gdz.top
Номер 20.9, страница 117 - 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 20. Взаимное расположение двух окружностей - номер 20.9, страница 117.
App\Models\Task {#1024 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1129847 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "117" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/20-9" "field_display_title" => "20.9" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#331 #items: array:1 [ 0 => App\Models\Term {#1029 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1026 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#342 #items: array:1 [ 0 => App\Models\Branch {#1023 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 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 {#333 #items: array:1 [ 0 => App\Models\Term {#1033 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "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 {#1030 #items: array:1 [ 0 => App\Models\Book {#1032 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 2907 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1034 #items: array:1 [ 0 => App\Models\Term {#1035 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6477 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Геометрия" "field_abbreviated_name" => null "field_cases" => array:6 [ "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 {#1036 #items: array:1 [ 0 => App\Models\Term {#1037 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5456 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "7" "field_cases" => array:6 [ "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 {#1038 #items: array:1 [ 0 => App\Models\Term {#1039 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "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 {#1040 #items: array:2 [ 0 => App\Models\Term {#1041 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 5801 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Смирнов" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Владимир" "field_patronymic" => "Алексеевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 5801 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Смирнов" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Владимир" "field_patronymic" => "Алексеевич" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1042 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7017 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Туяков" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Есенкельды" "field_patronymic" => "Алыбаевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 7017 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Туяков" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Есенкельды" "field_patronymic" => "Алыбаевич" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1043 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1044 #items: array:1 [ 0 => App\Models\Term {#1045 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "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 {#1046 #items: array:1 [ 0 => App\Models\Term {#1047 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ "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 {#1048 #items: array:1 [ 0 => App\Models\Term {#1049 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ "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 {#1050 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1051 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1052 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1053 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1054 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1055 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2017" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 7 класса общеобразовательной школы" "field_allowed" => "Утверждено Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "528" "field_priority" => "2" "field_default_folder" => "/geometrija_07/Smirnova-u17/" "field_isbn" => "978-601-07-0873-0" "field_cover" => array:1 [ 0 => "/media/geometrija_07/Smirnova-u17/covers/cover1.webp?ts=1753259592" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_07/Smirnova-u17/covers/cover1.webp?ts=1753259592" "alt" => "" "width" => "1632" "height" => "2472" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1056 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1057 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1058 #items: [] #escapeWhenCastingToString: false } "field_url" => "/7-klass/geometrija/smirnova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1059 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 378 "subject" => 543 "class_subject" => 389 ] ] #original: array:50 [ "id" => 2907 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1034} "field_class" => Illuminate\Database\Eloquent\Collection {#1036} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1038} "field_author" => Illuminate\Database\Eloquent\Collection {#1040} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1043} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1044} "field_country" => Illuminate\Database\Eloquent\Collection {#1046} "field_city" => Illuminate\Database\Eloquent\Collection {#1048} "field_series" => Illuminate\Database\Eloquent\Collection {#1050} "field_umk" => Illuminate\Database\Eloquent\Collection {#1051} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1052} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1053} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1054} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1055} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2017" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 7 класса общеобразовательной школы" "field_allowed" => "Утверждено Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "528" "field_priority" => "2" "field_default_folder" => "/geometrija_07/Smirnova-u17/" "field_isbn" => "978-601-07-0873-0" "field_cover" => array:1 [ 0 => "/media/geometrija_07/Smirnova-u17/covers/cover1.webp?ts=1753259592" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_07/Smirnova-u17/covers/cover1.webp?ts=1753259592" "alt" => "" "width" => "1632" "height" => "2472" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1056} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1057} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1058} "field_url" => "/7-klass/geometrija/smirnova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1059} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 378 "subject" => 543 "class_subject" => 389 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1060 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 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 {#333} "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 {#1030} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1060} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1061 #items: array:2 [ 0 => App\Models\Branch {#1023} 1 => App\Models\Branch {#1062 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1129382 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Взаимное расположение двух окружностей" "field_branch_order" => "20" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1063 #items: array:1 [ 0 => App\Models\Term {#1064 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ "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" => "112" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1065 #items: array:1 [ 0 => App\Models\Book {#1032} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1066 #items: array:1 [ 0 => App\Models\Branch {#1067 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 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 {#1068 #items: array:1 [ 0 => App\Models\Term {#1033} ] #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 {#1069 #items: array:1 [ 0 => App\Models\Book {#1032} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1070 #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 {#1068} "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 {#1069} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1070} ] #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" => 1129382 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Взаимное расположение двух окружностей" "field_branch_order" => "20" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1063} "field_page_start" => "112" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1065} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1066} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1071 #items: array:3 [ 0 => App\Models\Element {#1072 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 1319962 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1073 #items: array:1 [ 0 => App\Models\Edition {#1074 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5009 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1075 #items: array:1 [ 0 => 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:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #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 {#1077 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1078 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1079 #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 {#1075} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1077} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1078} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1079} "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" => "1129847" "type" => "task" ] "text" => "<p><strong>20.9.</strong> Расстояние между центрами двух окружностей равно $d$ и меньше разности $R_1 - R_2$ их радиусов ($R_1 > R_2$). Найдите наименьшее и наибольшее расстояния между точками, расположенными на данных окружностях.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "9-1.jpg" "alt" => null "width" => "1757" "height" => 321 "path" => "/media/geometrija_07/Smirnova-u17/0-20/9-1.webp?ts=1753982457" ] ] ] #original: array:7 [ "id" => 1319962 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1073} "task" => array:2 [ "refs" => "1129847" "type" => "task" ] "text" => "<p><strong>20.9.</strong> Расстояние между центрами двух окружностей равно $d$ и меньше разности $R_1 - R_2$ их радиусов ($R_1 > R_2$). Найдите наименьшее и наибольшее расстояния между точками, расположенными на данных окружностях.