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Задания, страница 19 - 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 - страница 19.
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" => 1129447 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "19" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/samost-4" "field_display_title" => "Задания" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1036 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1034 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1045 #items: array:1 [ 0 => App\Models\Branch {#1044 #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 {#1049 #items: array:1 [ 0 => App\Models\Term {#1046 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "field_page_check_not_needed" => null "field_short_name" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "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 {#1057 #items: array:1 [ 0 => App\Models\Book {#1056 #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 {#1061 #items: array:1 [ 0 => App\Models\Term {#1058 #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 [ "field_accusative_case" => "геометрию" "field_creative_case" => "геометрией" "field_dative_case" => "геометрии" "field_genitive_case" => "геометрии" "field_nominative_case" => "геометрия" "field_prepositional_case" => "геометрии" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "geometrija" ] #original: array:10 [ "id" => 6477 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Геометрия" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "геометрию" "field_creative_case" => "геометрией" "field_dative_case" => "геометрии" "field_genitive_case" => "геометрии" "field_nominative_case" => "геометрия" "field_prepositional_case" => "геометрии" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "geometrija" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1063 #items: array:1 [ 0 => App\Models\Term {#1060 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5456 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "7" "field_cases" => array:6 [ "field_accusative_case" => "седьмой" "field_creative_case" => "седьмым" "field_dative_case" => "седьмому" "field_genitive_case" => "седьмого" "field_nominative_case" => "седьмой" "field_prepositional_case" => "седьмом" ] "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 [ "field_accusative_case" => "седьмой" "field_creative_case" => "седьмым" "field_dative_case" => "седьмому" "field_genitive_case" => "седьмого" "field_nominative_case" => "седьмой" "field_prepositional_case" => "седьмом" ] "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 {#1065 #items: array:1 [ 0 => App\Models\Term {#1062 #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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #original: array:6 [ "id" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1059 #items: array:2 [ 0 => App\Models\Term {#1074 #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 {#1075 #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 {#1071 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1069 #items: array:1 [ 0 => App\Models\Term {#1066 #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 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1070 #items: array:1 [ 0 => App\Models\Term {#1073 #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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "kazakhstan" ] #original: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "kazakhstan" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1076 #items: array:1 [ 0 => App\Models\Term {#1067 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almaty" ] #original: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almaty" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1068 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1064 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1077 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1078 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1079 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1080 #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 {#1081 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1082 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1083 #items: [] #escapeWhenCastingToString: false } "field_url" => "/7-klass/geometrija/smirnova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1084 #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 {#1061} "field_class" => Illuminate\Database\Eloquent\Collection {#1063} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1065} "field_author" => Illuminate\Database\Eloquent\Collection {#1059} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1071} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1069} "field_country" => Illuminate\Database\Eloquent\Collection {#1070} "field_city" => Illuminate\Database\Eloquent\Collection {#1076} "field_series" => Illuminate\Database\Eloquent\Collection {#1068} "field_umk" => Illuminate\Database\Eloquent\Collection {#1064} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1077} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1078} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1079} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1080} "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 {#1081} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1082} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1083} "field_url" => "/7-klass/geometrija/smirnova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1084} "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 {#1047 #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 {#1049} "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 {#1057} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1047} ] #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 {#1106 #items: array:2 [ 0 => App\Models\Branch {#1114 #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 {#1115 #items: array:1 [ 0 => App\Models\Term {#1046} ] #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 {#1116 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1117 #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 {#1115} "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 {#1116} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1117} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Branch {#1118 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 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 {#1122 #items: array:1 [ 0 => App\Models\Term {#1119 #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 [ "field_accusative_case" => "Параграф" "field_creative_case" => "Параграфом" "field_dative_case" => "Параграфу" "field_genitive_case" => "Параграфа" "field_nominative_case" => "Параграф" "field_prepositional_case" => "Параграфе" ] "field_page_check_not_needed" => null "field_short_name" => "§" ] #original: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ "field_accusative_case" => "Параграф" "field_creative_case" => "Параграфом" "field_dative_case" => "Параграфу" "field_genitive_case" => "Параграфа" "field_nominative_case" => "Параграф" "field_prepositional_case" => "Параграфе" ] "field_page_check_not_needed" => null "field_short_name" => "§" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "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 {#1121 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1120 #items: array:1 [ 0 => App\Models\Branch {#1044} ] #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 {#1122} "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 {#1121} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1120} ] #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 {#1103 #items: array:3 [ 0 => App\Models\Element {#1096 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 1320094 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1086 #items: array:1 [ 0 => App\Models\Edition {#1088 #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 {#1054 #items: array:1 [ 0 => App\Models\Term {#1062} ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1048 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1052 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1053 #items: [] #escapeWhenCastingToString: false } "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 {#1054} "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 {#1048} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1052} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1053} "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" => "1129447" "type" => "task" ] "text" => "<p>Как вы думаете, на сколько частей эта прямая разбивает плоскость?</p><p>Попробуйте объяснить это самостоятельно.</p>" "img" => array:2 [ 0 => array:5 [ "name" => "4-1.jpg" "alt" => null "width" => "1610" "height" => 224 "path" => "/media/geometrija_07/Smirnova-u17/0-samost/4-1.webp?ts=1753982503" ] 1 => array:5 [ "name" => "4-2.jpg" "alt" => null "width" => "1158" "height" => 197 "path" => "/media/geometrija_07/Smirnova-u17/0-samost/4-2.webp?ts=1753982503" ] ] ] #original: array:7 [ "id" => 1320094 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1086} "task" => array:2 [ "refs" => "1129447" "type" => "task" ] "text" => "<p>Как вы думаете, на сколько частей эта прямая разбивает плоскость?</p><p>Попробуйте объяснить это самостоятельно.