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Номер 3.2, страница 75 - гдз по геометрии 8 класс учебник Шыныбеков, Шыныбеков
Авторы: Шыныбеков А. Н., Шыныбеков Д. А., Жумабаев Р. Н.
Тип: Учебник
Издательство: Атамұра
Год издания: 2018 - 2026
ISBN: 978-601-331-161-6
Рекомендовано Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 8 классе
Раздел 3. Площадь. 3.1. Площадь прямоугольника - номер 3.2, страница 75.
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" => 1122772 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "75" "field_page_end" => null "field_url" => "/8-klass/geometrija/shynybekov-uchebnik/3-2" "field_display_title" => "3.2" "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 {#1072 #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" => 1122385 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Площадь" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039 #items: array:1 [ 0 => App\Models\Term {#1038 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 30 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Раздел" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 30 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Раздел" "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" => "73" "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" => 2592 "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 {#1046 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2018" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 8 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "538" "field_priority" => "6" "field_default_folder" => "/geometrija_08/Shynybekov-u17/" "field_isbn" => "978-601-331-161-6" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1067 …2} "field_url" => "/8-klass/geometrija/shynybekov-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 2592 "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 {#1046 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2018" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 8 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "538" "field_priority" => "6" "field_default_folder" => "/geometrija_08/Shynybekov-u17/" "field_isbn" => "978-601-331-161-6" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1067 …2} "field_url" => "/8-klass/geometrija/shynybekov-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1069 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1122385 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Площадь" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039} "field_page_start" => "73" "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 {#1069} ] #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 {#1141 #items: array:2 [ 0 => App\Models\Branch {#1070 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1122385 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Площадь" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1073 #items: array:1 [ 0 => App\Models\Term {#1071 #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" => 30 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Раздел" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 30 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Раздел" "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" => "73" "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 {#1074 #items: array:1 [ 0 => App\Models\Book {#1075 #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" => 2592 "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 {#1076 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1078 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1080 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1082 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1094 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1096 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1098 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2018" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 8 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "538" "field_priority" => "6" "field_default_folder" => "/geometrija_08/Shynybekov-u17/" "field_isbn" => "978-601-331-161-6" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1100 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1101 …2} "field_url" => "/8-klass/geometrija/shynybekov-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1102 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 2592 "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 {#1076 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1078 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1080 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1082 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1094 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1096 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1098 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2018" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 8 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "538" "field_priority" => "6" "field_default_folder" => "/geometrija_08/Shynybekov-u17/" "field_isbn" => "978-601-331-161-6" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1100 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1101 …2} "field_url" => "/8-klass/geometrija/shynybekov-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1102 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1103 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1122385 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Площадь" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1073} "field_page_start" => "73" "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 {#1074} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1103} ] #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 {#1104 #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" => 1122386 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "3.1. Площадь прямоугольника" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1105 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "73" "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 {#1106 #items: array:1 [ 0 => App\Models\Book {#1107 #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" => 2592 "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 {#1108 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1110 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1112 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1114 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1118 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1125 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1127 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1128 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1129 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2018" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 8 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "538" "field_priority" => "6" "field_default_folder" => "/geometrija_08/Shynybekov-u17/" "field_isbn" => "978-601-331-161-6" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1131 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1132 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1133 …2} "field_url" => "/8-klass/geometrija/shynybekov-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1134 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 2592 "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 {#1108 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1110 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1112 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1114 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1118 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1125 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1127 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1128 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1129 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2018" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 8 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "538" "field_priority" => "6" "field_default_folder" => "/geometrija_08/Shynybekov-u17/" "field_isbn" => "978-601-331-161-6" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1131 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1132 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1133 …2} "field_url" => "/8-klass/geometrija/shynybekov-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1134 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1135 #items: array:1 [ 0 => App\Models\Branch {#1136 #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" => 1122385 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Площадь" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1137 …2} "field_page_start" => "73" "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 {#1139 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1140 …2} ] #original: array:24 [ "id" => 1122385 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Площадь" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1137 …2} "field_page_start" => "73" "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 {#1139 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1140 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1122386 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "3.1. Площадь прямоугольника" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1105} "field_page_start" => "73" "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 {#1106} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1135} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1152 #items: array:4 [ 0 => App\Models\Element {#1162 #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" => 1305223 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1171 #items: array:1 [ 0 => App\Models\Edition {#1163 #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" => 5006 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Учебник rus" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1165 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0rus-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1166 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1167 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1168 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5006 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Учебник rus" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1165 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0rus-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1166 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1167 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1168 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1122772" "type" => "task" ] "text" => "<p>3.2. Точки $N$, $E$, $L$ и $K$ – середины соответствующих сторон $AB$, $AD$, $BC$ и $CD$ прямоугольника $ABCD$ (рис. 3.7). Какую часть площади прямоугольника $ABCD$ составляют площади следующих фигур:</p><p><strong>1</strong>) $\triangle ABD$;</p><p><strong>2</strong>) $\triangle ABE$;</p><p><strong>3</strong>) $ABKD$;</p><p><strong>4</strong>) $ABLKD$;</p><p><strong>5</strong>) $ABLKE$;</p><p><strong>6</strong>) $LKEN$;</p><p><strong>7</strong>) $ALD$ и $BEC$?