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Задания, страница 11 - гдз по геометрии 11 класс учебник Смирнов, Туяков
Авторы: Смирнов В. А., Туяков Е. А.
Тип: Учебник
Издательство: Мектеп
Год издания: 2019 - 2026
Популярные ГДЗ в 11 классе
Глава I. Многогранники. Параграф 1. Понятие многогранника. Призма и ее элементы, виды призм. Развертка, площадь боковой и полной поверхностей призмы - страница 11.
App\Models\Task {#1003 // 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" => 1145936 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "field_page_start" => "11" "field_page_end" => null "field_url" => "/11-klass/geometrija/mektep-smirnov-uchebnik-dva-2019/samost-11" "field_display_title" => "Задания" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1008 #items: [] #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 {#1006 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1017 #items: array:1 [ 0 => App\Models\Branch {#1016 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:23 [ "id" => 1145812 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "field_display_title" => "Многогранники" "field_branch_order" => "I" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1021 #items: array:1 [ 0 => App\Models\Term {#1018 #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" => "2026-04-08 17:28:58" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 29 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "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" => "8" "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" => [] "book" => Illuminate\Database\Eloquent\Collection {#1029 #items: array:1 [ 0 => App\Models\Book {#1028 #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" => 4310 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1033 #items: array:1 [ 0 => App\Models\Term {#1030 #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 [ "id" => 6477 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "name" => "Геометрия" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "геометрию" "field_creative_case" => "геометрией" "field_dative_case" => "геометрии" "field_genitive_case" => "геометрии" "field_nominative_case" => "геометрия" "field_prepositional_case" => "геометрии" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "geometrija" ] #original: array:10 [ "id" => 6477 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "name" => "Геометрия" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "геометрию" "field_creative_case" => "геометрией" "field_dative_case" => "геометрии" "field_genitive_case" => "геометрии" "field_nominative_case" => "геометрия" "field_prepositional_case" => "геометрии" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "geometrija" ] #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_class" => Illuminate\Database\Eloquent\Collection {#1035 #items: array:1 [ 0 => App\Models\Term {#1032 #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" => 5460 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "name" => "11" "field_cases" => array:6 [ "field_accusative_case" => "одиннадцатый" "field_creative_case" => "одиннадцатым" "field_dative_case" => "одиннадцатому" "field_genitive_case" => "одинадцатого" "field_nominative_case" => "одинадцатый" "field_prepositional_case" => "одиннадцатом" ] "field_translit" => "odinadcatyj" ] #original: array:6 [ "id" => 5460 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "name" => "11" "field_cases" => array:6 [ "field_accusative_case" => "одиннадцатый" "field_creative_case" => "одиннадцатым" "field_dative_case" => "одиннадцатому" "field_genitive_case" => "одинадцатого" "field_nominative_case" => "одинадцатый" "field_prepositional_case" => "одиннадцатом" ] "field_translit" => "odinadcatyj" ] #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_publisher" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1034 #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" => 6996 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #original: array:6 [ "id" => 6996 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #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_author" => Illuminate\Database\Eloquent\Collection {#1031 #items: array:2 [ 0 => 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 [ "id" => 5801 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "name" => "Смирнов" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Владимир" "field_patronymic" => "Алексеевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 5801 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "name" => "Смирнов" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Владимир" "field_patronymic" => "Алексеевич" "field_surname_rp" => null ] #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\Term {#1047 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7017 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "name" => "Туяков" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Есенкельды" "field_patronymic" => "Алыбаевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 7017 