "142" "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 {#1081} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1117} ] #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 {#1118 #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" => 744401 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Сложение и вычитание векторов. Умножение вектора на число" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1119 #items: array:1 [ 0 => App\Models\Term {#1120 #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 [ …7] #original: array:7 [ …7] #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_page_start" => "145" "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 {#1121 #items: array:1 [ 0 => App\Models\Book {#1122 #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 } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1157 #items: array:1 [ 0 => App\Models\Branch {#1158 #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 [ …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:24 [ "id" => 744401 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Сложение и вычитание векторов. Умножение вектора на число" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1119} "field_page_start" => "145" "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 {#1121} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1157} ] #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 {#1164 #items: array:6 [ 0 => App\Models\Element {#1178 #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" => 939252 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1187 #items: array:1 [ 0 => App\Models\Edition {#1179 #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" => "744986" "type" => "task" ] "text" => """ <p><strong>571.</strong> Дан прямоугольный параллелепипед <em>KLMNK₁L₁M₁N₁</em>. Докажите, что:</p><figure><figure>\n <img src="/media/geometrija_10/atanasjan-u23/0-00/571.webp?ts=1745678707" alt="Дан прямоугольный параллелепипед, доказать" loading="lazy" width="1396" height="209">\n </figure>\n <figcaption> </figcaption></figure> """ "img" => array:1 [ 0 => array:5 [ "name" => "571-1.jpg" "alt" => null "width" => "1545" "height" => 242 "path" => "/media/geometrija_10/atanasjan-u23/0-00/571-1.webp?ts=1739436735" ] ] ] #original: array:7 [ "id" => 939252 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1187} "task" => array:2 [ "refs" => "744986" "type" => "task" ] "text" => """ <p><strong>571.</strong> Дан прямоугольный параллелепипед <em>KLMNK₁L₁M₁N₁</em>. Докажите, что:</p><figure><figure>\n <img src="/media/geometrija_10/atanasjan-u23/0-00/571.webp?ts=1745678707" alt="Дан прямоугольный параллелепипед, доказать" loading="lazy" width="1396" height="209">\n </figure>\n <figcaption> </figcaption></figure> """ "img" => array:1 [ 0 => array:5 [ "name" => "571-1.jpg" "alt" => null "width" => "1545" "height" => 242 "path" => "/media/geometrija_10/atanasjan-u23/0-00/571-1.webp?ts=1739436735" ] ] ] #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 {#1209 #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" => 942222 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1219 #items: array:1 [ 0 => App\Models\Edition {#1210 #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" => "744986" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "571-1.png" "alt" => null "width" => "694" "height" => 3667 "path" => "/media/geometrija_10/atanasjan-u23/2-00/571-1.webp?ts=1739441422" ] 1 => array:5 [ "name" => "571-2.png" "alt" => null "width" => "694" "height" => 3686 "path" => "/media/geometrija_10/atanasjan-u23/2-00/571-2.webp?ts=1739441422" ] 2 => array:5 [ "name" => "571-3.png" "alt" => null "width" => "694" "height" => 2833 "path" => "/media/geometrija_10/atanasjan-u23/2-00/571-3.webp?ts=1739441422" ] ] ] #original: array:6 [ "id" => 942222 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1219} "task" => array:2 [ "refs" => "744986" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "571-1.png" "alt" => null "width" => "694" "height" => 3667 "path" => "/media/geometrija_10/atanasjan-u23/2-00/571-1.webp?ts=1739441422" ] 