</p>" "img" => array:2 [ 0 => array:5 [ "name" => "4-1.jpg" "alt" => null "width" => "1610" "height" => 224 "path" => "/media/geometrija_07/Smirnova-u17/0-samost/4-1.webp?ts=1753982503" ] 1 => array:5 [ "name" => "4-2.jpg" "alt" => null "width" => "1158" "height" => 197 "path" => "/media/geometrija_07/Smirnova-u17/0-samost/4-2.webp?ts=1753982503" ] ] ] #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 {#1094 #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" => 1320481 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1050 #items: array:1 [ 0 => App\Models\Edition {#1042 #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 {#1104 #items: array:1 [ 0 => App\Models\Term {#1040 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6714 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Gdz" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "gdz" ] #original: array:6 [ "id" => 6714 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Gdz" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "gdz" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "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 {#1110 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1113 #items: [] #escapeWhenCastingToString: false } "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 {#1104} "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 {#1110} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1035} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1113} "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" => "1129447" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "4-1.jpg" "alt" => null "width" => "1124" "height" => 1199 "path" => "/media/geometrija_07/Smirnova-u17/1-samost/4-1.webp?ts=1753982622" ] ] ] #original: array:6 [ "id" => 1320481 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1050} "task" => array:2 [ "refs" => "1129447" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "4-1.jpg" "alt" => null "width" => "1124" "height" => 1199 "path" => "/media/geometrija_07/Smirnova-u17/1-samost/4-1.webp?ts=1753982622" ] ] ] #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 {#1095 #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" => 1541743 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1107 #items: array:1 [ 0 => App\Models\Edition {#1108 #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 {#1127 #items: array:1 [ 0 => App\Models\Term {#1123 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 7012 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Gemini 2.5 Pro" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "gemini 2.5 pro" ] #original: array:6 [ "id" => 7012 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Gemini 2.5 Pro" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "gemini 2.5 pro" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1131 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1124 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1128 #items: [] #escapeWhenCastingToString: false } "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 {#1127} "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 {#1131} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1124} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1128} "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" => "1129447" "type" => "task" ] "text" => "<p><strong>Как вы думаете, на сколько частей эта прямая разбивает плоскость?</strong></p><p>Любая прямая, проведенная на плоскости, разбивает эту плоскость ровно на две части. Эти части в геометрии носят название <strong>полуплоскости</strong>.<br>Ответ: на 2 части.</p><p><strong>Попробуйте объяснить это самостоятельно.</strong></p><p>Представим себе плоскость как бесконечный, идеально ровный лист бумаги. Прямая линия, проведенная на этой плоскости, также является бесконечной и простирается в обе стороны.</p><p>Эта прямая линия действует как непреодолимая граница. Все точки плоскости, которые не лежат на самой прямой, оказываются разделены этой границей на две отдельные области. Это можно охарактеризовать так:<br>1. Если взять любые две точки, принадлежащие одной и той же области, то их можно соединить отрезком, который <strong>не пересечет</strong> исходную прямую.<br>2. Если же взять одну точку из первой области, а другую — из второй, то любой отрезок, соединяющий эти две точки, <strong>обязательно пересечет</strong> прямую-границу.</p><p>Таким образом, прямая делит всю плоскость на две непересекающиеся части, которые называются <strong>полуплоскостями</strong>. Сама прямая является их <strong>общей границей</strong>.