</p>" "img" => array:1 [ 0 => array:5 [ "name" => "2-1.jpg" "alt" => null "width" => "1771" "height" => 515 "path" => "/media/geometrija_08/Shynybekov-u17/0rus-3/2-1.webp?ts=1753776431" ] ] ] #original: array:7 [ "id" => 1305223 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1171} "task" => array:2 [ "refs" => "1122772" "type" => "task" ] "text" => "<p>3.2. Точки $N$, $E$, $L$ и $K$ – середины соответствующих сторон $AB$, $AD$, $BC$ и $CD$ прямоугольника $ABCD$ (рис. 3.7). Какую часть площади прямоугольника $ABCD$ составляют площади следующих фигур:</p><p><strong>1</strong>) $\triangle ABD$;</p><p><strong>2</strong>) $\triangle ABE$;</p><p><strong>3</strong>) $ABKD$;</p><p><strong>4</strong>) $ABLKD$;</p><p><strong>5</strong>) $ABLKE$;</p><p><strong>6</strong>) $LKEN$;</p><p><strong>7</strong>) $ALD$ и $BEC$?</p>" "img" => array:1 [ 0 => array:5 [ "name" => "2-1.jpg" "alt" => null "width" => "1771" "height" => 515 "path" => "/media/geometrija_08/Shynybekov-u17/0rus-3/2-1.webp?ts=1753776431" ] ] ] #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 {#1169 #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" => 1305566 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1179 #items: array:1 [ 0 => App\Models\Edition {#1170 #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" => 5007 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Учебник kz" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1173 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "image" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0kz-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1174 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1175 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1176 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5007 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Учебник kz" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1173 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "image" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0kz-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1174 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1175 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1176 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1122772" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "2-1.jpg" "alt" => null "width" => "1731" "height" => 531 "path" => "/media/geometrija_08/Shynybekov-u17/0kz-3/2-1.webp?ts=1753776477" ] ] ] #original: array:6 [ "id" => 1305566 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1179} "task" => array:2 [ "refs" => "1122772" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "2-1.jpg" "alt" => null "width" => "1731" "height" => 531 "path" => "/media/geometrija_08/Shynybekov-u17/0kz-3/2-1.webp?ts=1753776477" ] ] ] #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 {#1177 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1306168 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1187 #items: array:1 [ 0 => App\Models\Edition {#1178 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5008 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1181 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1182 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1183 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1184 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5008 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1181 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1182 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1183 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1184 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1122772" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "2-1.jpg" "alt" => null "width" => "1536" "height" => 334 "path" => "/media/geometrija_08/Shynybekov-u17/1-3/2-1.webp?ts=1753776532" ] ] ] #original: array:6 [ "id" => 1306168 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1187} "task" => array:2 [ "refs" => "1122772" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "2-1.jpg" "alt" => null "width" => "1536" "height" => 334 "path" => "/media/geometrija_08/Shynybekov-u17/1-3/2-1.webp?ts=1753776532" ] ] ] #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" => 1481001 "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 [ "id" => 5435 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 rus" "field_order" => "4" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1189 …2} "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 {#1190 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1191 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1192 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5435 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 rus" "field_order" => "4" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1189 …2} "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 {#1190 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1191 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1192 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1122772" "type" => "task" ] "text" => "<p>Обозначим стороны прямоугольника $ABCD$ как $AD = a$ и $AB = b$. Тогда площадь прямоугольника равна $S_{ABCD} = a \cdot b$.