1 => array:5 [ "name" => "571-2.png" "alt" => null "width" => "694" "height" => 3686 "path" => "/media/geometrija_10/atanasjan-u23/2-00/571-2.webp?ts=1739441422" ] 2 => array:5 [ "name" => "571-3.png" "alt" => null "width" => "694" "height" => 2833 "path" => "/media/geometrija_10/atanasjan-u23/2-00/571-3.webp?ts=1739441422" ] ] ] #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 {#1201 #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" => 942164 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1211 #items: array:1 [ 0 => App\Models\Edition {#1202 #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" => "744986" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "571-1.jpg" "alt" => null "width" => "1362" "height" => 1343 "path" => "/media/geometrija_10/atanasjan-u23/3-00/571-1.webp?ts=1739437029" ] ] ] #original: array:6 [ "id" => 942164 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1211} "task" => array:2 [ "refs" => "744986" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "571-1.jpg" "alt" => null "width" => "1362" "height" => 1343 "path" => "/media/geometrija_10/atanasjan-u23/3-00/571-1.webp?ts=1739437029" ] ] ] #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 {#1185 #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" => 941412 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1195 #items: array:1 [ 0 => App\Models\Edition {#1186 #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" => "744986" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "571-1.jpg" "alt" => null "width" => "1124" "height" => 173 "path" => "/media/geometrija_10/atanasjan-u23/4-00/571-1.webp?ts=1739436631" ] 1 => array:5 [ "name" => "571-2.jpg" "alt" => null "width" => "1124" "height" => 599 "path" => "/media/geometrija_10/atanasjan-u23/4-00/571-2.webp?ts=1739436631" ] ] ] #original: array:6 [ "id" => 941412 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1195} "task" => array:2 [ "refs" => "744986" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "571-1.jpg" "alt" => null "width" => "1124" "height" => 173 "path" => "/media/geometrija_10/atanasjan-u23/4-00/571-1.webp?ts=1739436631" ] 1 => array:5 [ "name" => "571-2.jpg" "alt" => null "width" => "1124" "height" => 599 "path" => "/media/geometrija_10/atanasjan-u23/4-00/571-2.webp?ts=1739436631" ] ] ] #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 {#1193 #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" => 941528 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1203 #items: array:1 [ 0 => App\Models\Edition {#1194 #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" => "744986" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "571-1.jpg" "alt" => null "width" => "1329" "height" => 907 "path" => "/media/geometrija_10/atanasjan-u23/5-00/571-1.webp?ts=1739436631" ] ] ] #original: array:6 [ "id" => 941528 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1203} "task" => array:2 [ "refs" => "744986" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "571-1.jpg" "alt" => null "width" => "1329" "height" => 907 "path" => "/media/geometrija_10/atanasjan-u23/5-00/571-1.webp?ts=1739436631" ] ] ] #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 {#1217 #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" => 1352299 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1227 #items: array:1 [ 0 => App\Models\Edition {#1218 #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" => "744986" "type" => "task" ] "text" => "<p><strong>а) Докажите, что $|\vec{MK} + \vec{MM_1}| = |\vec{MK} - \vec{MM_1}|$</strong></p><p>Рассмотрим левую часть равенства: $|\vec{MK} + \vec{MM_1}|$.<br>Векторы $\vec{MK}$ и $\vec{MM_1}$ выходят из одной точки $M$. Так как $KLMNK_1L_1M_1N_1$ — прямоугольный параллелепипед, его грань $MKK_1M_1$ является прямоугольником. Векторы $\vec{MK}$ и $\vec{MM_1}$ являются смежными сторонами этого прямоугольника. По правилу параллелограмма для сложения векторов, их сумма равна вектору диагонали, исходящей из той же точки: $\vec{MK} + \vec{MM_1} = \vec{MK_1}$.<br>Следовательно, модуль их суммы равен длине этой диагонали: $|\vec{MK} + \vec{MM_1}| = |\vec{MK_1}| = MK_1$.