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "9-1.jpg" "alt" => null "width" => "1757" "height" => 321 "path" => "/media/geometrija_07/Smirnova-u17/0-20/9-1.webp?ts=1753982457" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1080 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1320414 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1081 #items: array:1 [ 0 => App\Models\Edition {#1082 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5010 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1083 #items: array:1 [ 0 => App\Models\Term {#1084 #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: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #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 {#1085 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1086 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1087 #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 {#1083} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1085} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1086} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1087} "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" => "1129847" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "9-1.jpg" "alt" => null "width" => "1453" "height" => 475 "path" => "/media/geometrija_07/Smirnova-u17/1-20/9-1.webp?ts=1753982572" ] ] ] #original: array:6 [ "id" => 1320414 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1081} "task" => array:2 [ "refs" => "1129847" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "9-1.jpg" "alt" => null "width" => "1453" "height" => 475 "path" => "/media/geometrija_07/Smirnova-u17/1-20/9-1.webp?ts=1753982572" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Element {#1088 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1541676 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1089 #items: array:1 [ 0 => App\Models\Edition {#1090 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5631 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1091 #items: array:1 [ 0 => App\Models\Term {#1092 #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: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #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 {#1093 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1094 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1095 #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 {#1091} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1093} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1094} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1095} "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" => "1129847" "type" => "task" ] "text" => "<p>Пусть $O_1$ и $O_2$ — центры двух окружностей, а $R_1$ и $R_2$ — их радиусы, причем по условию $R_1 > R_2$. Расстояние между центрами равно $d$, и выполняется условие $d < R_1 - R_2$.</p><p>Условие $d < R_1 - R_2$ означает, что окружность меньшего радиуса $R_2$ полностью находится внутри окружности большего радиуса $R_1$, и они не соприкасаются. Чтобы в этом убедиться, найдем максимальное расстояние от центра $O_1$ до любой точки на второй окружности. Это расстояние равно $d + R_2$. Условие $d < R_1 - R_2$ эквивалентно $d + R_2 < R_1$, что подтверждает, что каждая точка второй окружности находится внутри первой.</p><svg width="500" height="250" xmlns="http://www.w3.org/2000/svg" font-family="sans-serif" font-size="16"><defs><marker id="arrow" viewBox="0 0 10 10" refX="5" refY="5" markerWidth="6" markerHeight="6" orient="auto-start-reverse"><path d="M 0 0 L 10 5 L 0 10 z" fill="black"></path></marker></defs><line x1="20" y1="125" x2="480" y2="125" stroke="gray" stroke-width="1" stroke-dasharray="4,4"></line><circle cx="250" cy="125" r="120" stroke="#005A9C" stroke-width="2" fill="rgba(0, 90, 156, 0.1)"></circle><circle cx="250" cy="125" r="3" fill="#005A9C"></circle><text x="245" y="115" fill="#005A9C">O₁</text><circle cx="130" cy="125" r="4" fill="#2E8B57"></circle><text x="120" y="150" fill="#2E8B57">A₁</text><circle cx="370" cy="125" r="4" fill="#8A2BE2"></circle><text x="375" y="150" fill="#8A2BE2">A₂</text><circle cx="300" cy="125" r="40" stroke="#D83121" stroke-width="2" fill="rgba(216, 49, 33, 0.1)"></circle><circle cx="300" cy="125" r="3" fill="#D83121"></circle><text x="305" y="115" fill="#D83121">O₂</text><circle cx="260" cy="125" r="4" fill="black"></circle><text x="250" y="110" fill="black">B₁</text><circle cx="340" cy="125" r="4" fill="#FF4500"></circle><text x="345" y="150" fill="#FF4500">B₂</text><line x1="130" y1="170" x2="340" y2="170" stroke="#2E8B57" stroke-width="2" marker-start="url(#arrow)" marker-end="url(#arrow)"></line><text x="180" y="190" fill="#2E8B57">Наибольшее расстояние</text><line x1="340" y1="125" x2="370" y2="125" stroke="#8A2BE2" stroke-width="3"></line><line x1="340" y1="100" x2="370" y2="100" stroke="#8A2BE2" stroke-width="2" marker-start="url(#arrow)" marker-end="url(#arrow)"></line><text x="300" y="85" fill="#8A2BE2">Наименьшее расстояние</text><line x1="250" y1="40" x2="300" y2="40" stroke="black" stroke-width="1.5" marker-start="url(#arrow)" marker-end="url(#arrow)"></line><text x="270" y="35">d</text></svg><p>Рассмотрим произвольную точку $A$ на первой окружности (с центром $O_1$) и произвольную точку $B$ на второй окружности (с центром $O_2$). Экстремальные расстояния (наименьшее и наибольшее) достигаются для точек, лежащих на прямой, проходящей через центры окружностей $O_1$ и $O_2$.</p><p><strong>Наименьшее расстояние</strong></p><p>Наименьшее расстояние будет между точками, лежащими на линии центров. Так как вторая окружность находится внутри первой, ближайшие друг к другу точки ($A_{min}$ и $B_{min}$) будут расположены на линии центров с одной стороны от центра $O_1$.