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Вопросы, страница 109 - 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 4. Окружность. Геометрические построения. Параграф 19. Взаимное расположение прямой и окружности - страница 109.
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" => 1129820 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "109" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/vopr-19" "field_display_title" => "Вопросы" "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" => 1129379 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Окружность. Геометрические построения" "field_branch_order" => "4" "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" => "101" "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 {#1069 #items: array:1 [ 0 => App\Models\Term {#1068 #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 {#1059 #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 {#1062 #items: array:2 [ 0 => 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" => 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 {#1076 #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 {#1072 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1070 #items: array:1 [ 0 => App\Models\Term {#1063 #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 {#1071 #items: array:1 [ 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: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 {#1077 #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: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 {#1064 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1060 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1078 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1079 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1080 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1081 #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 {#1082 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1083 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1084 #items: [] #escapeWhenCastingToString: false } "field_url" => "/7-klass/geometrija/smirnova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1085 #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 {#1069} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1065} "field_author" => Illuminate\Database\Eloquent\Collection {#1062} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1072} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1070} "field_country" => Illuminate\Database\Eloquent\Collection {#1071} "field_city" => Illuminate\Database\Eloquent\Collection {#1077} "field_series" => Illuminate\Database\Eloquent\Collection {#1064} "field_umk" => Illuminate\Database\Eloquent\Collection {#1060} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1078} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1079} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1080} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1081} "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 {#1082} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1083} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1084} "field_url" => "/7-klass/geometrija/smirnova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1085} "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" => 1129379 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Окружность. Геометрические построения" "field_branch_order" => "4" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1049} "field_page_start" => "101" "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 {#1107 #items: array:2 [ 0 => App\Models\Branch {#1115 #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" => 1129379 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Окружность. Геометрические построения" "field_branch_order" => "4" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1116 #items: array:1 [ 0 => App\Models\Term {#1046} ] #escapeWhenCastingToString: false } "field_page_start" => "101" "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 {#1117 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1118 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1129379 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Окружность. Геометрические построения" "field_branch_order" => "4" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1116} "field_page_start" => "101" "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 {#1117} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1118} ] #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 {#1119 #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" => 1129381 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Взаимное расположение прямой и окружности" "field_branch_order" => "19" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1123 #items: array:1 [ 0 => App\Models\Term {#1120 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "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" => "107" "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 {#1122 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1121 #items: array:1 [ 0 => App\Models\Branch {#1044} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1129381 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Взаимное расположение прямой и окружности" "field_branch_order" => "19" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1123} "field_page_start" => "107" "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 {#1122} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1121} ] #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 {#1104 #items: array:3 [ 0 => App\Models\Element {#1097 #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" => 1320175 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1089 #items: array:1 [ 0 => App\Models\Edition {#1096 #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 {#1094 #items: array:1 [ 0 => App\Models\Term {#1095 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] …9 } ] #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 {#1092 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1093 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1090 #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 {#1094} "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 {#1092} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1093} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1090} "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" => "1129820" "type" => "task" ] "text" => "<p><strong>1.