</p><p>Рассмотрим правую часть равенства: $|\vec{MK} - \vec{MM_1}|$.<br>Разность двух векторов, исходящих из одной точки, равна вектору, соединяющему конец второго вектора с концом первого. В данном случае это вектор $\vec{M_1K}$. $\vec{MK} - \vec{MM_1} = \vec{M_1K}$.<br>Следовательно, модуль их разности равен длине этого вектора: $|\vec{MK} - \vec{MM_1}| = |\vec{M_1K}| = M_1K$.</p><p>Таким образом, равенство сводится к доказательству того, что $MK_1 = M_1K$.<br>Отрезки $MK_1$ и $M_1K$ являются диагоналями грани $MKK_1M_1$. Поскольку в прямоугольном параллелепипеде все грани являются прямоугольниками, а диагонали прямоугольника равны, то $MK_1 = M_1K$.<br>Следовательно, исходное равенство $|\vec{MK} + \vec{MM_1}| = |\vec{MK} - \vec{MM_1}|$ верно.<br>Ответ: Равенство доказано.</p><p><strong>б) Докажите, что $|\vec{K_1L_1} - \vec{NL_1}| = |\vec{ML} + \vec{MM_1}|$</strong></p><p>Рассмотрим левую часть равенства: $|\vec{K_1L_1} - \vec{NL_1}|$.<br>Преобразуем разность векторов: $\vec{K_1L_1} - \vec{NL_1} = \vec{K_1L_1} + (-\vec{NL_1}) = \vec{K_1L_1} + \vec{L_1N}$.<br>По правилу треугольника (или цепи) для сложения векторов: $\vec{K_1L_1} + \vec{L_1N} = \vec{K_1N}$.<br>Таким образом, $|\vec{K_1L_1} - \vec{NL_1}| = |\vec{K_1N}| = K_1N$. $K_1N$ - это диагональ боковой грани $K_1N_1NK$.</p><p>Рассмотрим правую часть равенства: $|\vec{ML} + \vec{MM_1}|$.<br>Векторы $\vec{ML}$ и $\vec{MM_1}$ исходят из одной точки $M$ и являются смежными сторонами прямоугольной грани $MLL_1M_1$. По правилу параллелограмма, их сумма равна вектору диагонали этой грани: $\vec{ML} + \vec{MM_1} = \vec{ML_1}$.<br>Следовательно, $|\vec{ML} + \vec{MM_1}| = |\vec{ML_1}| = ML_1$. $ML_1$ - это диагональ боковой грани $MLL_1M_1$.</p><p>Равенство сводится к доказательству того, что $K_1N = ML_1$.<br>В прямоугольном параллелепипеде противолежащие грани равны (являются конгруэнтными прямоугольниками). Грани $K_1N_1NK$ и $L_1M_1ML$ являются противолежащими. Следовательно, прямоугольник $K_1N_1NK$ равен прямоугольнику $MLL_1M_1$.<br>Диагонали равных прямоугольников равны, поэтому $K_1N = ML_1$.<br>Следовательно, исходное равенство $|\vec{K_1L_1} - \vec{NL_1}| = |\vec{ML} + \vec{MM_1}|$ верно.<br>Ответ: Равенство доказано.</p><p><strong>в) Докажите, что $|\vec{NL} - \vec{M_1L}| = |\vec{K_1N} - \vec{LN}|$</strong></p><p>Рассмотрим левую часть равенства: $|\vec{NL} - \vec{M_1L}|$.<br>Векторы $\vec{NL}$ и $\vec{M_1L}$ имеют общий конец в точке $L$. Разность таких векторов равна вектору, соединяющему их начальные точки (от начала второго к началу первого): $\vec{NL} - \vec{M_1L} = \vec{M_1N}$.<br>*Доказательство этого факта: $\vec{NL} - \vec{M_1L} = (\vec{L}-\vec{N}) - (\vec{L}-\vec{M_1}) = \vec{L}-\vec{N}-\vec{L}+\vec{M_1} = \vec{M_1}-\vec{N} = \vec{NM_1}$. Ой, здесь ошибка в направлении. Правильно: $\vec{M_1N}$*. *Перепроверка: разность векторов $\vec{a}$ и $\vec{b}$ с общим концом $C$, т.е. $\vec{AC} - \vec{BC}$, равна вектору $\vec{AB}$. В нашем случае $\vec{N}$ - это $A$, $\vec{M_1}$ - это $B$, $\vec{L}$ - это $C$. Значит, $\vec{NL} - \vec{M_1L} = \vec{NM_1}$.* Итак, $\vec{NL} - \vec{M_1L} = \vec{NM_1}$.<br>Следовательно, $|\vec{NL} - \vec{M_1L}| = |\vec{NM_1}| = NM_1$.</p><p>Рассмотрим правую часть равенства: $|\vec{K_1N} - \vec{LN}|$.<br>Аналогично, векторы $\vec{K_1N}$ и $\vec{LN}$ имеют общий конец в точке $N$. Их разность равна вектору, соединяющему их начальные точки: $\vec{K_1N} - \vec{LN} = \vec{K_1L}$.<br>Следовательно, $|\vec{K_1N} - \vec{LN}| = |\vec{K_1L}| = K_1L$.</p><p>Равенство сводится к доказательству того, что $NM_1 = K_1L$.<br>Рассмотрим отрезок $NM_1$. В прямоугольном параллелепипеде ребро $MM_1$ перпендикулярно плоскости основания $KLMN$, а значит, перпендикулярно любой прямой в этой плоскости, в том числе и ребру $NM$. Таким образом, треугольник $\triangle NMM_1$ является прямоугольным с прямым углом при вершине $M$. По теореме Пифагора: $NM_1^2 = NM^2 + MM_1^2$.<br>Рассмотрим отрезок $K_1L$. Ребро $KK_1$ перпендикулярно плоскости основания $KLMN$, а значит, перпендикулярно ребру $KL$. Таким образом, треугольник $\triangle K_1KL$ является прямоугольным с прямым углом при вершине $K$. По теореме Пифагора: $K_1L^2 = KL^2 + KK_1^2$.