</p><p>По условию задачи, точки $N, E, L, K$ являются серединами сторон $AB, AD, BC, CD$ соответственно. Следовательно:</p><ul><li>$N$ — середина $AB$, поэтому $AN = NB = \frac{b}{2}$</li><li>$E$ — середина $AD$, поэтому $AE = ED = \frac{a}{2}$</li><li>$L$ — середина $BC$, поэтому $BL = LC = \frac{a}{2}$ (так как $BC=AD$)</li><li>$K$ — середина $CD$, поэтому $CK = KD = \frac{b}{2}$ (так как $CD=AB$)</li></ul><p><strong>1</strong>) ΔABD;</p><p>Треугольник $ABD$ является прямоугольным, так как его стороны $AB$ и $AD$ являются сторонами прямоугольника. Его площадь равна половине произведения катетов $AB$ и $AD$.</p><p>$S_{\triangle ABD} = \frac{1}{2} \cdot AD \cdot AB = \frac{1}{2} a b$.</p><p>Отношение площади треугольника $ABD$ к площади прямоугольника $ABCD$ составляет:</p><p>$\frac{S_{\triangle ABD}}{S_{ABCD}} = \frac{\frac{1}{2}ab}{ab} = \frac{1}{2}$.</p><p>Ответ: $\frac{1}{2}$.</p><p><strong>2</strong>) ΔABE;</p><p>Треугольник $ABE$ является прямоугольным с прямым углом при вершине $A$. Его катеты — $AB$ и $AE$. Длина $AB = b$, а $AE = \frac{AD}{2} = \frac{a}{2}$.</p><p>Площадь треугольника $ABE$ равна:</p><p>$S_{\triangle ABE} = \frac{1}{2} \cdot AB \cdot AE = \frac{1}{2} \cdot b \cdot \frac{a}{2} = \frac{ab}{4}$.</p><p>Отношение его площади к площади прямоугольника:</p><p>$\frac{S_{\triangle ABE}}{S_{ABCD}} = \frac{\frac{1}{4}ab}{ab} = \frac{1}{4}$.</p><p>Ответ: $\frac{1}{4}$.</p><p><strong>3</strong>) ΔBKD;</p><p>Для нахождения площади треугольника $BKD$ примем за основание сторону $KD$. Длина основания $KD = \frac{CD}{2} = \frac{b}{2}$. Высотой, проведенной из вершины $B$ к основанию $KD$ (и прямой $CD$), будет сторона $BC$, длина которой равна $a$.</p><p>Площадь треугольника $BKD$ равна:</p><p>$S_{\triangle BKD} = \frac{1}{2} \cdot KD \cdot BC = \frac{1}{2} \cdot \frac{b}{2} \cdot a = \frac{ab}{4}$.</p><p>Отношение его площади к площади прямоугольника:</p><p>$\frac{S_{\triangle BKD}}{S_{ABCD}} = \frac{\frac{1}{4}ab}{ab} = \frac{1}{4}$.</p><p>Ответ: $\frac{1}{4}$.</p><p><strong>4</strong>) ABLKD;</p><p>Фигура $ABLKD$ — это пятиугольник. Его площадь можно найти, вычтя из площади всего прямоугольника $ABCD$ площадь треугольника $CLK$.</p><p>Треугольник $CLK$ является прямоугольным с прямым углом при вершине $C$. Его катеты — $CL$ и $CK$.</p><p>$CL = \frac{BC}{2} = \frac{a}{2}$ и $CK = \frac{CD}{2} = \frac{b}{2}$.</p><p>Площадь треугольника $CLK$ равна:</p><p>$S_{\triangle CLK} = \frac{1}{2} \cdot CL \cdot CK = \frac{1}{2} \cdot \frac{a}{2} \cdot \frac{b}{2} = \frac{ab}{8}$.</p><p>Площадь пятиугольника $ABLKD$ равна:</p><p>$S_{ABLKD} = S_{ABCD} - S_{\triangle CLK} = ab - \frac{ab}{8} = \frac{7}{8}ab$.</p><p>Отношение его площади к площади прямоугольника:</p><p>$\frac{S_{ABLKD}}{S_{ABCD}} = \frac{\frac{7}{8}ab}{ab} = \frac{7}{8}$.</p><p>Ответ: $\frac{7}{8}$.</p><p><strong>5</strong>) ABLKE;</p><p>Фигура $ABLKE$ — это пятиугольник. Его площадь можно найти, вычтя из площади прямоугольника $ABCD$ площади двух треугольников: $\triangle CLK$ и $\triangle EKD$.</p><p>Площадь $\triangle CLK$ мы уже нашли, она равна $\frac{ab}{8}$.</p><p>Треугольник $EKD$ является прямоугольным с прямым углом при вершине $D$. Его катеты — $ED$ и $KD$.</p><p>$ED = \frac{AD}{2} = \frac{a}{2}$ и $KD = \frac{CD}{2} = \frac{b}{2}$.</p><p>Площадь треугольника $EKD$ равна:</p><p>$S_{\triangle EKD} = \frac{1}{2} \cdot ED \cdot KD = \frac{1}{2} \cdot \frac{a}{2} \cdot \frac{b}{2} = \frac{ab}{8}$.</p><p>Площадь пятиугольника $ABLKE$ равна:</p><p>$S_{ABLKE} = S_{ABCD} - S_{\triangle CLK} - S_{\triangle EKD} = ab - \frac{ab}{8} - \frac{ab}{8} = ab - \frac{2ab}{8} = ab - \frac{ab}{4} = \frac{3}{4}ab$.</p><p>Отношение его площади к площади прямоугольника:</p><p>$\frac{S_{ABLKE}}{S_{ABCD}} = \frac{\frac{3}{4}ab}{ab} = \frac{3}{4}$.</p><p>Ответ: $\frac{3}{4}$.</p><p><strong>6</strong>) LKEN;</p><p>Четырехугольник $LKEN$ соединяет середины сторон прямоугольника. Его площадь можно найти, вычтя из площади прямоугольника $ABCD$ площади четырех угловых треугольников: $\triangle NAE$, $\triangle NBL$, $\triangle LCK$ и $\triangle KDE$.</p><p>Площади всех этих треугольников равны:</p><p>$S_{\triangle NAE} = S_{\triangle NBL} = S_{\triangle LCK} = S_{\triangle KDE} = \frac{1}{2} \cdot \frac{a}{2} \cdot \frac{b}{2} = \frac{ab}{8}$.</p><p>Сумма их площадей равна $4 \cdot \frac{ab}{8} = \frac{ab}{2}$.</p><p>Площадь четырехугольника $LKEN$ равна:</p><p>$S_{LKEN} = S_{ABCD} - 4 \cdot \frac{ab}{8} = ab - \frac{ab}{2} = \frac{1}{2}ab$.</p><p>Отношение его площади к площади прямоугольника:</p><p>$\frac{S_{LKEN}}{S_{ABCD}} = \frac{\frac{1}{2}ab}{ab} = \frac{1}{2}$.</p><p>Ответ: $\frac{1}{2}$.</p><p><strong>7</strong>) ALD и BEC?</p><p>Найдем площади треугольников $ALD$ и $BEC$.</p><p>Для треугольника $ALD$ основание $AD = a$. Высота, проведенная из вершины $L$ к основанию $AD$, равна стороне $AB=b$.</p><p>$S_{\triangle ALD} = \frac{1}{2} \cdot AD \cdot b = \frac{1}{2}ab$.</p><p>Для треугольника $BEC$ основание $BC = a$. Высота, проведенная из вершины $E$ к основанию $BC$, равна стороне $AB=b$.</p><p>$S_{\triangle BEC} = \frac{1}{2} \cdot BC \cdot b = \frac{1}{2}ab$.</p><p>Отношение площади каждого из этих треугольников к площади прямоугольника составляет $\frac{\frac{1}{2}ab}{ab} = \frac{1}{2}$.</p><p>Ответ: Площадь каждого из треугольников $\triangle ALD$ и $\triangle BEC$ составляет $\frac{1}{2}$ площади прямоугольника $ABCD$.</p>" ] #original: array:6 [ "id" => 1481001 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1195} "task" => array:2 [ "refs" => "1122772" "type" => "task" ] "text" => "<p>Обозначим стороны прямоугольника $ABCD$ как $AD = a$ и $AB = b$. Тогда площадь прямоугольника равна $S_{ABCD} = a \cdot b$.