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Номер 4.12, страница 22 - гдз по геометрии 7 класс учебник Смирнов, Туяков
Авторы: Смирнов В. А., Туяков Е. А.
Тип: Учебник
Издательство: Мектеп
Год издания: 2017 - 2026
ISBN: 978-601-07-0873-0
Утверждено Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 7 классе
Глава 1. Начальные геометрические сведения. Параграф 4. Полуплоскость и угол - номер 4.12, страница 22.
App\Models\Task {#1024 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1129460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "22" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/04-12" "field_display_title" => "4.12" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#331 #items: array:1 [ 0 => App\Models\Term {#1029 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1026 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#342 #items: array:1 [ 0 => App\Models\Branch {#1023 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1129359 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Начальные геометрические сведения" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333 #items: array:1 [ 0 => App\Models\Term {#1033 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1030 #items: array:1 [ 0 => App\Models\Book {#1032 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 2907 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1034 #items: array:1 [ 0 => App\Models\Term {#1035 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "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 {#1036 #items: array:1 [ 0 => App\Models\Term {#1037 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5456 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "7" "field_cases" => array:6 [ …6] "field_translit" => "sedmoj" ] #original: array:6 [ "id" => 5456 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "7" "field_cases" => array:6 [ …6] "field_translit" => "sedmoj" ] #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 {#1038 #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" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ …6] "field_translit" => "mektep" ] #original: array:6 [ "id" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ …6] "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 {#1040 #items: array:2 [ 0 => App\Models\Term {#1041 #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" => "2026-04-10 13:58:26" "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" => "2026-04-10 13:58:26" "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 {#1042 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7017 "created_at" => "2026-04-10 13:58:26" "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" => "2026-04-10 13:58:26" "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 {#1044 #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: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 {#1046 #items: array:1 [ 0 => App\Models\Term {#1047 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array: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 {#1048 #items: array:1 [ 0 => App\Models\Term {#1049 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array: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 {#1050 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1051 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1052 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1053 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1054 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1055 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2017" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 7 класса общеобразовательной школы" "field_allowed" => "Утверждено Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "528" "field_priority" => "2" "field_default_folder" => "/geometrija_07/Smirnova-u17/" "field_isbn" => "978-601-07-0873-0" "field_cover" => array:1 [ 0 => "/media/geometrija_07/Smirnova-u17/covers/cover1.webp?ts=1753259592" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_07/Smirnova-u17/covers/cover1.webp?ts=1753259592" "alt" => "" "width" => "1632" "height" => "2472" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1056 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1057 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1058 #items: [] #escapeWhenCastingToString: false } "field_url" => "/7-klass/geometrija/smirnova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1059 #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" => 378 "subject" => 543 "class_subject" => 389 ] ] #original: array:50 [ "id" => 2907 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1034} "field_class" => Illuminate\Database\Eloquent\Collection {#1036} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1038} "field_author" => Illuminate\Database\Eloquent\Collection {#1040} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1043} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1044} "field_country" => Illuminate\Database\Eloquent\Collection {#1046} "field_city" => Illuminate\Database\Eloquent\Collection {#1048} "field_series" => Illuminate\Database\Eloquent\Collection {#1050} "field_umk" => Illuminate\Database\Eloquent\Collection {#1051} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1052} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1053} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1054} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1055} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2017" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 7 класса общеобразовательной школы" "field_allowed" => "Утверждено Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "528" "field_priority" => "2" "field_default_folder" => "/geometrija_07/Smirnova-u17/" "field_isbn" => "978-601-07-0873-0" "field_cover" => array:1 [ 0 => "/media/geometrija_07/Smirnova-u17/covers/cover1.webp?ts=1753259592" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_07/Smirnova-u17/covers/cover1.webp?ts=1753259592" "alt" => "" "width" => "1632" "height" => "2472" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1056} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1057} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1058} "field_url" => "/7-klass/geometrija/smirnova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1059} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 