</p><p>Пусть $A_{min}$ — точка на большой окружности, а $B_{min}$ — точка на малой окружности, обе лежат на луче, выходящем из $O_1$ и проходящем через $O_2$. Расстояние от центра $O_1$ до точки $A_{min}$ равно $O_1A_{min} = R_1$. Расстояние от центра $O_1$ до точки $B_{min}$ равно $O_1B_{min} = O_1O_2 + O_2B_{min} = d + R_2$.</p><p>Поскольку обе точки лежат на одном луче из $O_1$, расстояние между ними равно разности их расстояний от $O_1$: $A_{min}B_{min} = |O_1A_{min} - O_1B_{min}| = |R_1 - (d + R_2)|$.</p><p>Из условия задачи $d < R_1 - R_2$, что эквивалентно $d + R_2 < R_1$. Поэтому выражение под знаком модуля положительно. Таким образом, наименьшее расстояние равно $R_1 - (d + R_2) = R_1 - d - R_2$.</p><p>Ответ: $R_1 - R_2 - d$.</p><p><strong>Наибольшее расстояние</strong></p><p>Наибольшее расстояние между точками двух окружностей будет между точками $A_{max}$ и $B_{max}$, которые лежат на линии центров и расположены так, чтобы максимизировать длину отрезка между ними. Это происходит, когда точки находятся на линии центров по разные стороны от отрезка $O_1O_2$.</p><p>Пусть точка $A_{max}$ лежит на первой окружности, а $B_{max}$ — на второй. Тогда расстояние $A_{max}B_{max}$ равно сумме расстояния от $A_{max}$ до $O_1$, расстояния между центрами $O_1O_2$ и расстояния от $O_2$ до $B_{max}$: $A_{max}B_{max} = A_{max}O_1 + O_1O_2 + O_2B_{max}$.</p><p>Поскольку $A_{max}$ лежит на первой окружности, $A_{max}O_1 = R_1$. Поскольку $B_{max}$ лежит на второй окружности, $O_2B_{max} = R_2$. Расстояние между центрами равно $d$. Следовательно, наибольшее расстояние равно $R_1 + d + R_2$.</p><p>Ответ: $R_1 + R_2 + d$.</p>" ] #original: array:6 [ "id" => 1541676 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1089} "task" => array:2 [ "refs" => "1129847" "type" => "task" ] "text" => "<p>Пусть $O_1$ и $O_2$ — центры двух окружностей, а $R_1$ и $R_2$ — их радиусы, причем по условию $R_1 > R_2$. Расстояние между центрами равно $d$, и выполняется условие $d < R_1 - R_2$.</p><p>Условие $d < R_1 - R_2$ означает, что окружность меньшего радиуса $R_2$ полностью находится внутри окружности большего радиуса $R_1$, и они не соприкасаются. Чтобы в этом убедиться, найдем максимальное расстояние от центра $O_1$ до любой точки на второй окружности. Это расстояние равно $d + R_2$. Условие $d < R_1 - R_2$ эквивалентно $d + R_2 < R_1$, что подтверждает, что каждая точка второй окружности находится внутри первой.</p><svg width="500" height="250" xmlns="http://www.w3.org/2000/svg" font-family="sans-serif" font-size="16"><defs><marker id="arrow" viewBox="0 0 10 10" refX="5" refY="5" markerWidth="6" markerHeight="6" orient="auto-start-reverse"><path d="M 0 0 L 10 5 L 0 10 z" fill="black"></path></marker></defs><line x1="20" y1="125" x2="480" y2="125" stroke="gray" stroke-width="1" stroke-dasharray="4,4"></line><circle cx="250" cy="125" r="120" stroke="#005A9C" stroke-width="2" fill="rgba(0, 90, 156, 0.1)"></circle><circle cx="250" cy="125" r="3" fill="#005A9C"></circle><text x="245" y="115" fill="#005A9C">O₁</text><circle cx="130" cy="125" r="4" fill="#2E8B57"></circle><text x="120" y="150" fill="#2E8B57">A₁</text><circle cx="370" cy="125" r="4" fill="#8A2BE2"></circle><text x="375" y="150" fill="#8A2BE2">A₂</text><circle cx="300" cy="125" r="40" stroke="#D83121" stroke-width="2" fill="rgba(216, 49, 33, 0.1)"></circle><circle cx="300" cy="125" r="3" fill="#D83121"></circle><text x="305" y="115" fill="#D83121">O₂</text><circle cx="260" cy="125" r="4" fill="black"></circle><text x="250" y="110" fill="black">B₁</text><circle cx="340" cy="125" r="4" fill="#FF4500"></circle><text x="345" y="150" fill="#FF4500">B₂</text><line x1="130" y1="170" x2="340" y2="170" stroke="#2E8B57" stroke-width="2" marker-start="url(#arrow)" marker-end="url(#arrow)"></line><text x="180" y="190" fill="#2E8B57">Наибольшее расстояние</text><line x1="340" y1="125" x2="370" y2="125" stroke="#8A2BE2" stroke-width="3"></line><line x1="340" y1="100" x2="370" y2="100" stroke="#8A2BE2" stroke-width="2" marker-start="url(#arrow)" marker-end="url(#arrow)"></line><text x="300" y="85" fill="#8A2BE2">Наименьшее расстояние</text><line x1="250" y1="40" x2="300" y2="40" stroke="black" stroke-width="1.5" marker-start="url(#arrow)" marker-end="url(#arrow)"></line><text x="270" y="35">d</text></svg><p>Рассмотрим произвольную точку $A$ на первой окружности (с центром $O_1$) и произвольную точку $B$ на второй окружности (с центром $O_2$). Экстремальные расстояния (наименьшее и наибольшее) достигаются для точек, лежащих на прямой, проходящей через центры окружностей $O_1$ и $O_2$.</p><p><strong>Наименьшее расстояние</strong></p><p>Наименьшее расстояние будет между точками, лежащими на линии центров. Так как вторая окружность находится внутри первой, ближайшие друг к другу точки ($A_{min}$ и $B_{min}$) будут расположены на линии центров с одной стороны от центра $O_1$.</p><p>Пусть $A_{min}$ — точка на большой окружности, а $B_{min}$ — точка на малой окружности, обе лежат на луче, выходящем из $O_1$ и проходящем через $O_2$. Расстояние от центра $O_1$ до точки $A_{min}$ равно $O_1A_{min} = R_1$. Расстояние от центра $O_1$ до точки $B_{min}$ равно $O_1B_{min} = O_1O_2 + O_2B_{min} = d + R_2$.</p><p>Поскольку обе точки лежат на одном луче из $O_1$, расстояние между ними равно разности их расстояний от $O_1$: $A_{min}B_{min} = |O_1A_{min} - O_1B_{min}| = |R_1 - (d + R_2)|$.</p><p>Из условия задачи $d < R_1 - R_2$, что эквивалентно $d + R_2 < R_1$. Поэтому выражение под знаком модуля положительно. Таким образом, наименьшее расстояние равно $R_1 - (d + R_2) = R_1 - d - R_2$.