</strong> Как могут быть расположены относительно друг друга прямая и окружность?</p><p><strong>2.</strong> Какая прямая называется: а) касательной к окружности; б) пересекающей окружность?</p><p><strong>3.</strong> В каком случае прямая и окружность не имеют общих точек?</p><p><strong>4.</strong> В каком случае прямая касается окружности?</p><p><strong>5.</strong> В каком случае прямая и окружность пересекаются?</p>" "img" => array:1 [ 0 => array:5 [ "name" => "19-1.jpg" "alt" => null "width" => "1765" "height" => 472 "path" => "/media/geometrija_07/Smirnova-u17/0-vopr/19-1.webp?ts=1753982528" ] ] ] #original: array:7 [ "id" => 1320175 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1089} "task" => array:2 [ "refs" => "1129820" "type" => "task" ] "text" => "<p><strong>1.</strong> Как могут быть расположены относительно друг друга прямая и окружность?</p><p><strong>2.</strong> Какая прямая называется: а) касательной к окружности; б) пересекающей окружность?</p><p><strong>3.</strong> В каком случае прямая и окружность не имеют общих точек?</p><p><strong>4.</strong> В каком случае прямая касается окружности?</p><p><strong>5.</strong> В каком случае прямая и окружность пересекаются?</p>" "img" => array:1 [ 0 => array:5 [ "name" => "19-1.jpg" "alt" => null "width" => "1765" "height" => 472 "path" => "/media/geometrija_07/Smirnova-u17/0-vopr/19-1.webp?ts=1753982528" ] ] ] #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 {#1091 #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" => 1320519 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1048 #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" => 5010 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1054 #items: array:1 [ 0 => App\Models\Term {#1087 …30} ] #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 {#1086 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1052 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1053 #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 {#1054} "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 {#1086} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1052} "field_process_formula" => "" "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" => "1129820" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "19-1.jpg" "alt" => null "width" => "1430" "height" => 934 "path" => "/media/geometrija_07/Smirnova-u17/1-vopr/19-1.webp?ts=1753982643" ] 1 => array:5 [ "name" => "19-2.jpg" "alt" => null "width" => "1472" "height" => 1568 "path" => "/media/geometrija_07/Smirnova-u17/1-vopr/19-2.webp?ts=1753982643" ] ] ] #original: array:6 [ "id" => 1320519 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1048} "task" => array:2 [ "refs" => "1129820" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "19-1.jpg" "alt" => null "width" => "1430" "height" => 934 "path" => "/media/geometrija_07/Smirnova-u17/1-vopr/19-1.webp?ts=1753982643" ] 1 => array:5 [ "name" => "19-2.jpg" "alt" => null "width" => "1472" "height" => 1568 "path" => "/media/geometrija_07/Smirnova-u17/1-vopr/19-2.webp?ts=1753982643" ] ] ] #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 {#1051 #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" => 1541781 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1110 #items: array:1 [ 0 => App\Models\Edition {#1050 #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 {#1040 #items: array:1 [ 0 => App\Models\Term {#1042 …30} ] #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 {#1035 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1038 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1106 #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 {#1040} "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 {#1035} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1038} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1106} "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" => "1129820" "type" => "task" ] "text" => "<p><strong>1. Как могут быть расположены относительно друг друга прямая и окружность?</strong><br>Взаимное расположение прямой и окружности на плоскости определяется количеством их общих точек. Это количество зависит от соотношения между радиусом окружности ($r$) и расстоянием от центра окружности до прямой ($d$).<br>Существует три возможных случая:<br>1. <strong>Прямая и окружность не имеют общих точек.</strong> Это происходит, когда расстояние от центра окружности до прямой больше радиуса ($d > r$).<br>2. <strong>Прямая и окружность имеют одну общую точку.</strong> В этом случае прямая называется <i>касательной</i> к окружности, а их общая точка — точкой касания. Это происходит, когда расстояние от центра до прямой равно радиусу ($d = r$).<br>3. <strong>Прямая и окружность имеют две общие точки.</strong> В этом случае прямая называется <i>секущей</i> (или пересекающей) окружность. Это происходит, когда расстояние от центра до прямой меньше радиуса ($d < r$).