378 "subject" => 543 "class_subject" => 389 ] ] #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 {#1060 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1129359 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Начальные геометрические сведения" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333} "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1030} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1060} ] #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 {#1061 #items: array:2 [ 0 => App\Models\Branch {#1023} 1 => App\Models\Branch {#1062 #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" => 1129363 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Полуплоскость и угол" "field_branch_order" => "4" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1063 #items: array:1 [ 0 => App\Models\Term {#1064 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => "§" ] #original: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => "§" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "19" "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 {#1065 #items: array:1 [ 0 => App\Models\Book {#1032} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1066 #items: array:1 [ 0 => App\Models\Branch {#1067 #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" => 1129359 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Начальные геометрические сведения" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_page_start" => "5" "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 {#1069 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1070 …2} ] #original: array:24 [ "id" => 1129359 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Начальные геометрические сведения" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_page_start" => "5" "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 {#1069 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1070 …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" => 1129363 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Полуплоскость и угол" "field_branch_order" => "4" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1063} "field_page_start" => "19" "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 {#1065} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1066} ] #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 {#1071 #items: array:3 [ 0 => App\Models\Element {#1072 #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" => 1319534 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1073 #items: array:1 [ 0 => App\Models\Edition {#1074 #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" => 5009 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1075 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1077 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1078 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1079 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5009 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1075 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1077 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1078 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1079 …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" => "1129460" "type" => "task" ] "text" => "<p><strong>4.12.</strong> Изобразите три попарно пересекающиеся прямые, не пересекающиеся в одной точке. На сколько частей они разбивают плоскость?</p>" "img" => array:1 [ 0 => array:5 [ "name" => "12-1.jpg" "alt" => null "width" => "1721" "height" => 182 "path" => "/media/geometrija_07/Smirnova-u17/0-04/12-1.webp?ts=1753982216" ] ] ] #original: array:7 [ "id" => 1319534 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1073} "task" => array:2 [ "refs" => "1129460" "type" => "task" ] "text" => "<p><strong>4.12.</strong> Изобразите три попарно пересекающиеся прямые, не пересекающиеся в одной точке. На сколько частей они разбивают плоскость?</p>" "img" => array:1 [ 0 => array:5 [ "name" => "12-1.jpg" "alt" => null "width" => "1721" "height" => 182 "path" => "/media/geometrija_07/Smirnova-u17/0-04/12-1.webp?ts=1753982216" ] ] ] #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 {#1080 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1319944 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1081 #items: array:1 [ 0 => App\Models\Edition {#1082 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5010 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1083 …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 {#1085 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5010 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1083 …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 {#1085 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1129460" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "12-1.jpg" "alt" => null "width" => "1333" "height" => 847 "path" => "/media/geometrija_07/Smirnova-u17/1-04/12-1.webp?ts=1753982328" ] ] ] #original: array:6 [ "id" => 1319944 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1081} "task" => array:2 [ "refs" => "1129460" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "12-1.jpg" "alt" => null "width" => "1333" "height" => 847 "path" => "/media/geometrija_07/Smirnova-u17/1-04/12-1.webp?ts=1753982328" ] ] ] #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 {#1088 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1541324 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1089 #items: array:1 [ 0 => App\Models\Edition {#1090 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5631 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_content_type" => "free" "field_content_mode" => "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 {#1093 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1094 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5631 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_content_type" => "free" "field_content_mode" => "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 {#1093 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1094 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1129460" "type" => "task" ] "text" => "<p>Для решения задачи необходимо изобразить три прямые, которые отвечают двум условиям:<br>1. <strong>Попарное пересечение:</strong> каждая прямая пересекает две другие. Это означает, что среди прямых нет параллельных.