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "name" => "Туяков" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Есенкельды" "field_patronymic" => "Алыбаевич" "field_surname_rp" => 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_author_foreign" => Illuminate\Database\Eloquent\Collection {#1043 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1041 #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:10 [ "id" => 6671 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #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_country" => Illuminate\Database\Eloquent\Collection {#1042 #items: array:1 [ 0 => 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:6 [ "id" => 6994 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "kazakhstan" ] #original: array:6 [ "id" => 6994 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "kazakhstan" ] #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_city" => Illuminate\Database\Eloquent\Collection {#1048 #items: array:1 [ 0 => App\Models\Term {#1039 #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" => 6995 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almaty" ] #original: array:6 [ "id" => 6995 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almaty" ] #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_series" => Illuminate\Database\Eloquent\Collection {#1040 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1036 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1049 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1050 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1051 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1052 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Естественно-математического направление" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1266" "field_priority" => "6" "field_default_folder" => "/geometrija_11/smirnov-2-u19/" "field_isbn" => null "field_cover" => array:1 [ 0 => "/media/geometrija_11/smirnov-2-u19/covers/cover1.webp" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_11/smirnov-2-u19/covers/cover1.webp" "alt" => "" "width" => "1632" "height" => "2472" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1053 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1054 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1055 #items: [] #escapeWhenCastingToString: false } "field_url" => "/11-klass/geometrija/mektep-smirnov-uchebnik-dva-2019" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1056 #items: [] #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" => 382 "subject" => 543 "class_subject" => 393 ] ] #original: array:50 [ "id" => 4310 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1033} "field_class" => Illuminate\Database\Eloquent\Collection {#1035} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1037} "field_author" => Illuminate\Database\Eloquent\Collection {#1031} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1043} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1041} "field_country" => Illuminate\Database\Eloquent\Collection {#1042} "field_city" => Illuminate\Database\Eloquent\Collection {#1048} "field_series" => Illuminate\Database\Eloquent\Collection {#1040} "field_umk" => Illuminate\Database\Eloquent\Collection {#1036} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1049} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1050} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1051} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1052} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Естественно-математического направление" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1266" "field_priority" => "6" "field_default_folder" => "/geometrija_11/smirnov-2-u19/" "field_isbn" => null "field_cover" => array:1 [ 0 => "/media/geometrija_11/smirnov-2-u19/covers/cover1.webp" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_11/smirnov-2-u19/covers/cover1.webp" "alt" => "" "width" => "1632" "height" => "2472" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1053} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1054} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1055} "field_url" => "/11-klass/geometrija/mektep-smirnov-uchebnik-dva-2019" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1056} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 382 "subject" => 543 "class_subject" => 393 ] ] #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 {#1019 #items: [] #escapeWhenCastingToString: false } ] #original: array:23 [ "id" => 1145812 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "field_display_title" => "Многогранники" "field_branch_order" => "I" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1021} "field_page_start" => "8" "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" => [] "book" => Illuminate\Database\Eloquent\Collection {#1029} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1019} ] #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 {#1078 #items: array:2 [ 0 => App\Models\Branch {#1086 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:23 [ "id" => 1145812 