</p><p>Это можно наглядно проиллюстрировать:<br><svg width="350" height="220" viewBox="0 0 350 220" xmlns="http://www.w3.org/2000/svg" style="margin: 10px 0;"><defs><clipPath id="clip_area_1"><polygon points="10,180 340,40 340,0 10,0"></polygon></clipPath><clipPath id="clip_area_2"><polygon points="10,180 340,40 340,220 10,220"></polygon></clipPath></defs><rect x="0" y="0" width="350" height="220" fill="#f9f9f9" stroke="#cccccc"></rect><rect x="0" y="0" width="350" height="220" fill="#e3f2fd" clip-path="url(#clip_area_1)"></rect><rect x="0" y="0" width="350" height="220" fill="#ffebee" clip-path="url(#clip_area_2)"></rect><line x1="10" y1="180" x2="340" y2="40" stroke="#d32f2f" stroke-width="3"></line><text x="120" y="80" font-family="sans-serif" font-size="18" fill="#1e88e5">Полуплоскость 1</text><text x="230" y="160" font-family="sans-serif" font-size="18" fill="#c62828">Полуплоскость 2</text><text x="175" y="125" font-family="sans-serif" font-size="14" fill="#333333" transform="rotate(-21.8 175,125)">Общая граница (прямая)</text></svg><br>На рисунке плоскость условно показана прямоугольником. Красная линия (прямая) разделяет её на две области, закрашенные разными цветами — это и есть две полуплоскости.</p><p>Ответ: Прямая, являясь бесконечной границей, разделяет всю плоскость на две отдельные области, называемые полуплоскостями.</p>" ] #original: array:6 [ "id" => 1541743 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1107} "task" => array:2 [ "refs" => "1129447" "type" => "task" ] "text" => "<p><strong>Как вы думаете, на сколько частей эта прямая разбивает плоскость?</strong></p><p>Любая прямая, проведенная на плоскости, разбивает эту плоскость ровно на две части. Эти части в геометрии носят название <strong>полуплоскости</strong>.<br>Ответ: на 2 части.</p><p><strong>Попробуйте объяснить это самостоятельно.</strong></p><p>Представим себе плоскость как бесконечный, идеально ровный лист бумаги. Прямая линия, проведенная на этой плоскости, также является бесконечной и простирается в обе стороны.</p><p>Эта прямая линия действует как непреодолимая граница. Все точки плоскости, которые не лежат на самой прямой, оказываются разделены этой границей на две отдельные области. Это можно охарактеризовать так:<br>1. Если взять любые две точки, принадлежащие одной и той же области, то их можно соединить отрезком, который <strong>не пересечет</strong> исходную прямую.<br>2. Если же взять одну точку из первой области, а другую — из второй, то любой отрезок, соединяющий эти две точки, <strong>обязательно пересечет</strong> прямую-границу.</p><p>Таким образом, прямая делит всю плоскость на две непересекающиеся части, которые называются <strong>полуплоскостями</strong>. Сама прямая является их <strong>общей границей</strong>.</p><p>Это можно наглядно проиллюстрировать:<br><svg width="350" height="220" viewBox="0 0 350 220" xmlns="http://www.w3.org/2000/svg" style="margin: 10px 0;"><defs><clipPath id="clip_area_1"><polygon points="10,180 340,40 340,0 10,0"></polygon></clipPath><clipPath id="clip_area_2"><polygon points="10,180 340,40 340,220 10,220"></polygon></clipPath></defs><rect x="0" y="0" width="350" height="220" fill="#f9f9f9" stroke="#cccccc"></rect><rect x="0" y="0" width="350" height="220" fill="#e3f2fd" clip-path="url(#clip_area_1)"></rect><rect x="0" y="0" width="350" height="220" fill="#ffebee" clip-path="url(#clip_area_2)"></rect><line x1="10" y1="180" x2="340" y2="40" stroke="#d32f2f" stroke-width="3"></line><text x="120" y="80" font-family="sans-serif" font-size="18" fill="#1e88e5">Полуплоскость 1</text><text x="230" y="160" font-family="sans-serif" font-size="18" fill="#c62828">Полуплоскость 2</text><text x="175" y="125" font-family="sans-serif" font-size="14" fill="#333333" transform="rotate(-21.8 175,125)">Общая граница (прямая)</text></svg><br>На рисунке плоскость условно показана прямоугольником. Красная линия (прямая) разделяет её на две области, закрашенные разными цветами — это и есть две полуплоскости.</p><p>Ответ: Прямая, являясь бесконечной границей, разделяет всю плоскость на две отдельные области, называемые полуплоскостями.</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" => "1129448" "type" => "task" ] "previous" => array:2 [ "refs" => "1129446" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1099 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1126 #items: array:1 [ 0 => App\Models\BookPage {#1085 #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" => 1114336 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "19" "field_url" => "/7-klass/geometrija/smirnova-uchebnik/page-19" "field_display_title" => "19" "field_folder" => "1" "field_image_name" => "19" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1129 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1132 #items: array:1 [ 0 => App\Models\Book {#1056} ] #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 {#1130 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1143 #items: array:1 [ 0 => App\Models\Element {#1142 #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" => 1290523 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1144 #items: array:1 [ 0 => App\Models\Edition {#1088} ] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1114336" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "19-1.jpg" "alt" => null "width" => "1906" "height" => 2818 "path" => "/media/geometrija_07/Smirnova-u17/0-1/19-1.webp?ts=1753260117" ] ] ] #original: array:6 [ "id" => 1290523 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1144} "book_page" => array:2 [ "refs" => "1114336" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "19-1.jpg" "alt" => null "width" => "1906" "height" => 2818 "path" => "/media/geometrija_07/Smirnova-u17/0-1/19-1.webp?ts=1753260117" ] ] ] #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" => "1114337" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1114335" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1211 #items: array:2 [ 0 => App\Models\Task {#1219 #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" => 1129446 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "19" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/03-22" "field_display_title" => "3.22" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1223 #items: array:1 [ 0 => App\Models\Term {#1220 #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 {#1222 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1221 #items: array:1 [ 0 => App\Models\Branch {#1114} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1258 #items: array:2 [ 0 => App\Models\Branch {#1266 #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 {#1267 #items: array:1 [ 0 => App\Models\Term {#1046} ] #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 {#1268 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1269 #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 {#1267} "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 {#1268} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1269} ] #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 {#1270 #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" => 1129362 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Измерение длин отрезков" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1271 #items: array:1 [ 0 => App\Models\Term {#1119} ] #escapeWhenCastingToString: false } "field_page_start" => "15" "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 {#1272 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1273 #items: array:1 [ 0 => App\Models\Branch {#1114} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1129362 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Измерение длин отрезков" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1271} "field_page_start" => "15" "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 {#1272} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1273} ] #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 {#1255 #items: array:3 [ 0 => App\Models\Element {#1248 #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" => 1319522 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1247 #items: array:1 [ 0 => App\Models\Edition {#1088} ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1129446" "type" => "task" ] "text" => "<p>3.22. На сколько частей разбивают плоскость:</p><p><strong>а)</strong> два луча;</p><p><strong>б)</strong> три луча;</p><p><strong>в)</strong> четыре луча;</p><p><strong>г)</strong> * $n$ лучей с общей вершиной?</p>" "img" => array:1 [ 0 => array:5 [ "name" => "22-1.jpg" "alt" => null "width" => "1705" "height" => 343 "path" => "/media/geometrija_07/Smirnova-u17/0-03/22-1.webp?ts=1753982216" ] ] ] #original: array:7 [ "id" => 1319522 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1247} "task" => array:2 [ "refs" => "1129446" "type" => "task" ] "text" => "<p>3.22. На сколько частей разбивают плоскость:</p><p><strong>а)</strong> два луча;</p><p><strong>б)</strong> три луча;</p><p><strong>в)</strong> четыре луча;</p><p><strong>г)</strong> * $n$ лучей с общей вершиной?</p>" "img" => array:1 [ 0 => array:5 [ "name" => "22-1.jpg" "alt" => null "width" => "1705" "height" => 343 "path" => "/media/geometrija_07/Smirnova-u17/0-03/22-1.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 {#1246 #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" => 1319914 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1245 #items: array:1 [ 0 => App\Models\Edition {#1042} ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1129446" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "22-1.jpg" "alt" => null "width" => "1048" "height" => 684 "path" => "/media/geometrija_07/Smirnova-u17/1-03/22-1.webp?ts=1753982328" ] ] ] #original: array:6 [ "id" => 1319914 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1245} "task" => array:2 [ "refs" => "1129446" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "22-1.jpg" "alt" => null "width" => "1048" "height" => 684 "path" => "/media/geometrija_07/Smirnova-u17/1-03/22-1.