</p><p>По условию задачи, точки $N, E, L, K$ являются серединами сторон $AB, AD, BC, CD$ соответственно. Следовательно:</p><ul><li>$N$ — середина $AB$, поэтому $AN = NB = \frac{b}{2}$</li><li>$E$ — середина $AD$, поэтому $AE = ED = \frac{a}{2}$</li><li>$L$ — середина $BC$, поэтому $BL = LC = \frac{a}{2}$ (так как $BC=AD$)</li><li>$K$ — середина $CD$, поэтому $CK = KD = \frac{b}{2}$ (так как $CD=AB$)</li></ul><p><strong>1</strong>) ΔABD;</p><p>Треугольник $ABD$ является прямоугольным, так как его стороны $AB$ и $AD$ являются сторонами прямоугольника. Его площадь равна половине произведения катетов $AB$ и $AD$.</p><p>$S_{\triangle ABD} = \frac{1}{2} \cdot AD \cdot AB = \frac{1}{2} a b$.</p><p>Отношение площади треугольника $ABD$ к площади прямоугольника $ABCD$ составляет:</p><p>$\frac{S_{\triangle ABD}}{S_{ABCD}} = \frac{\frac{1}{2}ab}{ab} = \frac{1}{2}$.</p><p>Ответ: $\frac{1}{2}$.</p><p><strong>2</strong>) ΔABE;</p><p>Треугольник $ABE$ является прямоугольным с прямым углом при вершине $A$. Его катеты — $AB$ и $AE$. Длина $AB = b$, а $AE = \frac{AD}{2} = \frac{a}{2}$.</p><p>Площадь треугольника $ABE$ равна:</p><p>$S_{\triangle ABE} = \frac{1}{2} \cdot AB \cdot AE = \frac{1}{2} \cdot b \cdot \frac{a}{2} = \frac{ab}{4}$.</p><p>Отношение его площади к площади прямоугольника:</p><p>$\frac{S_{\triangle ABE}}{S_{ABCD}} = \frac{\frac{1}{4}ab}{ab} = \frac{1}{4}$.</p><p>Ответ: $\frac{1}{4}$.</p><p><strong>3</strong>) ΔBKD;</p><p>Для нахождения площади треугольника $BKD$ примем за основание сторону $KD$. Длина основания $KD = \frac{CD}{2} = \frac{b}{2}$. Высотой, проведенной из вершины $B$ к основанию $KD$ (и прямой $CD$), будет сторона $BC$, длина которой равна $a$.</p><p>Площадь треугольника $BKD$ равна:</p><p>$S_{\triangle BKD} = \frac{1}{2} \cdot KD \cdot BC = \frac{1}{2} \cdot \frac{b}{2} \cdot a = \frac{ab}{4}$.</p><p>Отношение его площади к площади прямоугольника:</p><p>$\frac{S_{\triangle BKD}}{S_{ABCD}} = \frac{\frac{1}{4}ab}{ab} = \frac{1}{4}$.</p><p>Ответ: $\frac{1}{4}$.</p><p><strong>4</strong>) ABLKD;</p><p>Фигура $ABLKD$ — это пятиугольник. Его площадь можно найти, вычтя из площади всего прямоугольника $ABCD$ площадь треугольника $CLK$.</p><p>Треугольник $CLK$ является прямоугольным с прямым углом при вершине $C$. Его катеты — $CL$ и $CK$.</p><p>$CL = \frac{BC}{2} = \frac{a}{2}$ и $CK = \frac{CD}{2} = \frac{b}{2}$.</p><p>Площадь треугольника $CLK$ равна:</p><p>$S_{\triangle CLK} = \frac{1}{2} \cdot CL \cdot CK = \frac{1}{2} \cdot \frac{a}{2} \cdot \frac{b}{2} = \frac{ab}{8}$.</p><p>Площадь пятиугольника $ABLKD$ равна:</p><p>$S_{ABLKD} = S_{ABCD} - S_{\triangle CLK} = ab - \frac{ab}{8} = \frac{7}{8}ab$.</p><p>Отношение его площади к площади прямоугольника:</p><p>$\frac{S_{ABLKD}}{S_{ABCD}} = \frac{\frac{7}{8}ab}{ab} = \frac{7}{8}$.</p><p>Ответ: $\frac{7}{8}$.</p><p><strong>5</strong>) ABLKE;</p><p>Фигура $ABLKE$ — это пятиугольник. Его площадь можно найти, вычтя из площади прямоугольника $ABCD$ площади двух треугольников: $\triangle CLK$ и $\triangle EKD$.</p><p>Площадь $\triangle CLK$ мы уже нашли, она равна $\frac{ab}{8}$.</p><p>Треугольник $EKD$ является прямоугольным с прямым углом при вершине $D$. Его катеты — $ED$ и $KD$.</p><p>$ED = \frac{AD}{2} = \frac{a}{2}$ и $KD = \frac{CD}{2} = \frac{b}{2}$.</p><p>Площадь треугольника $EKD$ равна:</p><p>$S_{\triangle EKD} = \frac{1}{2} \cdot ED \cdot KD = \frac{1}{2} \cdot \frac{a}{2} \cdot \frac{b}{2} = \frac{ab}{8}$.</p><p>Площадь пятиугольника $ABLKE$ равна:</p><p>$S_{ABLKE} = S_{ABCD} - S_{\triangle CLK} - S_{\triangle EKD} = ab - \frac{ab}{8} - \frac{ab}{8} = ab - \frac{2ab}{8} = ab - \frac{ab}{4} = \frac{3}{4}ab$.</p><p>Отношение его площади к площади прямоугольника:</p><p>$\frac{S_{ABLKE}}{S_{ABCD}} = \frac{\frac{3}{4}ab}{ab} = \frac{3}{4}$.</p><p>Ответ: $\frac{3}{4}$.</p><p><strong>6</strong>) LKEN;</p><p>Четырехугольник $LKEN$ соединяет середины сторон прямоугольника. Его площадь можно найти, вычтя из площади прямоугольника $ABCD$ площади четырех угловых треугольников: $\triangle NAE$, $\triangle NBL$, $\triangle LCK$ и $\triangle KDE$.</p><p>Площади всех этих треугольников равны:</p><p>$S_{\triangle NAE} = S_{\triangle NBL} = S_{\triangle LCK} = S_{\triangle KDE} = \frac{1}{2} \cdot \frac{a}{2} \cdot \frac{b}{2} = \frac{ab}{8}$.</p><p>Сумма их площадей равна $4 \cdot \frac{ab}{8} = \frac{ab}{2}$.</p><p>Площадь четырехугольника $LKEN$ равна:</p><p>$S_{LKEN} = S_{ABCD} - 4 \cdot \frac{ab}{8} = ab - \frac{ab}{2} = \frac{1}{2}ab$.</p><p>Отношение его площади к площади прямоугольника:</p><p>$\frac{S_{LKEN}}{S_{ABCD}} = \frac{\frac{1}{2}ab}{ab} = \frac{1}{2}$.</p><p>Ответ: $\frac{1}{2}$.</p><p><strong>7</strong>) ALD и BEC?</p><p>Найдем площади треугольников $ALD$ и $BEC$.</p><p>Для треугольника $ALD$ основание $AD = a$. Высота, проведенная из вершины $L$ к основанию $AD$, равна стороне $AB=b$.</p><p>$S_{\triangle ALD} = \frac{1}{2} \cdot AD \cdot b = \frac{1}{2}ab$.</p><p>Для треугольника $BEC$ основание $BC = a$. Высота, проведенная из вершины $E$ к основанию $BC$, равна стороне $AB=b$.</p><p>$S_{\triangle BEC} = \frac{1}{2} \cdot BC \cdot b = \frac{1}{2}ab$.</p><p>Отношение площади каждого из этих треугольников к площади прямоугольника составляет $\frac{\frac{1}{2}ab}{ab} = \frac{1}{2}$.</p><p>Ответ: Площадь каждого из треугольников $\triangle ALD$ и $\triangle BEC$ составляет $\frac{1}{2}$ площади прямоугольника $ABCD$.