<br>В параллелепипеде противолежащие ребра равны: $NM = KL$ и $MM_1 = KK_1$. Следовательно, $NM^2 + MM_1^2 = KL^2 + KK_1^2$, что означает $NM_1^2 = K_1L^2$, и $NM_1 = K_1L$.<br>Таким образом, исходное равенство $|\vec{NL} - \vec{M_1L}| = |\vec{K_1N} - \vec{LN}|$ верно.<br>Ответ: Равенство доказано.</p>" ] #original: array:6 [ "id" => 1352299 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1227} "task" => array:2 [ "refs" => "744986" "type" => "task" ] "text" => "<p><strong>а) Докажите, что $|\vec{MK} + \vec{MM_1}| = |\vec{MK} - \vec{MM_1}|$</strong></p><p>Рассмотрим левую часть равенства: $|\vec{MK} + \vec{MM_1}|$.<br>Векторы $\vec{MK}$ и $\vec{MM_1}$ выходят из одной точки $M$. Так как $KLMNK_1L_1M_1N_1$ — прямоугольный параллелепипед, его грань $MKK_1M_1$ является прямоугольником. Векторы $\vec{MK}$ и $\vec{MM_1}$ являются смежными сторонами этого прямоугольника. По правилу параллелограмма для сложения векторов, их сумма равна вектору диагонали, исходящей из той же точки: $\vec{MK} + \vec{MM_1} = \vec{MK_1}$.<br>Следовательно, модуль их суммы равен длине этой диагонали: $|\vec{MK} + \vec{MM_1}| = |\vec{MK_1}| = MK_1$.</p><p>Рассмотрим правую часть равенства: $|\vec{MK} - \vec{MM_1}|$.<br>Разность двух векторов, исходящих из одной точки, равна вектору, соединяющему конец второго вектора с концом первого. В данном случае это вектор $\vec{M_1K}$. $\vec{MK} - \vec{MM_1} = \vec{M_1K}$.<br>Следовательно, модуль их разности равен длине этого вектора: $|\vec{MK} - \vec{MM_1}| = |\vec{M_1K}| = M_1K$.</p><p>Таким образом, равенство сводится к доказательству того, что $MK_1 = M_1K$.<br>Отрезки $MK_1$ и $M_1K$ являются диагоналями грани $MKK_1M_1$. Поскольку в прямоугольном параллелепипеде все грани являются прямоугольниками, а диагонали прямоугольника равны, то $MK_1 = M_1K$.<br>Следовательно, исходное равенство $|\vec{MK} + \vec{MM_1}| = |\vec{MK} - \vec{MM_1}|$ верно.<br>Ответ: Равенство доказано.</p><p><strong>б) Докажите, что $|\vec{K_1L_1} - \vec{NL_1}| = |\vec{ML} + \vec{MM_1}|$</strong></p><p>Рассмотрим левую часть равенства: $|\vec{K_1L_1} - \vec{NL_1}|$.<br>Преобразуем разность векторов: $\vec{K_1L_1} - \vec{NL_1} = \vec{K_1L_1} + (-\vec{NL_1}) = \vec{K_1L_1} + \vec{L_1N}$.<br>По правилу треугольника (или цепи) для сложения векторов: $\vec{K_1L_1} + \vec{L_1N} = \vec{K_1N}$.<br>Таким образом, $|\vec{K_1L_1} - \vec{NL_1}| = |\vec{K_1N}| = K_1N$. $K_1N$ - это диагональ боковой грани $K_1N_1NK$.</p><p>Рассмотрим правую часть равенства: $|\vec{ML} + \vec{MM_1}|$.<br>Векторы $\vec{ML}$ и $\vec{MM_1}$ исходят из одной точки $M$ и являются смежными сторонами прямоугольной грани $MLL_1M_1$. По правилу параллелограмма, их сумма равна вектору диагонали этой грани: $\vec{ML} + \vec{MM_1} = \vec{ML_1}$.<br>Следовательно, $|\vec{ML} + \vec{MM_1}| = |\vec{ML_1}| = ML_1$. $ML_1$ - это диагональ боковой грани $MLL_1M_1$.</p><p>Равенство сводится к доказательству того, что $K_1N = ML_1$.<br>В прямоугольном параллелепипеде противолежащие грани равны (являются конгруэнтными прямоугольниками). Грани $K_1N_1NK$ и $L_1M_1ML$ являются противолежащими. Следовательно, прямоугольник $K_1N_1NK$ равен прямоугольнику $MLL_1M_1$.<br>Диагонали равных прямоугольников равны, поэтому $K_1N = ML_1$.<br>Следовательно, исходное равенство $|\vec{K_1L_1} - \vec{NL_1}| = |\vec{ML} + \vec{MM_1}|$ верно.<br>Ответ: Равенство доказано.</p><p><strong>в) Докажите, что $|\vec{NL} - \vec{M_1L}| = |\vec{K_1N} - \vec{LN}|$</strong></p><p>Рассмотрим левую часть равенства: $|\vec{NL} - \vec{M_1L}|$.<br>Векторы $\vec{NL}$ и $\vec{M_1L}$ имеют общий конец в точке $L$. Разность таких векторов равна вектору, соединяющему их начальные точки (от начала второго к началу первого): $\vec{NL} - \vec{M_1L} = \vec{M_1N}$.<br>*Доказательство этого факта: $\vec{NL} - \vec{M_1L} = (\vec{L}-\vec{N}) - (\vec{L}-\vec{M_1}) = \vec{L}-\vec{N}-\vec{L}+\vec{M_1} = \vec{M_1}-\vec{N} = \vec{NM_1}$. Ой, здесь ошибка в направлении. Правильно: $\vec{M_1N}$*. *Перепроверка: разность векторов $\vec{a}$ и $\vec{b}$ с общим концом $C$, т.