</p><p>Ответ: $R_1 - R_2 - d$.</p><p><strong>Наибольшее расстояние</strong></p><p>Наибольшее расстояние между точками двух окружностей будет между точками $A_{max}$ и $B_{max}$, которые лежат на линии центров и расположены так, чтобы максимизировать длину отрезка между ними. Это происходит, когда точки находятся на линии центров по разные стороны от отрезка $O_1O_2$.</p><p>Пусть точка $A_{max}$ лежит на первой окружности, а $B_{max}$ — на второй. Тогда расстояние $A_{max}B_{max}$ равно сумме расстояния от $A_{max}$ до $O_1$, расстояния между центрами $O_1O_2$ и расстояния от $O_2$ до $B_{max}$: $A_{max}B_{max} = A_{max}O_1 + O_1O_2 + O_2B_{max}$.</p><p>Поскольку $A_{max}$ лежит на первой окружности, $A_{max}O_1 = R_1$. Поскольку $B_{max}$ лежит на второй окружности, $O_2B_{max} = R_2$. Расстояние между центрами равно $d$. Следовательно, наибольшее расстояние равно $R_1 + d + R_2$.</p><p>Ответ: $R_1 + R_2 + d$.</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" => "1129848" "type" => "task" ] "previous" => array:2 [ "refs" => "1129846" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1096 #items: array:1 [ 0 => App\Models\Book {#1032} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1111 #items: array:1 [ 0 => App\Models\BookPage {#1110 #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" => 1114434 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "117" "field_url" => "/7-klass/geometrija/smirnova-uchebnik/page-117" "field_display_title" => "117" "field_folder" => "1" "field_image_name" => "117" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1112 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1142 #items: array:1 [ 0 => App\Models\Book {#1113 #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 {#1115 #items: array:1 [ 0 => App\Models\Term {#1114 #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 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false …4 } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1116 #items: array:1 [ 0 => App\Models\Term {#1117 …30} ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1118 #items: array:1 [ 0 => App\Models\Term {#1119 …30} ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1120 #items: array:2 [ 0 => App\Models\Term {#1121 …30} 1 => App\Models\Term {#1122 …30} ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1123 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1124 #items: array:1 [ 0 => App\Models\Term {#1125 …30} ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1126 #items: array:1 [ 0 => App\Models\Term {#1127 …30} ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1128 #items: array:1 [ 0 => App\Models\Term {#1129 …30} ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1130 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1131 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1132 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1133 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1134 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1135 #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 {#1136 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1137 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1138 #items: [] #escapeWhenCastingToString: false } "field_url" => "/7-klass/geometrija/smirnova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1139 #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 {#1115} "field_class" => Illuminate\Database\Eloquent\Collection {#1116} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1118} "field_author" => Illuminate\Database\Eloquent\Collection {#1120} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1123} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1124} "field_country" => Illuminate\Database\Eloquent\Collection {#1126} "field_city" => Illuminate\Database\Eloquent\Collection {#1128} "field_series" => Illuminate\Database\Eloquent\Collection {#1130} "field_umk" => Illuminate\Database\Eloquent\Collection {#1131} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1132} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1133} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1134} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1135} "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 {#1136} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1137} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1138} "field_url" => "/7-klass/geometrija/smirnova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1139} "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 } "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 {#1140 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1151 #items: array:1 [ 0 => App\Models\Element {#1150 #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" => 1290621 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1160 #items: array:1 [ 0 => App\Models\Edition {#1152 …30} ] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1114434" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "117-1.jpg" "alt" => null "width" => "1935" "height" => 2838 "path" => "/media/geometrija_07/Smirnova-u17/0-1/117-1.webp?ts=1753260190" ] ] ] #original: array:6 [ "id" => 1290621 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1160} "book_page" => array:2 [ "refs" => "1114434" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "117-1.jpg" "alt" => null "width" => "1935" "height" => 2838 "path" => "/media/geometrija_07/Smirnova-u17/0-1/117-1.