<br><svg width="450" height="180" viewBox="0 0 450 180" xmlns="http://www.w3.org/2000/svg" style="font-family: sans-serif; font-size: 14px; text-anchor: middle;"><g id="case1"><circle cx="75" cy="100" r="40" stroke="black" stroke-width="2" fill="#f0f0f0"></circle><line x1="15" y1="155" x2="135" y2="155" stroke="#007bff" stroke-width="2"></line><text x="75" y="30">Нет общих точек</text><text x="75" y="50">($d > r$)</text></g><g id="case2"><circle cx="225" cy="100" r="40" stroke="black" stroke-width="2" fill="#f0f0f0"></circle><line x1="165" y1="140" x2="285" y2="140" stroke="#dc3545" stroke-width="2"></line><circle cx="225" cy="140" r="2.5" fill="#dc3545"></circle><text x="225" y="30">Одна общая точка</text><text x="225" y="50">(касательная, $d = r$)</text></g><g id="case3"><circle cx="375" cy="100" r="40" stroke="black" stroke-width="2" fill="#f0f0f0"></circle><line x1="315" y1="120" x2="435" y2="120" stroke="#28a745" stroke-width="2"></line><circle cx="349" cy="120" r="2.5" fill="#28a745"></circle><circle cx="401" cy="120" r="2.5" fill="#28a745"></circle><text x="375" y="30">Две общие точки</text><text x="375" y="50">(секущая, $d < r$)</text></g></svg><br><strong>Ответ:</strong> Прямая и окружность могут не иметь общих точек, иметь одну общую точку (касаться) или иметь две общие точки (пересекаться).</p><p><strong>2. Какая прямая называется: а) касательной к окружности; б) пересекающей окружность?</strong><br><strong>а) касательной к окружности</strong><br>Касательной к окружности называется прямая, которая имеет с окружностью ровно одну общую точку. Эта точка называется точкой касания. Свойство касательной: она перпендикулярна радиусу, проведенному в точку касания.<br><strong>б) пересекающей окружность</strong><br>Пересекающей окружность (или секущей) называется прямая, которая имеет с окружностью две различные общие точки. Отрезок секущей, лежащий внутри окружности, называется хордой.<br><strong>Ответ:</strong> а) Касательная – прямая, имеющая с окружностью одну общую точку. б) Пересекающая (секущая) – прямая, имеющая с окружностью две общие точки.</p><p><strong>3. В каком случае прямая и окружность не имеют общих точек?</strong><br>Прямая и окружность не имеют общих точек, если расстояние от центра окружности до этой прямой больше, чем радиус окружности. Если обозначить радиус окружности как $r$, а расстояние от ее центра (точка O) до прямой как $d$, то условие отсутствия общих точек записывается как $d > r$.<br><svg width="200" height="150" viewBox="0 0 200 150" xmlns="http://www.w3.org/2000/svg" style="font-family: sans-serif; font-size: 14px;"><circle cx="100" cy="70" r="40" stroke="black" stroke-width="2" fill="#f0f0f0"></circle><line x1="20" y1="130" x2="180" y2="130" stroke="#007bff" stroke-width="2"></line><line x1="100" y1="70" x2="100" y2="130" stroke="#dc3545" stroke-width="1.5" stroke-dasharray="4 2"></line><text x="105" y="65">O</text><circle cx="100" cy="70" r="2" fill="black"></circle><text x="80" y="105" fill="#dc3545">d</text><line x1="100" y1="70" x2="140" y2="70" stroke="#28a745" stroke-width="1.5"></line><text x="115" y="65" fill="#28a745">r</text><text x="70" y="20" style="font-size: 16px;">$d > r$</text></svg><br><strong>Ответ:</strong> Прямая и окружность не имеют общих точек, если расстояние от центра окружности до прямой больше ее радиуса ($d > r$).</p><p><strong>4. В каком случае прямая касается окружности?</strong><br>Прямая касается окружности, если она имеет с ней ровно одну общую точку. Это происходит в том случае, когда расстояние от центра окружности до прямой в точности равно радиусу окружности. Если $r$ – радиус, а $d$ – расстояние от центра (точка O) до прямой, то условие касания: $d = r$.<br><svg width="200" height="150" viewBox="0 0 200 150" xmlns="http://www.w3.org/2000/svg" style="font-family: sans-serif; font-size: 14px;"><circle cx="100" cy="70" r="40" stroke="black" stroke-width="2" fill="#f0f0f0"></circle><line x1="20" y1="110" x2="180" y2="110" stroke="#dc3545" stroke-width="2"></line><line x1="100" y1="70" x2="100" y2="110" stroke="#28a745" stroke-width="1.5"></line><text x="105" y="65">O</text><circle cx="100" cy="70" r="2" fill="black"></circle><circle cx="100" cy="110" r="2.5" fill="#dc3545"></circle><text x="80" y="95" fill="#28a745">d=r</text><text x="70" y="20" style="font-size: 16px;">$d = r$</text></svg><br><strong>Ответ:</strong> Прямая касается окружности, если расстояние от центра окружности до прямой равно ее радиусу ($d = r$).</p><p><strong>5. В каком случае прямая и окружность пересекаются?</strong><br>Прямая и окружность пересекаются, если они имеют две общие точки. Это возможно, когда расстояние от центра окружности до прямой меньше радиуса окружности. Математически это условие записывается как $d < r$, где $r$ – радиус, а $d$ – расстояние от центра (точка O) до прямой.<br><svg width="200" height="150" viewBox="0 0 200 150" xmlns="http://www.w3.org/2000/svg" style="font-family: sans-serif; font-size: 14px;"><circle cx="100" cy="70" r="40" stroke="black" stroke-width="2" fill="#f0f0f0"></circle><line x1="20" y1="90" x2="180" y2="90" stroke="#28a745" stroke-width="2"></line><line x1="100" y1="70" x2="100" y2="90" stroke="#dc3545" stroke-width="1.5" stroke-dasharray="4 2"></line><text x="105" y="65">O</text><circle cx="100" cy="70" r="2" fill="black"></circle><text x="105" y="85" fill="#dc3545">d</text><line x1="100" y1="70" x2="134.64" y2="90" stroke="#28a745" stroke-width="1.5"></line><text x="110" y="75" fill="#28a745">r</text><circle cx="65.36" cy="90" r="2.5" fill="#28a745"></circle><circle cx="134.64" cy="90" r="2.5" fill="#28a745"></circle><text x="70" y="20" style="font-size: 16px;">$d < r$</text></svg><br><strong>Ответ:</strong> Прямая и окружность пересекаются, если расстояние от центра окружности до прямой меньше ее радиуса ($d < r$).