е. $\vec{AC} - \vec{BC}$, равна вектору $\vec{AB}$. В нашем случае $\vec{N}$ - это $A$, $\vec{M_1}$ - это $B$, $\vec{L}$ - это $C$. Значит, $\vec{NL} - \vec{M_1L} = \vec{NM_1}$.* Итак, $\vec{NL} - \vec{M_1L} = \vec{NM_1}$.<br>Следовательно, $|\vec{NL} - \vec{M_1L}| = |\vec{NM_1}| = NM_1$.</p><p>Рассмотрим правую часть равенства: $|\vec{K_1N} - \vec{LN}|$.<br>Аналогично, векторы $\vec{K_1N}$ и $\vec{LN}$ имеют общий конец в точке $N$. Их разность равна вектору, соединяющему их начальные точки: $\vec{K_1N} - \vec{LN} = \vec{K_1L}$.<br>Следовательно, $|\vec{K_1N} - \vec{LN}| = |\vec{K_1L}| = K_1L$.</p><p>Равенство сводится к доказательству того, что $NM_1 = K_1L$.<br>Рассмотрим отрезок $NM_1$. В прямоугольном параллелепипеде ребро $MM_1$ перпендикулярно плоскости основания $KLMN$, а значит, перпендикулярно любой прямой в этой плоскости, в том числе и ребру $NM$. Таким образом, треугольник $\triangle NMM_1$ является прямоугольным с прямым углом при вершине $M$. По теореме Пифагора: $NM_1^2 = NM^2 + MM_1^2$.<br>Рассмотрим отрезок $K_1L$. Ребро $KK_1$ перпендикулярно плоскости основания $KLMN$, а значит, перпендикулярно ребру $KL$. Таким образом, треугольник $\triangle K_1KL$ является прямоугольным с прямым углом при вершине $K$. По теореме Пифагора: $K_1L^2 = KL^2 + KK_1^2$.<br>В параллелепипеде противолежащие ребра равны: $NM = KL$ и $MM_1 = KK_1$. Следовательно, $NM^2 + MM_1^2 = KL^2 + KK_1^2$, что означает $NM_1^2 = K_1L^2$, и $NM_1 = K_1L$.<br>Таким образом, исходное равенство $|\vec{NL} - \vec{M_1L}| = |\vec{K_1N} - \vec{LN}|$ верно.<br>Ответ: Равенство доказано.</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" => "744987" "type" => "task" ] "previous" => array:2 [ "refs" => "744985" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1251 #items: array:1 [ 0 => App\Models\Book {#1176 #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 {#1165 #items: array:1 [ 0 => App\Models\Term {#1166 #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 {#1175 #items: array:2 [ 0 => App\Models\Term {#1174 #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 {#1172 #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 {#1171 #items: array:1 [ 0 => App\Models\Term {#1173 #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 {#1169 #items: array:5 [ 0 => App\Models\Term {#1167 #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 {#1170 #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 {#1168 #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 {#1225 #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 {#1226 #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 {#1228 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1229 #items: array:1 [ 0 => App\Models\Term {#1230 #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 {#1231 #items: array:1 [ 0 => App\Models\Term {#1232 #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 {#1233 #items: array:1 [ 0 => App\Models\Term {#1234 #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 {#1235 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1236 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1237 #items: array:1 [ 0 => App\Models\Term {#1238 #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 {#1239 #items: array:1 [ 0 => App\Models\Term {#1240 #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 {#1241 #items: array:1 [ 0 => App\Models\Term {#1242 #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 {#1243 #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 {#1244 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1245 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1246 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1247 #items: array:1 [ 0 => App\Models\Term {#1248 #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 {#1165} "field_class" => Illuminate\Database\Eloquent\Collection {#1175} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1171} "field_author" => Illuminate\Database\Eloquent\Collection {#1169} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1228} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1229} "field_country" => Illuminate\Database\Eloquent\Collection {#1231} "field_city" => Illuminate\Database\Eloquent\Collection {#1233} "field_series" => Illuminate\Database\Eloquent\Collection {#1235} "field_umk" => Illuminate\Database\Eloquent\Collection {#1236} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1237} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1239} "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1241} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1243} "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 {#1244} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1245} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1246} "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1247} "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: 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#escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1258 #items: array:1 [ 0 => App\Models\Book {#1176} ] #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 {#1259 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1269 #items: array:1 [ 0 => App\Models\Element {#1268 #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" => "933932" "type" => "book_page" ] "previous" => array:2 [ "refs" => "933930" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1481 #items: array:15 [ 0 => App\Models\Task {#1503 #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 {#1510 #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 {#1517 #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 {#1520 #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 {#1519 #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 {#1521 #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 {#1541 #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 {#1561 #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 {#1581 #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 {#1601 #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 {#1621 #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 {#1641 #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 {#1661 #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 {#1681 #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 {#1701 #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" => 933931 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "149" "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/page-149" "field_display_title" => "149" "field_folder" => "1" "field_image_name" => "149" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1257} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1258} "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 {#1259} "content" => Illuminate\Database\Eloquent\Collection {#1269} "next" => array:2 [ "refs" => "933932" "type" => "book_page" ] "previous" => array:2 [ "refs" => "933930" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1481} ] #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" => 744986 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "149" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/00-571" "field_display_title" => "571" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037} "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} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1079} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1163} "content" => Illuminate\Database\Eloquent\Collection {#1164} "next" => array:2 [ "refs" => "744987" "type" => "task" ] "previous" => array:2 [ "refs" => "744985" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1251} "page" => array:2 [ "refs" => "933931" "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 => "*" ] }
№571 (с. 149)
Условие. №571 (с. 149)
скриншот условия
571. Дан прямоугольный параллелепипед KLMNK₁L₁M₁N₁. Докажите, что:
Решение 2. №571 (с. 149)
Решение 3. №571 (с. 149)
Решение 4. №571 (с. 149)
Решение 5. №571 (с. 149)
Решение 6. №571 (с. 149)
а) Докажите, что $|\vec{MK} + \vec{MM_1}| = |\vec{MK} - \vec{MM_1}|$
Рассмотрим левую часть равенства: $|\vec{MK} + \vec{MM_1}|$.