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "field_display_title" => "Многогранники" "field_branch_order" => "I" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1087 #items: array:1 [ 0 => App\Models\Term {#1018} ] #escapeWhenCastingToString: false } "field_page_start" => "8" "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" => [] "book" => Illuminate\Database\Eloquent\Collection {#1088 #items: array:1 [ 0 => App\Models\Book {#1028} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1089 #items: [] #escapeWhenCastingToString: false } ] #original: array:23 [ "id" => 1145812 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "field_display_title" => "Многогранники" "field_branch_order" => "I" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1087} "field_page_start" => "8" "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" => [] "book" => Illuminate\Database\Eloquent\Collection {#1088} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1089} ] #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 {#1090 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:23 [ "id" => 1145813 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "field_display_title" => "Понятие многогранника. Призма и ее элементы, виды призм. Развертка, площадь боковой и полной поверхностей призмы" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1094 #items: array:1 [ 0 => App\Models\Term {#1091 #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" => "2026-04-08 17:28:58" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ "field_accusative_case" => "Параграф" "field_creative_case" => "Параграфом" "field_dative_case" => "Параграфу" "field_genitive_case" => "Параграфа" "field_nominative_case" => "Параграф" "field_prepositional_case" => "Параграфе" ] "field_page_check_not_needed" => null "field_short_name" => "§" ] #original: array:7 [ "id" => 32 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ "field_accusative_case" => "Параграф" "field_creative_case" => "Параграфом" "field_dative_case" => "Параграфу" "field_genitive_case" => "Параграфа" "field_nominative_case" => "Параграф" "field_prepositional_case" => "Параграфе" ] "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" => "8" "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" => [] "book" => Illuminate\Database\Eloquent\Collection {#1093 #items: array:1 [ 0 => App\Models\Book {#1028} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1092 #items: array:1 [ 0 => App\Models\Branch {#1016} ] #escapeWhenCastingToString: false } ] #original: array:23 [ "id" => 1145813 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "field_display_title" => "Понятие многогранника. Призма и ее элементы, виды призм. Развертка, площадь боковой и полной поверхностей призмы" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1094} "field_page_start" => "8" "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" => [] "book" => Illuminate\Database\Eloquent\Collection {#1093} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1092} ] #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 {#1009 #items: array:2 [ 0 => App\Models\Element {#1069 #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" => 1390778 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1059 #items: array:1 [ 0 => App\Models\Edition {#1068 #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" => 5035 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => 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 [ "id" => 6996 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #original: array:6 [ "id" => 6996 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #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_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 {#1065 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1063 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1064 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5035 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1066} "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 {#1065} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1063} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1064} "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" => "1145936" "type" => "task" ] "text" => "<p>Докажите самостоятельно, что пересечение (общая часть) двух выпуклых фигур является выпуклой фигурой.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "11-1.jpg" "alt" => null "width" => "1724" "height" => 214 "path" => "/media/geometrija_11/smirnov-2-u19/0-samost/11-1.webp" ] ] ] #original: array:7 [ "id" => 1390778 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1059} "task" => array:2 [ "refs" => "1145936" "type" => "task" ] "text" => "<p>Докажите самостоятельно, что пересечение (общая часть) двух выпуклых фигур является выпуклой фигурой.