<br>2. <strong>Непересечение в одной точке:</strong> все три прямые не проходят через одну общую точку.</p><p>Такое расположение прямых образует на плоскости треугольник, ограниченный отрезками этих прямых.<br>Ниже представлено изображение такой конфигурации. Цифрами обозначены части (области), на которые плоскость разбивается этими прямыми.</p><svg width="250" height="200" viewBox="0 0 250 200" xmlns="http://www.w3.org/2000/svg" style="display: block; margin: 1em auto;"><style> .line { stroke: #000000; stroke-width: 2; } .label { font-family: sans-serif; font-size: 18px; font-weight: bold; fill: #d9534f; text-anchor: middle; dominant-baseline: middle; } </style><line class="line" x1="10" y1="180" x2="240" y2="180"></line><line class="line" x1="30" y1="20" x2="210" y2="190"></line><line class="line" x1="220" y1="20" x2="40" y2="190"></line><text x="125" y="155" class="label">1</text><text x="80" y="110" class="label">2</text><text x="170" y="110" class="label">3</text><text x="125" y="60" class="label">4</text><text x="25" y="160" class="label">5</text><text x="225" y="160" class="label">6</text><text x="125" y="190" class="label">7</text></svg><p>Подсчитаем количество частей, на которые прямые разбивают плоскость, последовательно добавляя прямые:</p><p><strong>1.</strong> <strong>Первая прямая</strong> делит бесконечную плоскость на <strong>2</strong> части.</p><p><strong>2.</strong> <strong>Вторая прямая</strong> пересекает первую. При этом она сама делится точкой пересечения на два луча. Каждый из этих лучей проходит через одну из двух существующих областей и делит её надвое. Таким образом, вторая прямая добавляет ещё 2 новые области. Общее число частей становится $2 + 2 = 4$.</p><p><strong>3.</strong> <strong>Третья прямая</strong>, согласно условию, пересекает первые две прямые в двух различных точках. Эти две точки пересечения делят третью прямую на три части (один отрезок и два луча). Каждая из этих трёх частей прямой проходит через одну из четырёх существующих областей, разделяя её на две. Следовательно, третья прямая добавляет ещё 3 новые области. Общее число частей становится $4 + 3 = 7$.</p><p>Этот результат можно также проверить с помощью общей формулы для максимального числа областей $L_n$, на которые $n$ прямых могут разделить плоскость (при условии, что никакие две прямые не параллельны и никакие три не пересекаются в одной точке):<br>$L_n = \frac{n(n+1)}{2} + 1$<br>Для нашего случая, где $n=3$:<br>$L_3 = \frac{3(3+1)}{2} + 1 = \frac{3 \times 4}{2} + 1 = 6 + 1 = 7$</p><p><strong>Ответ:</strong> три попарно пересекающиеся прямые, не пересекающиеся в одной точке, разбивают плоскость на 7 частей.</p>" ] #original: array:6 [ "id" => 1541324 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1089} "task" => array:2 [ "refs" => "1129460" "type" => "task" ] "text" => "<p>Для решения задачи необходимо изобразить три прямые, которые отвечают двум условиям:<br>1. <strong>Попарное пересечение:</strong> каждая прямая пересекает две другие. Это означает, что среди прямых нет параллельных.<br>2. <strong>Непересечение в одной точке:</strong> все три прямые не проходят через одну общую точку.</p><p>Такое расположение прямых образует на плоскости треугольник, ограниченный отрезками этих прямых.<br>Ниже представлено изображение такой конфигурации. Цифрами обозначены части (области), на которые плоскость разбивается этими прямыми.</p><svg width="250" height="200" viewBox="0 0 250 200" xmlns="http://www.w3.org/2000/svg" style="display: block; margin: 1em auto;"><style> .line { stroke: #000000; stroke-width: 2; } .label { font-family: sans-serif; font-size: 18px; font-weight: bold; fill: #d9534f; text-anchor: middle; dominant-baseline: middle; } </style><line class="line" x1="10" y1="180" x2="240" y2="180"></line><line class="line" x1="30" y1="20" x2="210" y2="190"></line><line class="line" x1="220" y1="20" x2="40" y2="190"></line><text x="125" y="155" class="label">1</text><text x="80" y="110" class="label">2</text><text x="170" y="110" class="label">3</text><text x="125" y="60" class="label">4</text><text x="25" y="160" class="label">5</text><text x="225" y="160" class="label">6</text><text x="125" y="190" class="label">7</text></svg><p>Подсчитаем количество частей, на которые прямые разбивают плоскость, последовательно добавляя прямые:</p><p><strong>1.</strong> <strong>Первая прямая</strong> делит бесконечную плоскость на <strong>2</strong> части.</p><p><strong>2.</strong> <strong>Вторая прямая</strong> пересекает первую. При этом она сама делится точкой пересечения на два луча. Каждый из этих лучей проходит через одну из двух существующих областей и делит её надвое. Таким образом, вторая прямая добавляет ещё 2 новые области. Общее число частей становится $2 + 2 = 4$.</p><p><strong>3.</strong> <strong>Третья прямая</strong>, согласно условию, пересекает первые две прямые в двух различных точках. Эти две точки пересечения делят третью прямую на три части (один отрезок и два луча). Каждая из этих трёх частей прямой проходит через одну из четырёх существующих областей, разделяя её на две. Следовательно, третья прямая добавляет ещё 3 новые области. Общее число частей становится $4 + 3 = 7$.</p><p>Этот результат можно также проверить с помощью общей формулы для максимального числа областей $L_n$, на которые $n$ прямых могут разделить плоскость (при условии, что никакие две прямые не параллельны и никакие три не пересекаются в одной точке):<br>$L_n = \frac{n(n+1)}{2} + 1$<br>Для нашего случая, где $n=3$:<br>$L_3 = \frac{3(3+1)}{2} + 1 = \frac{3 \times 4}{2} + 1 = 6 + 1 = 7$</p><p><strong>Ответ:</strong> три попарно пересекающиеся прямые, не пересекающиеся в одной точке, разбивают плоскость на 7 частей.