Векторы $\vec{MK}$ и $\vec{MM_1}$ выходят из одной точки $M$. Так как $KLMNK_1L_1M_1N_1$ — прямоугольный параллелепипед, его грань $MKK_1M_1$ является прямоугольником. Векторы $\vec{MK}$ и $\vec{MM_1}$ являются смежными сторонами этого прямоугольника. По правилу параллелограмма для сложения векторов, их сумма равна вектору диагонали, исходящей из той же точки: $\vec{MK} + \vec{MM_1} = \vec{MK_1}$.
Следовательно, модуль их суммы равен длине этой диагонали: $|\vec{MK} + \vec{MM_1}| = |\vec{MK_1}| = MK_1$.
Рассмотрим правую часть равенства: $|\vec{MK} - \vec{MM_1}|$.
Разность двух векторов, исходящих из одной точки, равна вектору, соединяющему конец второго вектора с концом первого. В данном случае это вектор $\vec{M_1K}$. $\vec{MK} - \vec{MM_1} = \vec{M_1K}$.
Следовательно, модуль их разности равен длине этого вектора: $|\vec{MK} - \vec{MM_1}| = |\vec{M_1K}| = M_1K$.
Таким образом, равенство сводится к доказательству того, что $MK_1 = M_1K$.
Отрезки $MK_1$ и $M_1K$ являются диагоналями грани $MKK_1M_1$. Поскольку в прямоугольном параллелепипеде все грани являются прямоугольниками, а диагонали прямоугольника равны, то $MK_1 = M_1K$.
Следовательно, исходное равенство $|\vec{MK} + \vec{MM_1}| = |\vec{MK} - \vec{MM_1}|$ верно.
Ответ: Равенство доказано.
б) Докажите, что $|\vec{K_1L_1} - \vec{NL_1}| = |\vec{ML} + \vec{MM_1}|$
Рассмотрим левую часть равенства: $|\vec{K_1L_1} - \vec{NL_1}|$.
Преобразуем разность векторов: $\vec{K_1L_1} - \vec{NL_1} = \vec{K_1L_1} + (-\vec{NL_1}) = \vec{K_1L_1} + \vec{L_1N}$.
По правилу треугольника (или цепи) для сложения векторов: $\vec{K_1L_1} + \vec{L_1N} = \vec{K_1N}$.
Таким образом, $|\vec{K_1L_1} - \vec{NL_1}| = |\vec{K_1N}| = K_1N$. $K_1N$ - это диагональ боковой грани $K_1N_1NK$.
Рассмотрим правую часть равенства: $|\vec{ML} + \vec{MM_1}|$.
Векторы $\vec{ML}$ и $\vec{MM_1}$ исходят из одной точки $M$ и являются смежными сторонами прямоугольной грани $MLL_1M_1$. По правилу параллелограмма, их сумма равна вектору диагонали этой грани: $\vec{ML} + \vec{MM_1} = \vec{ML_1}$.