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "11-1.jpg" "alt" => null "width" => "1724" "height" => 214 "path" => "/media/geometrija_11/smirnov-2-u19/0-samost/11-1.webp" ] ] ] #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 {#1061 #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" => 1566082 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1022 #items: array:1 [ 0 => App\Models\Edition {#1062 #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" => 5689 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1058 #items: array:1 [ 0 => App\Models\Term {#1060 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 7012 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "name" => "Gemini 2.5 Pro" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "gemini 2.5 pro" ] #original: array:6 [ "id" => 7012 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "name" => "Gemini 2.5 Pro" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "gemini 2.5 pro" ] #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_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1026 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1057 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1024 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5689 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1058} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1026} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1057} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1024} "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" => "1145936" "type" => "task" ] "text" => "<strong>Решение</strong><p>Вспомним определение выпуклой фигуры. Фигура (или множество точек) называется <strong>выпуклой</strong>, если для любых двух точек этой фигуры отрезок, соединяющий эти точки, полностью ей принадлежит.</p><p>Пусть даны две выпуклые фигуры, $F_1$ и $F_2$. Обозначим их пересечение (общую часть) как фигуру $F$. Таким образом, $F = F_1 \cap F_2$. Нам необходимо доказать, что фигура $F$ также является выпуклой.</p><p>Для доказательства выберем две произвольные точки $A$ и $B$, которые принадлежат фигуре $F$.</p><p>Согласно определению пересечения, если точка принадлежит пересечению, то она принадлежит каждой из фигур, образующих это пересечение. Следовательно:<br>- раз точка $A$ принадлежит $F$ ($A \in F$), то она принадлежит и $F_1$ ($A \in F_1$), и $F_2$ ($A \in F_2$).<br>- раз точка $B$ принадлежит $F$ ($B \in F$), то она принадлежит и $F_1$ ($B \in F_1$), и $F_2$ ($B \in F_2$).</p><p>Теперь воспользуемся свойством выпуклости исходных фигур $F_1$ и $F_2$.</p><p>Поскольку $F_1$ — выпуклая фигура, и точки $A$ и $B$ принадлежат ей, то по определению выпуклости весь отрезок $[AB]$ также принадлежит фигуре $F_1$.</p><p>Аналогично, поскольку $F_2$ — выпуклая фигура, и точки $A$ и $B$ принадлежат ей, то весь отрезок $[AB]$ также принадлежит фигуре $F_2$.</p><p>Итак, мы установили, что отрезок $[AB]$ полностью содержится и в фигуре $F_1$, и в фигуре $F_2$. Это означает, что отрезок $[AB]$ содержится в их общей части, то есть в пересечении $F$.</p><p>Таким образом, мы показали, что для любых двух точек $A$ и $B$ из фигуры $F$ соединяющий их отрезок $[AB]$ целиком принадлежит фигуре $F$. Это в точности соответствует определению выпуклой фигуры.</p><p>Следовательно, пересечение двух (и, по аналогии, любого количества) выпуклых фигур является выпуклой фигурой. Что и требовалось доказать.</p><p><strong>Ответ:</strong> Утверждение доказано. Пересечение двух выпуклых фигур является выпуклой фигурой. Это следует из определения выпуклой фигуры: если взять любые две точки, принадлежащие пересечению, то они принадлежат и каждой из исходных фигур. Так как исходные фигуры выпуклы, отрезок, соединяющий эти точки, принадлежит каждой из них. Следовательно, этот отрезок принадлежит и их пересечению, что доказывает выпуклость пересечения.</p>" ] #original: array:6 [ "id" => 1566082 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1022} "task" => array:2 [ "refs" => "1145936" "type" => "task" ] "text" => "<strong>Решение</strong><p>Вспомним определение выпуклой фигуры. Фигура (или множество точек) называется <strong>выпуклой</strong>, если для любых двух точек этой фигуры отрезок, соединяющий эти точки, полностью ей принадлежит.</p><p>Пусть даны две выпуклые фигуры, $F_1$ и $F_2$. Обозначим их пересечение (общую часть) как фигуру $F$. Таким образом, $F = F_1 \cap F_2$. Нам необходимо доказать, что фигура $F$ также является выпуклой.</p><p>Для доказательства выберем две произвольные точки $A$ и $B$, которые принадлежат фигуре $F$.</p><p>Согласно определению пересечения, если точка принадлежит пересечению, то она принадлежит каждой из фигур, образующих это пересечение. Следовательно:<br>- раз точка $A$ принадлежит $F$ ($A \in F$), то она принадлежит и $F_1$ ($A \in F_1$), и $F_2$ ($A \in F_2$).<br>- раз точка $B$ принадлежит $F$ ($B \in F$), то она принадлежит и $F_1$ ($B \in F_1$), и $F_2$ ($B \in F_2$).</p><p>Теперь воспользуемся свойством выпуклости исходных фигур $F_1$ и $F_2$.</p><p>Поскольку $F_1$ — выпуклая фигура, и точки $A$ и $B$ принадлежат ей, то по определению выпуклости весь отрезок $[AB]$ также принадлежит фигуре $F_1$.</p><p>Аналогично, поскольку $F_2$ — выпуклая фигура, и точки $A$ и $B$ принадлежат ей, то весь отрезок $[AB]$ также принадлежит фигуре $F_2$.