</p>" ] #original: array:6 [ "id" => 1541781 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1110} "task" => array:2 [ "refs" => "1129820" "type" => "task" ] "text" => "<p><strong>1. Как могут быть расположены относительно друг друга прямая и окружность?</strong><br>Взаимное расположение прямой и окружности на плоскости определяется количеством их общих точек. Это количество зависит от соотношения между радиусом окружности ($r$) и расстоянием от центра окружности до прямой ($d$).<br>Существует три возможных случая:<br>1. <strong>Прямая и окружность не имеют общих точек.</strong> Это происходит, когда расстояние от центра окружности до прямой больше радиуса ($d > r$).<br>2. <strong>Прямая и окружность имеют одну общую точку.</strong> В этом случае прямая называется <i>касательной</i> к окружности, а их общая точка — точкой касания. Это происходит, когда расстояние от центра до прямой равно радиусу ($d = r$).<br>3. <strong>Прямая и окружность имеют две общие точки.</strong> В этом случае прямая называется <i>секущей</i> (или пересекающей) окружность. Это происходит, когда расстояние от центра до прямой меньше радиуса ($d < r$).<br><svg width="450" height="180" viewBox="0 0 450 180" xmlns="http://www.w3.org/2000/svg" style="font-family: sans-serif; font-size: 14px; text-anchor: middle;"><g id="case1"><circle cx="75" cy="100" r="40" stroke="black" stroke-width="2" fill="#f0f0f0"></circle><line x1="15" y1="155" x2="135" y2="155" stroke="#007bff" stroke-width="2"></line><text x="75" y="30">Нет общих точек</text><text x="75" y="50">($d > r$)</text></g><g id="case2"><circle cx="225" cy="100" r="40" stroke="black" stroke-width="2" fill="#f0f0f0"></circle><line x1="165" y1="140" x2="285" y2="140" stroke="#dc3545" stroke-width="2"></line><circle cx="225" cy="140" r="2.5" fill="#dc3545"></circle><text x="225" y="30">Одна общая точка</text><text x="225" y="50">(касательная, $d = r$)</text></g><g id="case3"><circle cx="375" cy="100" r="40" stroke="black" stroke-width="2" fill="#f0f0f0"></circle><line x1="315" y1="120" x2="435" y2="120" stroke="#28a745" stroke-width="2"></line><circle cx="349" cy="120" r="2.5" fill="#28a745"></circle><circle cx="401" cy="120" r="2.5" fill="#28a745"></circle><text x="375" y="30">Две общие точки</text><text x="375" y="50">(секущая, $d < r$)</text></g></svg><br><strong>Ответ:</strong> Прямая и окружность могут не иметь общих точек, иметь одну общую точку (касаться) или иметь две общие точки (пересекаться).</p><p><strong>2. Какая прямая называется: а) касательной к окружности; б) пересекающей окружность?</strong><br><strong>а) касательной к окружности</strong><br>Касательной к окружности называется прямая, которая имеет с окружностью ровно одну общую точку. Эта точка называется точкой касания. Свойство касательной: она перпендикулярна радиусу, проведенному в точку касания.<br><strong>б) пересекающей окружность</strong><br>Пересекающей окружность (или секущей) называется прямая, которая имеет с окружностью две различные общие точки. Отрезок секущей, лежащий внутри окружности, называется хордой.<br><strong>Ответ:</strong> а) Касательная – прямая, имеющая с окружностью одну общую точку. б) Пересекающая (секущая) – прямая, имеющая с окружностью две общие точки.</p><p><strong>3. В каком случае прямая и окружность не имеют общих точек?</strong><br>Прямая и окружность не имеют общих точек, если расстояние от центра окружности до этой прямой больше, чем радиус окружности. Если обозначить радиус окружности как $r$, а расстояние от ее центра (точка O) до прямой как $d$, то условие отсутствия общих точек записывается как $d > r$.<br><svg width="200" height="150" viewBox="0 0 200 150" xmlns="http://www.w3.org/2000/svg" style="font-family: sans-serif; font-size: 14px;"><circle cx="100" cy="70" r="40" stroke="black" stroke-width="2" fill="#f0f0f0"></circle><line x1="20" y1="130" x2="180" y2="130" stroke="#007bff" stroke-width="2"></line><line x1="100" y1="70" x2="100" y2="130" stroke="#dc3545" stroke-width="1.5" stroke-dasharray="4 2"></line><text x="105" y="65">O</text><circle cx="100" cy="70" r="2" fill="black"></circle><text x="80" y="105" fill="#dc3545">d</text><line x1="100" y1="70" x2="140" y2="70" stroke="#28a745" stroke-width="1.5"></line><text x="115" y="65" fill="#28a745">r</text><text x="70" y="20" style="font-size: 16px;">$d > r$</text></svg><br><strong>Ответ:</strong> Прямая и окружность не имеют общих точек, если расстояние от центра окружности до прямой больше ее радиуса ($d > r$).</p><p><strong>4. В каком случае прямая касается окружности?</strong><br>Прямая касается окружности, если она имеет с ней ровно одну общую точку. Это происходит в том случае, когда расстояние от центра окружности до прямой в точности равно радиусу окружности. Если $r$ – радиус, а $d$ – расстояние от центра (точка O) до прямой, то условие касания: $d = r$.<br><svg width="200" height="150" viewBox="0 0 200 150" xmlns="http://www.w3.org/2000/svg" style="font-family: sans-serif; font-size: 14px;"><circle cx="100" cy="70" r="40" stroke="black" stroke-width="2" fill="#f0f0f0"></circle><line x1="20" y1="110" x2="180" y2="110" stroke="#dc3545" stroke-width="2"></line><line x1="100" y1="70" x2="100" y2="110" stroke="#28a745" stroke-width="1.5"></line><text x="105" y="65">O</text><circle cx="100" cy="70" r="2" fill="black"></circle><circle cx="100" cy="110" r="2.5" fill="#dc3545"></circle><text x="80" y="95" fill="#28a745">d=r</text><text x="70" y="20" style="font-size: 16px;">$d = r$</text></svg><br><strong>Ответ:</strong> Прямая касается окружности, если расстояние от центра окружности до прямой равно ее радиусу ($d = r$).</p><p><strong>5. В каком случае прямая и окружность пересекаются?