Следовательно, $|\vec{ML} + \vec{MM_1}| = |\vec{ML_1}| = ML_1$. $ML_1$ - это диагональ боковой грани $MLL_1M_1$.
Равенство сводится к доказательству того, что $K_1N = ML_1$.
В прямоугольном параллелепипеде противолежащие грани равны (являются конгруэнтными прямоугольниками). Грани $K_1N_1NK$ и $L_1M_1ML$ являются противолежащими. Следовательно, прямоугольник $K_1N_1NK$ равен прямоугольнику $MLL_1M_1$.
Диагонали равных прямоугольников равны, поэтому $K_1N = ML_1$.
Следовательно, исходное равенство $|\vec{K_1L_1} - \vec{NL_1}| = |\vec{ML} + \vec{MM_1}|$ верно.
Ответ: Равенство доказано.
в) Докажите, что $|\vec{NL} - \vec{M_1L}| = |\vec{K_1N} - \vec{LN}|$
Рассмотрим левую часть равенства: $|\vec{NL} - \vec{M_1L}|$.
Векторы $\vec{NL}$ и $\vec{M_1L}$ имеют общий конец в точке $L$. Разность таких векторов равна вектору, соединяющему их начальные точки (от начала второго к началу первого): $\vec{NL} - \vec{M_1L} = \vec{M_1N}$.
*Доказательство этого факта: $\vec{NL} - \vec{M_1L} = (\vec{L}-\vec{N}) - (\vec{L}-\vec{M_1}) = \vec{L}-\vec{N}-\vec{L}+\vec{M_1} = \vec{M_1}-\vec{N} = \vec{NM_1}$. Ой, здесь ошибка в направлении. Правильно: $\vec{M_1N}$*. *Перепроверка: разность векторов $\vec{a}$ и $\vec{b}$ с общим концом $C$, т.е. $\vec{AC} - \vec{BC}$, равна вектору $\vec{AB}$. В нашем случае $\vec{N}$ - это $A$, $\vec{M_1}$ - это $B$, $\vec{L}$ - это $C$. Значит, $\vec{NL} - \vec{M_1L} = \vec{NM_1}$.* Итак, $\vec{NL} - \vec{M_1L} = \vec{NM_1}$.
Следовательно, $|\vec{NL} - \vec{M_1L}| = |\vec{NM_1}| = NM_1$.
Рассмотрим правую часть равенства: $|\vec{K_1N} - \vec{LN}|$.
Аналогично, векторы $\vec{K_1N}$ и $\vec{LN}$ имеют общий конец в точке $N$. Их разность равна вектору, соединяющему их начальные точки: $\vec{K_1N} - \vec{LN} = \vec{K_1L}$.
Следовательно, $|\vec{K_1N} - \vec{LN}| = |\vec{K_1L}| = K_1L$.
Равенство сводится к доказательству того, что $NM_1 = K_1L$.
Рассмотрим отрезок $NM_1$. В прямоугольном параллелепипеде ребро $MM_1$ перпендикулярно плоскости основания $KLMN$, а значит, перпендикулярно любой прямой в этой плоскости, в том числе и ребру $NM$. Таким образом, треугольник $\triangle NMM_1$ является прямоугольным с прямым углом при вершине $M$. По теореме Пифагора: $NM_1^2 = NM^2 + MM_1^2$.
Рассмотрим отрезок $K_1L$. Ребро $KK_1$ перпендикулярно плоскости основания $KLMN$, а значит, перпендикулярно ребру $KL$. Таким образом, треугольник $\triangle K_1KL$ является прямоугольным с прямым углом при вершине $K$. По теореме Пифагора: $K_1L^2 = KL^2 + KK_1^2$.
В параллелепипеде противолежащие ребра равны: $NM = KL$ и $MM_1 = KK_1$. Следовательно, $NM^2 + MM_1^2 = KL^2 + KK_1^2$, что означает $NM_1^2 = K_1L^2$, и $NM_1 = K_1L$.
Таким образом, исходное равенство $|\vec{NL} - \vec{M_1L}| = |\vec{K_1N} - \vec{LN}|$ верно.
Ответ: Равенство доказано.
Другие задания:
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