</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" => "1122773" "type" => "task" ] "previous" => array:2 [ "refs" => "1122771" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1206 #items: array:1 [ 0 => App\Models\Book {#1160 #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" => 2592 "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 {#1144 #items: array:1 [ 0 => App\Models\Term {#1143 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6477 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Геометрия" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "geometrija" ] #original: array:10 [ "id" => 6477 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Геометрия" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "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 {#1142 #items: array:1 [ 0 => App\Models\Term {#1149 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5457 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "8" "field_cases" => array:6 [ …6] "field_translit" => "vosmoj" ] #original: array:6 [ "id" => 5457 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "8" "field_cases" => array:6 [ …6] "field_translit" => "vosmoj" ] #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 {#1148 #items: array:1 [ 0 => App\Models\Term {#1147 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6991 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Атамұра" "field_cases" => array:6 [ …6] "field_translit" => "atamu'ra" ] #original: array:6 [ "id" => 6991 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Атамұра" "field_cases" => array:6 [ …6] "field_translit" => "atamu'ra" ] #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 {#1145 #items: array:3 [ 0 => App\Models\Term {#1146 #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" => 6992 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Шыныбеков" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Абдухали" "field_patronymic" => "Насырович" "field_surname_rp" => null ] #original: array:12 [ "id" => 6992 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Шыныбеков" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Абдухали" "field_patronymic" => "Насырович" "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 {#1151 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 6993 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Шыныбеков" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Данияр" "field_patronymic" => "Абдухалиевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 6993 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Шыныбеков" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Данияр" "field_patronymic" => "Абдухалиевич" "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 => "*" ] } 2 => App\Models\Term {#1150 #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" => 7001 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумабаев" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Ринат" "field_patronymic" => "Нурланович" "field_surname_rp" => null ] #original: array:12 [ "id" => 7001 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумабаев" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Ринат" "field_patronymic" => "Нурланович" "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 {#1158 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1157 #items: array:1 [ 0 => App\Models\Term {#1159 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "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 {#1155 #items: array:1 [ 0 => App\Models\Term {#1153 #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" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ …6] "field_translit" => "kazakhstan" ] #original: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ …6] "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 {#1156 #items: array:1 [ 0 => App\Models\Term {#1154 #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" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ …6] "field_translit" => "almaty" ] #original: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ …6] "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 {#1193 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1194 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1196 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1197 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1198 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1199 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2018" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 8 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "538" "field_priority" => "6" "field_default_folder" => "/geometrija_08/Shynybekov-u17/" "field_isbn" => "978-601-331-161-6" "field_cover" => array:1 [ 0 => "/media/geometrija_08/Shynybekov-u17/covers/cover1.webp?ts=1753258124" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_08/Shynybekov-u17/covers/cover1.webp?ts=1753258124" "alt" => "" "width" => "1632" "height" => "2472" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1200 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1201 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1202 #items: [] #escapeWhenCastingToString: false } "field_url" => "/8-klass/geometrija/shynybekov-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1203 #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" => 379 "subject" => 543 "class_subject" => 390 ] ] #original: array:50 [ "id" => 2592 "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 {#1144} "field_class" => Illuminate\Database\Eloquent\Collection {#1142} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1148} "field_author" => Illuminate\Database\Eloquent\Collection {#1145} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1158} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1157} "field_country" => Illuminate\Database\Eloquent\Collection {#1155} "field_city" => Illuminate\Database\Eloquent\Collection {#1156} "field_series" => Illuminate\Database\Eloquent\Collection {#1193} "field_umk" => Illuminate\Database\Eloquent\Collection {#1194} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1196} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1197} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1198} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1199} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2018" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 8 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "538" "field_priority" => "6" "field_default_folder" => "/geometrija_08/Shynybekov-u17/" "field_isbn" => "978-601-331-161-6" "field_cover" => array:1 [ 0 => "/media/geometrija_08/Shynybekov-u17/covers/cover1.webp?ts=1753258124" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_08/Shynybekov-u17/covers/cover1.webp?ts=1753258124" "alt" => "" "width" => "1632" "height" => "2472" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1200} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1201} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1202} "field_url" => "/8-klass/geometrija/shynybekov-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1203} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 379 "subject" => 543 "class_subject" => 390 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1211 #items: array:1 [ 0 => App\Models\BookPage {#1210 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 1114277 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "75" "field_url" => "/8-klass/geometrija/shynybekov-uchebnik/page-75" "field_display_title" => "75" "field_folder" => "1" "field_image_name" => "75" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1212 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1213 #items: array:1 [ 0 => App\Models\Book {#1160} ] #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 {#1214 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1216 #items: array:2 [ 0 => App\Models\Element {#1225 #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" => 1290412 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1226 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1290412 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1226 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #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 {#1227 #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" => 1290438 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1228 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1290438 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1228 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #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" => "1114278" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1114276" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1393 #items: array:5 [ 0 => App\Models\Task {#1405 #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" => 1122768 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "75" "field_page_end" => null "field_url" => "/8-klass/geometrija/shynybekov-uchebnik/vopr-75" "field_display_title" => "Вопросы" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1406 …2} "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 {#1407 …2} "top_parent_branch" => 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+usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] }
№3.2 (с. 75)
Учебник rus. №3.2 (с. 75)
скриншот условия
3.2. Точки $N$, $E$, $L$ и $K$ – середины соответствующих сторон $AB$, $AD$, $BC$ и $CD$ прямоугольника $ABCD$ (рис. 3.7). Какую часть площади прямоугольника $ABCD$ составляют площади следующих фигур:
1) $\triangle ABD$;
2) $\triangle ABE$;
3) $ABKD$;
4) $ABLKD$;
5) $ABLKE$;
6) $LKEN$;
7) $ALD$ и $BEC$?
Учебник kz. №3.2 (с. 75)
Решение. №3.2 (с. 75)
Решение 2 rus. №3.2 (с. 75)
Обозначим стороны прямоугольника $ABCD$ как $AD = a$ и $AB = b$. Тогда площадь прямоугольника равна $S_{ABCD} = a \cdot b$.
По условию задачи, точки $N, E, L, K$ являются серединами сторон $AB, AD, BC, CD$ соответственно. Следовательно:
- $N$ — середина $AB$, поэтому $AN = NB = \frac{b}{2}$
- $E$ — середина $AD$, поэтому $AE = ED = \frac{a}{2}$
- $L$ — середина $BC$, поэтому $BL = LC = \frac{a}{2}$ (так как $BC=AD$)
- $K$ — середина $CD$, поэтому $CK = KD = \frac{b}{2}$ (так как $CD=AB$)
1) ΔABD;
Треугольник $ABD$ является прямоугольным, так как его стороны $AB$ и $AD$ являются сторонами прямоугольника. Его площадь равна половине произведения катетов $AB$ и $AD$.
$S_{\triangle ABD} = \frac{1}{2} \cdot AD \cdot AB = \frac{1}{2} a b$.
Отношение площади треугольника $ABD$ к площади прямоугольника $ABCD$ составляет:
$\frac{S_{\triangle ABD}}{S_{ABCD}} = \frac{\frac{1}{2}ab}{ab} = \frac{1}{2}$.
Ответ: $\frac{1}{2}$.
2) ΔABE;
Треугольник $ABE$ является прямоугольным с прямым углом при вершине $A$. Его катеты — $AB$ и $AE$. Длина $AB = b$, а $AE = \frac{AD}{2} = \frac{a}{2}$.
Площадь треугольника $ABE$ равна:
$S_{\triangle ABE} = \frac{1}{2} \cdot AB \cdot AE = \frac{1}{2} \cdot b \cdot \frac{a}{2} = \frac{ab}{4}$.
Отношение его площади к площади прямоугольника:
$\frac{S_{\triangle ABE}}{S_{ABCD}} = \frac{\frac{1}{4}ab}{ab} = \frac{1}{4}$.
Ответ: $\frac{1}{4}$.
3) ΔBKD;
Для нахождения площади треугольника $BKD$ примем за основание сторону $KD$. Длина основания $KD = \frac{CD}{2} = \frac{b}{2}$. Высотой, проведенной из вершины $B$ к основанию $KD$ (и прямой $CD$), будет сторона $BC$, длина которой равна $a$.
Площадь треугольника $BKD$ равна:
$S_{\triangle BKD} = \frac{1}{2} \cdot KD \cdot BC = \frac{1}{2} \cdot \frac{b}{2} \cdot a = \frac{ab}{4}$.