</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" => "1129461" "type" => "task" ] "previous" => array:2 [ "refs" => "1129459" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1096 #items: array:1 [ 0 => App\Models\Book {#1032} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1315 #items: array:1 [ 0 => App\Models\BookPage {#1101 #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" => 1114339 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "22" "field_url" => "/7-klass/geometrija/smirnova-uchebnik/page-22" "field_display_title" => "22" "field_folder" => "1" "field_image_name" => "22" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1104 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1103 #items: array:1 [ 0 => App\Models\Book {#1105 #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" => 2907 "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 {#1106 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1108 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1110 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1112 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1115 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1116 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1118 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1120 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1122 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1125 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1127 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2017" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 7 класса общеобразовательной школы" "field_allowed" => "Утверждено Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "528" "field_priority" => "2" "field_default_folder" => "/geometrija_07/Smirnova-u17/" "field_isbn" => "978-601-07-0873-0" "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 {#1128 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1129 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_url" => "/7-klass/geometrija/smirnova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1131 …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" => 2907 "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 {#1106 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1108 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1110 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1112 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1115 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1116 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1118 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1120 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1122 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1125 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1127 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2017" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 7 класса общеобразовательной школы" "field_allowed" => "Утверждено Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "528" "field_priority" => "2" "field_default_folder" => "/geometrija_07/Smirnova-u17/" "field_isbn" => "978-601-07-0873-0" "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 {#1128 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1129 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_url" => "/7-klass/geometrija/smirnova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1131 …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 } "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 {#1132 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1133 #items: array:1 [ 0 => App\Models\Element {#1134 #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" => 1290526 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1135 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1290526 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1135 …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" => "1114340" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1114338" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1142 #items: array:11 [ 0 => App\Models\Task {#1143 #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" => 1129454 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "22" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/04-6" "field_display_title" => "4.6" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1144 …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 {#1146 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1147 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1153 …2} "content" => Illuminate\Database\Eloquent\Collection {#1163 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1182 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1129454 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "22" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/04-6" "field_display_title" => "4.6" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1144 …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 {#1146 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1147 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1153 …2} "content" => Illuminate\Database\Eloquent\Collection {#1163 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1182 …2} "page" => array:2 [ …2] ] #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\Task {#1183 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1129455 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "22" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/04-7" "field_display_title" => "4.7" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1184 …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 {#1185 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1186 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1187 …2} "content" => Illuminate\Database\Eloquent\Collection {#1188 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1195 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1129455 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "22" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/04-7" "field_display_title" => "4.7" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1184 …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 {#1185 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1186 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1187 …2} "content" => Illuminate\Database\Eloquent\Collection {#1188 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1195 …2} "page" => array:2 [ …2] ] #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\Task {#1196 #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" => 1129456 