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 {#1243 #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" => 1541312 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1244 #items: array:1 [ 0 => App\Models\Edition {#1108} ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1129446" "type" => "task" ] "text" => "<p><strong>а)</strong> Два луча, исходящие из одной точки (общей вершины), образуют угол. Эти лучи делят всю плоскость на две части: внутреннюю область угла и внешнюю область угла.<br><svg width="150" height="100" viewBox="0 0 150 100" xmlns="http://www.w3.org/2000/svg"><title>Два луча делят плоскость на две части</title><circle cx="50" cy="50" r="2.5" fill="black"></circle><text x="40" y="48" font-size="14">O</text><line x1="50" y1="50" x2="140" y2="30" stroke="black" stroke-width="2"></line><line x1="50" y1="50" x2="140" y2="80" stroke="black" stroke-width="2"></line><text x="90" y="55" font-size="14" fill="#00008B">Часть 1</text><text x="20" y="75" font-size="14" fill="#8B0000">Часть 2</text></svg><br>Таким образом, два луча с общей вершиной разбивают плоскость на 2 части.<br>Ответ: 2.</p><p><strong>б)</strong> Рассмотрим три луча с общей вершиной. Первые два луча делят плоскость на две части. Третий луч, проведенный из той же вершины, пройдет через одну из этих частей и разделит ее на две новые. В результате общее число частей станет $2 - 1 + 2 = 3$.<br><svg width="150" height="120" viewBox="0 0 150 120" xmlns="http://www.w3.org/2000/svg"><title>Три луча делят плоскость на три части</title><circle cx="75" cy="60" r="2.5" fill="black"></circle><text x="65" y="58" font-size="14">O</text><line x1="75" y1="60" x2="140" y2="40" stroke="black" stroke-width="2"></line><line x1="75" y1="60" x2="40" y2="110" stroke="black" stroke-width="2"></line><line x1="75" y1="60" x2="110" y2="110" stroke="black" stroke-width="2"></line><text x="100" y="70" font-size="14" fill="#00008B">Часть 1</text><text x="50" y="70" font-size="14" fill="#8B0000">Часть 2</text><text x="70" y="30" font-size="14" fill="#006400">Часть 3</text></svg><br>Три луча с общей вершиной разбивают плоскость на 3 части, образуя три угла вокруг общей вершины.<br>Ответ: 3.</p><p><strong>в)</strong> Рассмотрим четыре луча с общей вершиной. По аналогии с предыдущими пунктами, первые три луча делят плоскость на 3 части. Четвертый луч, исходящий из той же вершины, разделит одну из существующих трех частей на две.<br>Следовательно, общее количество частей станет $3 - 1 + 2 = 4$.<br><svg width="150" height="150" viewBox="0 0 150 150" xmlns="http://www.w3.org/2000/svg"><title>Четыре луча делят плоскость на четыре части</title><circle cx="75" cy="75" r="2.5" fill="black"></circle><text x="65" y="73" font-size="14">O</text><line x1="75" y1="75" x2="145" y2="60" stroke="black" stroke-width="2"></line><line x1="75" y1="75" x2="5" y2="60" stroke="black" stroke-width="2"></line><line x1="75" y1="75" x2="60" y2="145" stroke="black" stroke-width="2"></line><line x1="75" y1="75" x2="90" y2="5" stroke="black" stroke-width="2"></line><text x="95" y="50" font-size="14" fill="#00008B">1</text><text x="45" y="50" font-size="14" fill="#8B0000">2</text><text x="45" y="105" font-size="14" fill="#006400">3</text><text x="95" y="105" font-size="14" fill="#4B0082">4</text></svg><br>Четыре луча с общей вершиной разбивают плоскость на 4 части.<br>Ответ: 4.</p><p><strong>г)</strong> Обобщая результаты предыдущих пунктов, можно заметить закономерность: количество частей, на которые лучи с общей вершиной делят плоскость, равно количеству лучей.<br>- 2 луча → 2 части<br>- 3 луча → 3 части<br>- 4 луча → 4 части<br>Этот результат можно доказать методом математической индукции.<br><i>База индукции:</i> для $n=2$ лучей утверждение верно, они делят плоскость на 2 части.<br><i>Шаг индукции:</i> предположим, что $k$ лучей делят плоскость на $k$ частей. Добавим $(k+1)$-й луч. Он пройдет через одну из $k$ существующих частей и разделит ее на две. Общее число частей станет равным $(k-1) + 2 = k+1$.<br>Таким образом, утверждение верно и для $k+1$.<br>Следовательно, при условии, что никакие два луча не совпадают и не являются продолжением друг друга (не образуют прямую), $n$ лучей с общей вершиной разбивают плоскость на $n$ частей.<br>Ответ: $n$.</p>" ] #original: array:6 [ "id" => 1541312 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1244} "task" => array:2 [ "refs" => "1129446" "type" => "task" ] "text" => "<p><strong>а)</strong> Два луча, исходящие из одной точки (общей вершины), образуют угол. Эти лучи делят всю плоскость на две части: внутреннюю область угла и внешнюю область угла.