</strong><br>Прямая и окружность пересекаются, если они имеют две общие точки. Это возможно, когда расстояние от центра окружности до прямой меньше радиуса окружности. Математически это условие записывается как $d < r$, где $r$ – радиус, а $d$ – расстояние от центра (точка O) до прямой.<br><svg width="200" height="150" viewBox="0 0 200 150" xmlns="http://www.w3.org/2000/svg" style="font-family: sans-serif; font-size: 14px;"><circle cx="100" cy="70" r="40" stroke="black" stroke-width="2" fill="#f0f0f0"></circle><line x1="20" y1="90" x2="180" y2="90" stroke="#28a745" stroke-width="2"></line><line x1="100" y1="70" x2="100" y2="90" stroke="#dc3545" stroke-width="1.5" stroke-dasharray="4 2"></line><text x="105" y="65">O</text><circle cx="100" cy="70" r="2" fill="black"></circle><text x="105" y="85" fill="#dc3545">d</text><line x1="100" y1="70" x2="134.64" y2="90" stroke="#28a745" stroke-width="1.5"></line><text x="110" y="75" fill="#28a745">r</text><circle cx="65.36" cy="90" r="2.5" fill="#28a745"></circle><circle cx="134.64" cy="90" r="2.5" fill="#28a745"></circle><text x="70" y="20" style="font-size: 16px;">$d < r$</text></svg><br><strong>Ответ:</strong> Прямая и окружность пересекаются, если расстояние от центра окружности до прямой меньше ее радиуса ($d < r$).</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" => "1129821" "type" => "task" ] "previous" => array:2 [ "refs" => "1129819" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1100 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1126 #items: array:1 [ 0 => App\Models\BookPage {#1125 #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" => 1114426 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "109" "field_url" => "/7-klass/geometrija/smirnova-uchebnik/page-109" "field_display_title" => "109" "field_folder" => "1" "field_image_name" => "109" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1127 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1128 #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 {#1129 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1139 #items: array:1 [ 0 => App\Models\Element {#1138 #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" => 1290613 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1140 #items: array:1 [ 0 => App\Models\Edition {#1096} ] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1114426" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "109-1.jpg" "alt" => null "width" => "1935" "height" => 2838 "path" => "/media/geometrija_07/Smirnova-u17/0-1/109-1.webp?ts=1753260190" ] ] ] #original: array:6 [ "id" => 1290613 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1140} "book_page" => array:2 [ "refs" => "1114426" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "109-1.jpg" "alt" => null "width" => "1935" "height" => 2838 "path" => "/media/geometrija_07/Smirnova-u17/0-1/109-1.webp?ts=1753260190" ] ] ] #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" => "1114427" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1114425" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1207 #items: array:7 [ 0 => App\Models\Task {#1221 #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" => 1129819 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "107" "field_page_end" => "109" "field_url" => "/7-klass/geometrija/smirnova-uchebnik/samost-19" "field_display_title" => "Задания" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1222 #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 {#1223 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1224 #items: array:1 [ 0 => App\Models\Branch {#1115} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1225 #items: array:2 [ 0 => App\Models\Branch {#1115} 1 => App\Models\Branch {#1119} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1228 #items: array:3 [ 0 => App\Models\Element {#1237 …30} 1 => App\Models\Element {#1239 …30} 2 => App\Models\Element {#1241 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1129820" "type" => "task" ] "previous" => array:2 [ "refs" => "1129818" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1234 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1114424" "type" => "book_page" ] ] #original: array:24 [ "id" => 1129819 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "107" "field_page_end" => "109" "field_url" => "/7-klass/geometrija/smirnova-uchebnik/samost-19" "field_display_title" => "Задания" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1222} "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 {#1223} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1224} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1225} "content" => Illuminate\Database\Eloquent\Collection {#1228} "next" => array:2 [ "refs" => "1129820" "type" => "task" ] "previous" => array:2 [ "refs" => "1129818" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1234} "page" => array:2 [ "refs" => "1114424" "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 {#1227 #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" => 1129820 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "109" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/vopr-19" "field_display_title" => "Вопросы" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1226 #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 {#1233 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1235 #items: array:1 [ 0 => App\Models\Branch {#1115} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1231 #items: array:2 [ 0 => App\Models\Branch {#1115} 1 => App\Models\Branch {#1119} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1229 #items: array:3 [ 0 => App\Models\Element {#1097} 1 => App\Models\Element {#1091} 2 => App\Models\Element {#1051} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1129821" "type" => "task" ] "previous" => array:2 [ "refs" => "1129819" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1232 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1114426" "type" => "book_page" ] ] #original: array:24 [ "id" => 1129820 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "109" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/vopr-19" "field_display_title" => "Вопросы" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1226} "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 {#1233} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1235} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1231} "content" => Illuminate\Database\Eloquent\Collection {#1229} "next" => array:2 [ "refs" => "1129821" "type" => "task" ] "previous" => array:2 [ "refs" => "1129819" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1232} "page" => array:2 [ "refs" => "1114426" "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 => "*" ] } 2 => App\Models\Task {#1230 #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" => 1129821 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "109" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/19-1" "field_display_title" => "19.1" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1246 #items: array:1 [ 0 => App\Models\Term {#1243 …30} ] #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 {#1245 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1244 #items: array:1 [ 0 => App\Models\Branch {#1115} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1247 #items: array:2 [ 0 => App\Models\Branch {#1115} 1 => App\Models\Branch {#1119} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1250 #items: array:3 [ 0 => App\Models\Element {#1259 …30} 1 => App\Models\Element {#1261 …30} 2 => App\Models\Element {#1263 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1129822" "type" => "task" ] "previous" => array:2 [ "refs" => "1129820" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1256 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1114426" "type" => "book_page" ] ] #original: array:24 [ "id" => 1129821 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "109" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/19-1" "field_display_title" => "19.1" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1246} "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 {#1245} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1244} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1247} "content" => Illuminate\Database\Eloquent\Collection {#1250} "next" => array:2 [ "refs" => "1129822" "type" => "task" ] "previous" => array:2 [ "refs" => "1129820" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1256} "page" => array:2 [ "refs" => "1114426" "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 => "*" ] } 3 => App\Models\Task {#1249 #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" => 1129822 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "109" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/19-2" "field_display_title" => "19.2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1248 #items: array:1 [ 0 => App\Models\Term {#1243 …30} ] #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 {#1255 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1257 #items: array:1 [ 0 => App\Models\Branch {#1115} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1253 #items: array:2 [ 0 => App\Models\Branch {#1115} 1 => App\Models\Branch {#1119} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1252 #items: array:3 [ 0 => App\Models\Element {#1273 …30} 1 => App\Models\Element {#1275 …30} 2 => App\Models\Element {#1277 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1129823" "type" => "task" ] "previous" => array:2 [ "refs" => "1129821" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1270 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1114426" "type" => "book_page" ] ] #original: array:24 [ "id" => 1129822 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "109" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/19-2" "field_display_title" => "19.2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1248} "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 {#1255} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1257} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1253} "content" => Illuminate\Database\Eloquent\Collection {#1252} "next" => array:2 [ "refs" => "1129823" "type" => "task" ] "previous" => array:2 [ "refs" => "1129821" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1270} "page" => array:2 [ "refs" => "1114426" "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 => App\Models\Task {#1254 #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" => 1129823 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "109" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/19-3" "field_display_title" => "19.3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1251 #items: array:1 [ 0 => App\Models\Term {#1243 …30} ] #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 {#1269 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1271 #items: array:1 [ 0 => App\Models\Branch {#1115} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1267 #items: array:2 [ 0 => App\Models\Branch {#1115} 1 => App\Models\Branch {#1119} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1266 #items: array:3 [ 0 => App\Models\Element {#1287 …30} 1 => App\Models\Element {#1289 …30} 2 => App\Models\Element {#1291 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1129824" "type" => "task" ] "previous" => array:2 [ "refs" => "1129822" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1284 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1114426" "type" => "book_page" ] ] #original: array:24 [ "id" => 1129823 