Отношение его площади к площади прямоугольника:
$\frac{S_{\triangle BKD}}{S_{ABCD}} = \frac{\frac{1}{4}ab}{ab} = \frac{1}{4}$.
Ответ: $\frac{1}{4}$.
4) ABLKD;
Фигура $ABLKD$ — это пятиугольник. Его площадь можно найти, вычтя из площади всего прямоугольника $ABCD$ площадь треугольника $CLK$.
Треугольник $CLK$ является прямоугольным с прямым углом при вершине $C$. Его катеты — $CL$ и $CK$.
$CL = \frac{BC}{2} = \frac{a}{2}$ и $CK = \frac{CD}{2} = \frac{b}{2}$.
Площадь треугольника $CLK$ равна:
$S_{\triangle CLK} = \frac{1}{2} \cdot CL \cdot CK = \frac{1}{2} \cdot \frac{a}{2} \cdot \frac{b}{2} = \frac{ab}{8}$.
Площадь пятиугольника $ABLKD$ равна:
$S_{ABLKD} = S_{ABCD} - S_{\triangle CLK} = ab - \frac{ab}{8} = \frac{7}{8}ab$.
Отношение его площади к площади прямоугольника:
$\frac{S_{ABLKD}}{S_{ABCD}} = \frac{\frac{7}{8}ab}{ab} = \frac{7}{8}$.
Ответ: $\frac{7}{8}$.
5) ABLKE;
Фигура $ABLKE$ — это пятиугольник. Его площадь можно найти, вычтя из площади прямоугольника $ABCD$ площади двух треугольников: $\triangle CLK$ и $\triangle EKD$.
Площадь $\triangle CLK$ мы уже нашли, она равна $\frac{ab}{8}$.
Треугольник $EKD$ является прямоугольным с прямым углом при вершине $D$. Его катеты — $ED$ и $KD$.
$ED = \frac{AD}{2} = \frac{a}{2}$ и $KD = \frac{CD}{2} = \frac{b}{2}$.
Площадь треугольника $EKD$ равна:
$S_{\triangle EKD} = \frac{1}{2} \cdot ED \cdot KD = \frac{1}{2} \cdot \frac{a}{2} \cdot \frac{b}{2} = \frac{ab}{8}$.
Площадь пятиугольника $ABLKE$ равна:
$S_{ABLKE} = S_{ABCD} - S_{\triangle CLK} - S_{\triangle EKD} = ab - \frac{ab}{8} - \frac{ab}{8} = ab - \frac{2ab}{8} = ab - \frac{ab}{4} = \frac{3}{4}ab$.
Отношение его площади к площади прямоугольника:
$\frac{S_{ABLKE}}{S_{ABCD}} = \frac{\frac{3}{4}ab}{ab} = \frac{3}{4}$.
Ответ: $\frac{3}{4}$.
6) LKEN;
Четырехугольник $LKEN$ соединяет середины сторон прямоугольника. Его площадь можно найти, вычтя из площади прямоугольника $ABCD$ площади четырех угловых треугольников: $\triangle NAE$, $\triangle NBL$, $\triangle LCK$ и $\triangle KDE$.
Площади всех этих треугольников равны:
$S_{\triangle NAE} = S_{\triangle NBL} = S_{\triangle LCK} = S_{\triangle KDE} = \frac{1}{2} \cdot \frac{a}{2} \cdot \frac{b}{2} = \frac{ab}{8}$.
Сумма их площадей равна $4 \cdot \frac{ab}{8} = \frac{ab}{2}$.
Площадь четырехугольника $LKEN$ равна:
$S_{LKEN} = S_{ABCD} - 4 \cdot \frac{ab}{8} = ab - \frac{ab}{2} = \frac{1}{2}ab$.
Отношение его площади к площади прямоугольника:
$\frac{S_{LKEN}}{S_{ABCD}} = \frac{\frac{1}{2}ab}{ab} = \frac{1}{2}$.
Ответ: $\frac{1}{2}$.
7) ALD и BEC?
Найдем площади треугольников $ALD$ и $BEC$.
Для треугольника $ALD$ основание $AD = a$. Высота, проведенная из вершины $L$ к основанию $AD$, равна стороне $AB=b$.
$S_{\triangle ALD} = \frac{1}{2} \cdot AD \cdot b = \frac{1}{2}ab$.
Для треугольника $BEC$ основание $BC = a$. Высота, проведенная из вершины $E$ к основанию $BC$, равна стороне $AB=b$.
$S_{\triangle BEC} = \frac{1}{2} \cdot BC \cdot b = \frac{1}{2}ab$.
Отношение площади каждого из этих треугольников к площади прямоугольника составляет $\frac{\frac{1}{2}ab}{ab} = \frac{1}{2}$.
Ответ: Площадь каждого из треугольников $\triangle ALD$ и $\triangle BEC$ составляет $\frac{1}{2}$ площади прямоугольника $ABCD$.
Другие задания:
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