<br><svg width="150" height="100" viewBox="0 0 150 100" xmlns="http://www.w3.org/2000/svg"><title>Два луча делят плоскость на две части</title><circle cx="50" cy="50" r="2.5" fill="black"></circle><text x="40" y="48" font-size="14">O</text><line x1="50" y1="50" x2="140" y2="30" stroke="black" stroke-width="2"></line><line x1="50" y1="50" x2="140" y2="80" stroke="black" stroke-width="2"></line><text x="90" y="55" font-size="14" fill="#00008B">Часть 1</text><text x="20" y="75" font-size="14" fill="#8B0000">Часть 2</text></svg><br>Таким образом, два луча с общей вершиной разбивают плоскость на 2 части.<br>Ответ: 2.</p><p><strong>б)</strong> Рассмотрим три луча с общей вершиной. Первые два луча делят плоскость на две части. Третий луч, проведенный из той же вершины, пройдет через одну из этих частей и разделит ее на две новые. В результате общее число частей станет $2 - 1 + 2 = 3$.<br><svg width="150" height="120" viewBox="0 0 150 120" xmlns="http://www.w3.org/2000/svg"><title>Три луча делят плоскость на три части</title><circle cx="75" cy="60" r="2.5" fill="black"></circle><text x="65" y="58" font-size="14">O</text><line x1="75" y1="60" x2="140" y2="40" stroke="black" stroke-width="2"></line><line x1="75" y1="60" x2="40" y2="110" stroke="black" stroke-width="2"></line><line x1="75" y1="60" x2="110" y2="110" stroke="black" stroke-width="2"></line><text x="100" y="70" font-size="14" fill="#00008B">Часть 1</text><text x="50" y="70" font-size="14" fill="#8B0000">Часть 2</text><text x="70" y="30" font-size="14" fill="#006400">Часть 3</text></svg><br>Три луча с общей вершиной разбивают плоскость на 3 части, образуя три угла вокруг общей вершины.<br>Ответ: 3.</p><p><strong>в)</strong> Рассмотрим четыре луча с общей вершиной. По аналогии с предыдущими пунктами, первые три луча делят плоскость на 3 части. Четвертый луч, исходящий из той же вершины, разделит одну из существующих трех частей на две.<br>Следовательно, общее количество частей станет $3 - 1 + 2 = 4$.<br><svg width="150" height="150" viewBox="0 0 150 150" xmlns="http://www.w3.org/2000/svg"><title>Четыре луча делят плоскость на четыре части</title><circle cx="75" cy="75" r="2.5" fill="black"></circle><text x="65" y="73" font-size="14">O</text><line x1="75" y1="75" x2="145" y2="60" stroke="black" stroke-width="2"></line><line x1="75" y1="75" x2="5" y2="60" stroke="black" stroke-width="2"></line><line x1="75" y1="75" x2="60" y2="145" stroke="black" stroke-width="2"></line><line x1="75" y1="75" x2="90" y2="5" stroke="black" stroke-width="2"></line><text x="95" y="50" font-size="14" fill="#00008B">1</text><text x="45" y="50" font-size="14" fill="#8B0000">2</text><text x="45" y="105" font-size="14" fill="#006400">3</text><text x="95" y="105" font-size="14" fill="#4B0082">4</text></svg><br>Четыре луча с общей вершиной разбивают плоскость на 4 части.<br>Ответ: 4.</p><p><strong>г)</strong> Обобщая результаты предыдущих пунктов, можно заметить закономерность: количество частей, на которые лучи с общей вершиной делят плоскость, равно количеству лучей.<br>- 2 луча → 2 части<br>- 3 луча → 3 части<br>- 4 луча → 4 части<br>Этот результат можно доказать методом математической индукции.<br><i>База индукции:</i> для $n=2$ лучей утверждение верно, они делят плоскость на 2 части.<br><i>Шаг индукции:</i> предположим, что $k$ лучей делят плоскость на $k$ частей. Добавим $(k+1)$-й луч. Он пройдет через одну из $k$ существующих частей и разделит ее на две. Общее число частей станет равным $(k-1) + 2 = k+1$.<br>Таким образом, утверждение верно и для $k+1$.<br>Следовательно, при условии, что никакие два луча не совпадают и не являются продолжением друг друга (не образуют прямую), $n$ лучей с общей вершиной разбивают плоскость на $n$ частей.<br>Ответ: $n$.</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" => "1129447" "type" => "task" ] "previous" => array:2 [ "refs" => "1129445" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1251 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1114336" "type" => "book_page" ] ] #original: array:24 [ "id" => 1129446 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "19" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/03-22" "field_display_title" => "3.22" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1223} "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 {#1222} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1221} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1258} "content" => Illuminate\Database\Eloquent\Collection {#1255} "next" => array:2 [ "refs" => "1129447" "type" => "task" ] "previous" => array:2 [ "refs" => "1129445" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1251} "page" => array:2 [ "refs" => "1114336" "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 => "*" ] } 1 => App\Models\Task {#1224 #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" => 1129447 