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "22" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/04-8" "field_display_title" => "4.8" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1197 …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 {#1198 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1199 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1200 …2} "content" => Illuminate\Database\Eloquent\Collection {#1201 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1208 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1129456 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "22" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/04-8" "field_display_title" => "4.8" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1197 …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 {#1198 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1199 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1200 …2} "content" => Illuminate\Database\Eloquent\Collection {#1201 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1208 …2} "page" => array:2 [ …2] ] #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\Task {#1209 #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" => 1129457 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "22" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/04-9" "field_display_title" => "4.9" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1210 …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 {#1211 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1212 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1213 …2} "content" => Illuminate\Database\Eloquent\Collection {#1214 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1221 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1129457 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "22" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/04-9" "field_display_title" => "4.9" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1210 …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 {#1211 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1212 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1213 …2} "content" => Illuminate\Database\Eloquent\Collection {#1214 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1221 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Task {#1222 #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" => 1129458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "22" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/04-10" "field_display_title" => "4.10" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1223 …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 {#1224 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1225 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1226 …2} "content" => Illuminate\Database\Eloquent\Collection {#1227 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1234 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1129458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "22" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/04-10" "field_display_title" => "4.10" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1223 …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 {#1224 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1225 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1226 …2} "content" => Illuminate\Database\Eloquent\Collection {#1227 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1234 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 5 => App\Models\Task {#1235 #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" => 1129459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "22" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/04-11" "field_display_title" => "4.11" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1236 …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 {#1237 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1238 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1239 …2} "content" => Illuminate\Database\Eloquent\Collection {#1240 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1247 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1129459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "22" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/04-11" "field_display_title" => "4.11" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1236 …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 {#1237 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1238 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1239 …2} "content" => Illuminate\Database\Eloquent\Collection {#1240 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1247 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 6 => App\Models\Task {#1248 #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" => 1129460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "22" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/04-12" "field_display_title" => "4.12" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1249 …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 {#1250 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1251 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1252 …2} "content" => Illuminate\Database\Eloquent\Collection {#1253 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1260 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1129460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "22" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/04-12" "field_display_title" => "4.12" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1249 …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 {#1250 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1251 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1252 …2} "content" => Illuminate\Database\Eloquent\Collection {#1253 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1260 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 7 => App\Models\Task {#1261 #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" => 1129461 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "22" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/04-13" "field_display_title" => "4.13" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1262 …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 {#1263 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1264 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1265 …2} "content" => Illuminate\Database\Eloquent\Collection {#1266 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1273 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1129461 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "22" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/04-13" "field_display_title" => "4.13" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1262 …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 {#1263 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1264 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1265 …2} "content" => Illuminate\Database\Eloquent\Collection {#1266 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1273 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 8 => App\Models\Task {#1274 #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" => 1129462 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "22" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/04-14" "field_display_title" => "4.14" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1275 …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 {#1276 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1277 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1278 …2} "content" => Illuminate\Database\Eloquent\Collection {#1279 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1286 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1129462 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "22" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/04-14" "field_display_title" => "4.14" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1275 …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 {#1276 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1277 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1278 …2} "content" => Illuminate\Database\Eloquent\Collection {#1279 …2} "next" => array:2 [ …2] …3 ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 9 => App\Models\Task {#1287 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 10 => App\Models\Task {#1300 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 1114339 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "22" "field_url" => "/7-klass/geometrija/smirnova-uchebnik/page-22" "field_display_title" => "22" "field_folder" => "1" "field_image_name" => "22" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1104} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1103} "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 {#1132} "content" => Illuminate\Database\Eloquent\Collection {#1133} "next" => array:2 [ "refs" => "1114340" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1114338" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1142} ] #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" => 1129460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "22" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/04-12" "field_display_title" => "4.12" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#331} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1026} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#342} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1061} "content" => Illuminate\Database\Eloquent\Collection {#1071} "next" => array:2 [ "refs" => "1129461" "type" => "task" ] "previous" => array:2 [ "refs" => "1129459" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1096} "page" => array:2 [ "refs" => "1114339" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] }
№4.12 (с. 22)
Условие. №4.12 (с. 22)
скриншот условия
4.12. Изобразите три попарно пересекающиеся прямые, не пересекающиеся в одной точке. На сколько частей они разбивают плоскость?
Решение. №4.12 (с. 22)
Решение 2. №4.12 (с. 22)
Для решения задачи необходимо изобразить три прямые, которые отвечают двум условиям:
1. Попарное пересечение: каждая прямая пересекает две другие. Это означает, что среди прямых нет параллельных.
2. Непересечение в одной точке: все три прямые не проходят через одну общую точку.
Такое расположение прямых образует на плоскости треугольник, ограниченный отрезками этих прямых.
Ниже представлено изображение такой конфигурации. Цифрами обозначены части (области), на которые плоскость разбивается этими прямыми.
Подсчитаем количество частей, на которые прямые разбивают плоскость, последовательно добавляя прямые:
1. Первая прямая делит бесконечную плоскость на 2 части.
2. Вторая прямая пересекает первую. При этом она сама делится точкой пересечения на два луча. Каждый из этих лучей проходит через одну из двух существующих областей и делит её надвое. Таким образом, вторая прямая добавляет ещё 2 новые области. Общее число частей становится $2 + 2 = 4$.
3. Третья прямая, согласно условию, пересекает первые две прямые в двух различных точках. Эти две точки пересечения делят третью прямую на три части (один отрезок и два луча). Каждая из этих трёх частей прямой проходит через одну из четырёх существующих областей, разделяя её на две. Следовательно, третья прямая добавляет ещё 3 новые области. Общее число частей становится $4 + 3 = 7$.
Этот результат можно также проверить с помощью общей формулы для максимального числа областей $L_n$, на которые $n$ прямых могут разделить плоскость (при условии, что никакие две прямые не параллельны и никакие три не пересекаются в одной точке):
$L_n = \frac{n(n+1)}{2} + 1$
Для нашего случая, где $n=3$:
$L_3 = \frac{3(3+1)}{2} + 1 = \frac{3 \times 4}{2} + 1 = 6 + 1 = 7$
Ответ: три попарно пересекающиеся прямые, не пересекающиеся в одной точке, разбивают плоскость на 7 частей.
Другие задания:
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ПрисоединитьсяМы подготовили для вас ответ c подробным объяснением домашего задания по геометрии за 7 класс, для упражнения номер 4.12 расположенного на странице 22 к учебнику 2017 года издания для учащихся школ и гимназий.
Теперь на нашем сайте ГДЗ.ТОП вы всегда легко и бесплатно найдёте условие с правильным ответом на вопрос «Как решить ДЗ» и «Как сделать» задание по геометрии к упражнению №4.12 (с. 22), авторов: Смирнов (Владимир Алексеевич), Туяков (Есенкельды Алыбаевич), учебного пособия издательства Мектеп.