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "109" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/19-3" "field_display_title" => "19.3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1251} "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 {#1269} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1271} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1267} "content" => Illuminate\Database\Eloquent\Collection {#1266} "next" => array:2 [ "refs" => "1129824" "type" => "task" ] "previous" => array:2 [ "refs" => "1129822" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1284} "page" => array:2 [ "refs" => "1114426" "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 => "*" ] } 5 => App\Models\Task {#1268 #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" => 1129824 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "109" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/19-4" "field_display_title" => "19.4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1265 #items: array:1 [ 0 => App\Models\Term {#1243 …30} ] #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 {#1283 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1285 #items: array:1 [ 0 => App\Models\Branch {#1115} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1281 #items: array:2 [ 0 => App\Models\Branch {#1115} 1 => App\Models\Branch {#1119} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1280 #items: array:3 [ 0 => App\Models\Element {#1301 …30} 1 => App\Models\Element {#1303 …30} 2 => App\Models\Element {#1305 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1129825" "type" => "task" ] "previous" => array:2 [ "refs" => "1129823" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1298 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1114426" "type" => "book_page" ] ] #original: array:24 [ "id" => 1129824 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "109" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/19-4" "field_display_title" => "19.4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1265} "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 {#1283} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1285} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1281} "content" => Illuminate\Database\Eloquent\Collection {#1280} "next" => array:2 [ "refs" => "1129825" "type" => "task" ] "previous" => array:2 [ "refs" => "1129823" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1298} "page" => array:2 [ "refs" => "1114426" "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 => "*" ] } 6 => App\Models\Task {#1282 #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" => 1129825 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "109" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/19-5" "field_display_title" => "19.5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1279 #items: array:1 [ 0 => App\Models\Term {#1243 …30} ] #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 {#1297 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1299 #items: array:1 [ 0 => App\Models\Branch {#1115} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1295 #items: array:2 [ 0 => App\Models\Branch {#1115} 1 => App\Models\Branch {#1119} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1294 #items: array:3 [ 0 => App\Models\Element {#1315 …30} 1 => App\Models\Element {#1317 …30} 2 => App\Models\Element {#1319 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1129826" "type" => "task" ] "previous" => array:2 [ "refs" => "1129824" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1312 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1114426" "type" => "book_page" ] ] #original: array:24 [ "id" => 1129825 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "109" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/19-5" "field_display_title" => "19.5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1279} "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 {#1297} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1299} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1295} "content" => Illuminate\Database\Eloquent\Collection {#1294} "next" => array:2 [ "refs" => "1129826" "type" => "task" ] "previous" => array:2 [ "refs" => "1129824" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1312} "page" => array:2 [ "refs" => "1114426" "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" => 1114426 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "109" "field_url" => "/7-klass/geometrija/smirnova-uchebnik/page-109" "field_display_title" => "109" "field_folder" => "1" "field_image_name" => "109" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1127} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1128} "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 {#1129} "content" => Illuminate\Database\Eloquent\Collection {#1139} "next" => array:2 [ "refs" => "1114427" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1114425" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1207} ] #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" => 1129820 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "109" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/vopr-19" "field_display_title" => "Вопросы" "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 {#1107} "content" => Illuminate\Database\Eloquent\Collection {#1104} "next" => array:2 [ "refs" => "1129821" "type" => "task" ] "previous" => array:2 [ "refs" => "1129819" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1100} "page" => array:2 [ "refs" => "1114426" "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 => "*" ] }
Вопросы (с. 109)
Условие. Вопросы (с. 109)
скриншот условия
1. Как могут быть расположены относительно друг друга прямая и окружность?