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "19" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/samost-4" "field_display_title" => "Задания" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1263 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1252 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1250 #items: array:1 [ 0 => App\Models\Branch {#1266} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1254 #items: array:2 [ 0 => App\Models\Branch {#1266} 1 => App\Models\Branch {#1118} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1227 #items: array:3 [ 0 => App\Models\Element {#1096} 1 => App\Models\Element {#1094} 2 => App\Models\Element {#1095} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1129448" "type" => "task" ] "previous" => array:2 [ "refs" => "1129446" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1253 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1114336" "type" => "book_page" ] ] #original: array:24 [ "id" => 1129447 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "19" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/samost-4" "field_display_title" => "Задания" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1263} "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 {#1252} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1250} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1254} "content" => Illuminate\Database\Eloquent\Collection {#1227} "next" => array:2 [ "refs" => "1129448" "type" => "task" ] "previous" => array:2 [ "refs" => "1129446" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1253} "page" => array:2 [ "refs" => "1114336" "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 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 1114336 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "19" "field_url" => "/7-klass/geometrija/smirnova-uchebnik/page-19" "field_display_title" => "19" "field_folder" => "1" "field_image_name" => "19" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1129} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1132} "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 {#1130} "content" => Illuminate\Database\Eloquent\Collection {#1143} "next" => array:2 [ "refs" => "1114337" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1114335" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1211} ] #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" => 1129447 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "19" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/samost-4" "field_display_title" => "Задания" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1036} "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 {#1034} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1045} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1106} "content" => Illuminate\Database\Eloquent\Collection {#1103} "next" => array:2 [ "refs" => "1129448" "type" => "task" ] "previous" => array:2 [ "refs" => "1129446" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1099} "page" => array:2 [ "refs" => "1114336" "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 => "*" ] }
Задания (с. 19)
Условие. Задания (с. 19)
скриншот условия
Как вы думаете, на сколько частей эта прямая разбивает плоскость?
Попробуйте объяснить это самостоятельно.
Решение. Задания (с. 19)
Решение 2. Задания (с. 19)
Как вы думаете, на сколько частей эта прямая разбивает плоскость?
Любая прямая, проведенная на плоскости, разбивает эту плоскость ровно на две части. Эти части в геометрии носят название полуплоскости.
Ответ: на 2 части.
Попробуйте объяснить это самостоятельно.
Представим себе плоскость как бесконечный, идеально ровный лист бумаги. Прямая линия, проведенная на этой плоскости, также является бесконечной и простирается в обе стороны.
Эта прямая линия действует как непреодолимая граница. Все точки плоскости, которые не лежат на самой прямой, оказываются разделены этой границей на две отдельные области. Это можно охарактеризовать так:
1. Если взять любые две точки, принадлежащие одной и той же области, то их можно соединить отрезком, который не пересечет исходную прямую.
2. Если же взять одну точку из первой области, а другую — из второй, то любой отрезок, соединяющий эти две точки, обязательно пересечет прямую-границу.
Таким образом, прямая делит всю плоскость на две непересекающиеся части, которые называются полуплоскостями. Сама прямая является их общей границей.
Это можно наглядно проиллюстрировать:
На рисунке плоскость условно показана прямоугольником. Красная линия (прямая) разделяет её на две области, закрашенные разными цветами — это и есть две полуплоскости.
Ответ: Прямая, являясь бесконечной границей, разделяет всю плоскость на две отдельные области, называемые полуплоскостями.
Другие задания:
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