</p><p>Итак, мы установили, что отрезок $[AB]$ полностью содержится и в фигуре $F_1$, и в фигуре $F_2$. Это означает, что отрезок $[AB]$ содержится в их общей части, то есть в пересечении $F$.</p><p>Таким образом, мы показали, что для любых двух точек $A$ и $B$ из фигуры $F$ соединяющий их отрезок $[AB]$ целиком принадлежит фигуре $F$. Это в точности соответствует определению выпуклой фигуры.</p><p>Следовательно, пересечение двух (и, по аналогии, любого количества) выпуклых фигур является выпуклой фигурой. Что и требовалось доказать.</p><p><strong>Ответ:</strong> Утверждение доказано. Пересечение двух выпуклых фигур является выпуклой фигурой. Это следует из определения выпуклой фигуры: если взять любые две точки, принадлежащие пересечению, то они принадлежат и каждой из исходных фигур. Так как исходные фигуры выпуклы, отрезок, соединяющий эти точки, принадлежит каждой из них. Следовательно, этот отрезок принадлежит и их пересечению, что доказывает выпуклость пересечения.</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" => "1145937" "type" => "task" ] "previous" => array:2 [ "refs" => "1145935" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1072 #items: array:1 [ 0 => App\Models\Book {#1028} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1076 #items: array:1 [ 0 => App\Models\BookPage {#1077 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false 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#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" => 1294461 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1103 #items: array:1 [ 0 => App\Models\Edition {#1068} ] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1116196" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "11-1.jpg" "alt" => null "width" => "2009" "height" => 2912 "path" => "/media/geometrija_11/smirnov-2-u19/0-1/11-1.webp" ] ] ] #original: array:6 [ "id" => 1294461 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1103} "book_page" => array:2 [ "refs" => "1116196" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "11-1.jpg" "alt" => null "width" => "2009" "height" => 2912 "path" => "/media/geometrija_11/smirnov-2-u19/0-1/11-1.webp" ] ] ] #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" => "1116197" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1116195" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1080 #items: array:1 [ 0 => App\Models\Task {#1003} ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 1116196 "created_at" => "2026-04-08 17:28:58" "updated_at" => null "field_page_start" => "11" "field_url" => "/11-klass/geometrija/mektep-smirnov-uchebnik-dva-2019/page-11" "field_display_title" => "11" "field_folder" => "1" "field_image_name" => 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Задания (с. 11)
Условие. Задания (с. 11)
скриншот условия
Докажите самостоятельно, что пересечение (общая часть) двух выпуклых фигур является выпуклой фигурой.
Решение 2 (rus). Задания (с. 11)
Решение
Вспомним определение выпуклой фигуры. Фигура (или множество точек) называется выпуклой, если для любых двух точек этой фигуры отрезок, соединяющий эти точки, полностью ей принадлежит.
Пусть даны две выпуклые фигуры, $F_1$ и $F_2$. Обозначим их пересечение (общую часть) как фигуру $F$. Таким образом, $F = F_1 \cap F_2$. Нам необходимо доказать, что фигура $F$ также является выпуклой.
Для доказательства выберем две произвольные точки $A$ и $B$, которые принадлежат фигуре $F$.
Согласно определению пересечения, если точка принадлежит пересечению, то она принадлежит каждой из фигур, образующих это пересечение. Следовательно:
- раз точка $A$ принадлежит $F$ ($A \in F$), то она принадлежит и $F_1$ ($A \in F_1$), и $F_2$ ($A \in F_2$).
- раз точка $B$ принадлежит $F$ ($B \in F$), то она принадлежит и $F_1$ ($B \in F_1$), и $F_2$ ($B \in F_2$).
Теперь воспользуемся свойством выпуклости исходных фигур $F_1$ и $F_2$.
Поскольку $F_1$ — выпуклая фигура, и точки $A$ и $B$ принадлежат ей, то по определению выпуклости весь отрезок $[AB]$ также принадлежит фигуре $F_1$.
Аналогично, поскольку $F_2$ — выпуклая фигура, и точки $A$ и $B$ принадлежат ей, то весь отрезок $[AB]$ также принадлежит фигуре $F_2$.
Итак, мы установили, что отрезок $[AB]$ полностью содержится и в фигуре $F_1$, и в фигуре $F_2$. Это означает, что отрезок $[AB]$ содержится в их общей части, то есть в пересечении $F$.
Таким образом, мы показали, что для любых двух точек $A$ и $B$ из фигуры $F$ соединяющий их отрезок $[AB]$ целиком принадлежит фигуре $F$. Это в точности соответствует определению выпуклой фигуры.
Следовательно, пересечение двух (и, по аналогии, любого количества) выпуклых фигур является выпуклой фигурой. Что и требовалось доказать.
Ответ: Утверждение доказано. Пересечение двух выпуклых фигур является выпуклой фигурой. Это следует из определения выпуклой фигуры: если взять любые две точки, принадлежащие пересечению, то они принадлежат и каждой из исходных фигур. Так как исходные фигуры выпуклы, отрезок, соединяющий эти точки, принадлежит каждой из них. Следовательно, этот отрезок принадлежит и их пересечению, что доказывает выпуклость пересечения.
Другие задания:
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