2. Какая прямая называется: а) касательной к окружности; б) пересекающей окружность?
3. В каком случае прямая и окружность не имеют общих точек?
4. В каком случае прямая касается окружности?
5. В каком случае прямая и окружность пересекаются?
Решение. Вопросы (с. 109)
Решение 2. Вопросы (с. 109)
1. Как могут быть расположены относительно друг друга прямая и окружность?
Взаимное расположение прямой и окружности на плоскости определяется количеством их общих точек. Это количество зависит от соотношения между радиусом окружности ($r$) и расстоянием от центра окружности до прямой ($d$).
Существует три возможных случая:
1. Прямая и окружность не имеют общих точек. Это происходит, когда расстояние от центра окружности до прямой больше радиуса ($d > r$).
2. Прямая и окружность имеют одну общую точку. В этом случае прямая называется касательной к окружности, а их общая точка — точкой касания. Это происходит, когда расстояние от центра до прямой равно радиусу ($d = r$).
3. Прямая и окружность имеют две общие точки. В этом случае прямая называется секущей (или пересекающей) окружность. Это происходит, когда расстояние от центра до прямой меньше радиуса ($d < r$).
Ответ: Прямая и окружность могут не иметь общих точек, иметь одну общую точку (касаться) или иметь две общие точки (пересекаться).
2. Какая прямая называется: а) касательной к окружности; б) пересекающей окружность?
а) касательной к окружности
Касательной к окружности называется прямая, которая имеет с окружностью ровно одну общую точку. Эта точка называется точкой касания. Свойство касательной: она перпендикулярна радиусу, проведенному в точку касания.
б) пересекающей окружность
Пересекающей окружность (или секущей) называется прямая, которая имеет с окружностью две различные общие точки. Отрезок секущей, лежащий внутри окружности, называется хордой.
Ответ: а) Касательная – прямая, имеющая с окружностью одну общую точку. б) Пересекающая (секущая) – прямая, имеющая с окружностью две общие точки.
3. В каком случае прямая и окружность не имеют общих точек?
Прямая и окружность не имеют общих точек, если расстояние от центра окружности до этой прямой больше, чем радиус окружности. Если обозначить радиус окружности как $r$, а расстояние от ее центра (точка O) до прямой как $d$, то условие отсутствия общих точек записывается как $d > r$.
Ответ: Прямая и окружность не имеют общих точек, если расстояние от центра окружности до прямой больше ее радиуса ($d > r$).
4. В каком случае прямая касается окружности?
Прямая касается окружности, если она имеет с ней ровно одну общую точку. Это происходит в том случае, когда расстояние от центра окружности до прямой в точности равно радиусу окружности. Если $r$ – радиус, а $d$ – расстояние от центра (точка O) до прямой, то условие касания: $d = r$.
Ответ: Прямая касается окружности, если расстояние от центра окружности до прямой равно ее радиусу ($d = r$).
5. В каком случае прямая и окружность пересекаются?
Прямая и окружность пересекаются, если они имеют две общие точки. Это возможно, когда расстояние от центра окружности до прямой меньше радиуса окружности. Математически это условие записывается как $d < r$, где $r$ – радиус, а $d$ – расстояние от центра (точка O) до прямой.
Ответ: Прямая и окружность пересекаются, если расстояние от центра окружности до прямой меньше ее радиуса ($d < r$).
Другие задания:
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