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" => "1114435" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1114433" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1444 #items: array:4 [ 0 => App\Models\Task {#1143 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1129847 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "117" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/20-9" "field_display_title" => "20.9" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1141 #items: array:1 [ 0 => App\Models\Term {#1144 …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 {#1147 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1145 #items: array:1 [ 0 => App\Models\Branch {#1148 …30} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1189 #items: array:2 [ 0 => App\Models\Branch {#1148 …30} 1 => App\Models\Branch {#1190 …30} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1199 #items: array:3 [ 0 => App\Models\Element {#1200 …30} 1 => App\Models\Element {#1208 …30} 2 => App\Models\Element {#1216 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1129848" "type" => "task" ] "previous" => array:2 [ "refs" => "1129846" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1224 #items: array:1 [ 0 => App\Models\Book {#1161 …30} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1114434" "type" => "book_page" ] ] #original: array:24 [ "id" => 1129847 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "117" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/20-9" "field_display_title" => "20.9" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1141} "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 {#1147} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1145} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1189} "content" => Illuminate\Database\Eloquent\Collection {#1199} "next" => array:2 [ "refs" => "1129848" "type" => "task" ] "previous" => array:2 [ "refs" => "1129846" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1224} "page" => array:2 [ "refs" => "1114434" "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 {#1225 #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" => 1129848 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "117" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/20-10" "field_display_title" => "20.10" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1226 #items: array:1 [ 0 => App\Models\Term {#1227 …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 {#1228 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1229 #items: array:1 [ 0 => App\Models\Branch {#1230 …30} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1262 #items: array:2 [ 0 => App\Models\Branch {#1230 …30} 1 => App\Models\Branch {#1263 …30} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1272 #items: array:3 [ 0 => App\Models\Element {#1273 …30} 1 => App\Models\Element {#1281 …30} 2 => App\Models\Element {#1289 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1129849" "type" => "task" ] "previous" => array:2 [ "refs" => "1129847" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1297 #items: array:1 [ 0 => App\Models\Book {#1234 …30} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1114434" "type" => "book_page" ] ] #original: array:24 [ "id" => 1129848 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "117" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/20-10" "field_display_title" => "20.10" "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 {#1228} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1229} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1262} "content" => Illuminate\Database\Eloquent\Collection {#1272} "next" => array:2 [ "refs" => "1129849" "type" => "task" ] "previous" => array:2 [ "refs" => "1129847" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1297} "page" => array:2 [ "refs" => "1114434" "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 {#1298 #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" => 1129849 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "117" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/20-11" "field_display_title" => "20.11" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1299 #items: array:1 [ 0 => App\Models\Term {#1300 …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 {#1301 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1302 #items: array:1 [ 0 => App\Models\Branch {#1303 …30} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1335 #items: array:2 [ 0 => App\Models\Branch {#1303 …30} 1 => App\Models\Branch {#1336 …30} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1345 #items: array:3 [ 0 => App\Models\Element {#1346 …30} 1 => App\Models\Element {#1354 …30} 2 => App\Models\Element {#1362 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1129850" "type" => "task" ] "previous" => array:2 [ "refs" => "1129848" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1370 #items: array:1 [ 0 => App\Models\Book {#1307 …30} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1114434" "type" => "book_page" ] ] #original: array:24 [ "id" => 1129849 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "117" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/20-11" "field_display_title" => "20.11" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1299} "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 {#1301} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1302} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1335} "content" => Illuminate\Database\Eloquent\Collection {#1345} "next" => array:2 [ "refs" => "1129850" "type" => "task" ] "previous" => array:2 [ "refs" => "1129848" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1370} "page" => array:2 [ "refs" => "1114434" "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 {#1371 #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" => 1129850 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "117" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/20-12" "field_display_title" => "20.12" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1372 #items: array:1 [ 0 => App\Models\Term {#1373 …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 {#1374 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1375 #items: array:1 [ 0 => App\Models\Branch {#1376 …30} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1408 #items: array:2 [ 0 => App\Models\Branch {#1376 …30} 1 => App\Models\Branch {#1409 …30} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1418 #items: array:3 [ 0 => App\Models\Element {#1419 …30} 1 => App\Models\Element {#1427 …30} 2 => App\Models\Element {#1435 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1129851" "type" => "task" ] "previous" => array:2 [ "refs" => "1129849" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1443 #items: array:1 [ 0 => App\Models\Book {#1380 …30} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1114434" "type" => "book_page" ] ] #original: array:24 [ "id" => 1129850 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "117" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/20-12" "field_display_title" => "20.12" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1372} "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 {#1374} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1375} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1408} "content" => Illuminate\Database\Eloquent\Collection {#1418} "next" => array:2 [ "refs" => "1129851" "type" => "task" ] "previous" => array:2 [ "refs" => "1129849" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1443} "page" => array:2 [ "refs" => "1114434" "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" => 1114434 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "117" "field_url" => "/7-klass/geometrija/smirnova-uchebnik/page-117" "field_display_title" => "117" "field_folder" => "1" "field_image_name" => "117" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1112} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1142} "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 {#1140} "content" => Illuminate\Database\Eloquent\Collection {#1151} "next" => array:2 [ "refs" => "1114435" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1114433" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1444} ] #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" => 1129847 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "117" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/20-9" "field_display_title" => "20.9" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#331} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1026} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#342} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1061} "content" => Illuminate\Database\Eloquent\Collection {#1071} "next" => array:2 [ "refs" => "1129848" "type" => "task" ] "previous" => array:2 [ "refs" => "1129846" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1096} "page" => array:2 [ "refs" => "1114434" "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 => "*" ] }
№20.9 (с. 117)
Условие. №20.9 (с. 117)
скриншот условия
20.9. Расстояние между центрами двух окружностей равно $d$ и меньше разности $R_1 - R_2$ их радиусов ($R_1 > R_2$). Найдите наименьшее и наибольшее расстояния между точками, расположенными на данных окружностях.
Решение. №20.9 (с. 117)
Решение 2. №20.9 (с. 117)
Пусть $O_1$ и $O_2$ — центры двух окружностей, а $R_1$ и $R_2$ — их радиусы, причем по условию $R_1 > R_2$. Расстояние между центрами равно $d$, и выполняется условие $d < R_1 - R_2$.
Условие $d < R_1 - R_2$ означает, что окружность меньшего радиуса $R_2$ полностью находится внутри окружности большего радиуса $R_1$, и они не соприкасаются. Чтобы в этом убедиться, найдем максимальное расстояние от центра $O_1$ до любой точки на второй окружности. Это расстояние равно $d + R_2$. Условие $d < R_1 - R_2$ эквивалентно $d + R_2 < R_1$, что подтверждает, что каждая точка второй окружности находится внутри первой.
Рассмотрим произвольную точку $A$ на первой окружности (с центром $O_1$) и произвольную точку $B$ на второй окружности (с центром $O_2$). Экстремальные расстояния (наименьшее и наибольшее) достигаются для точек, лежащих на прямой, проходящей через центры окружностей $O_1$ и $O_2$.
Наименьшее расстояние
Наименьшее расстояние будет между точками, лежащими на линии центров. Так как вторая окружность находится внутри первой, ближайшие друг к другу точки ($A_{min}$ и $B_{min}$) будут расположены на линии центров с одной стороны от центра $O_1$.
Пусть $A_{min}$ — точка на большой окружности, а $B_{min}$ — точка на малой окружности, обе лежат на луче, выходящем из $O_1$ и проходящем через $O_2$. Расстояние от центра $O_1$ до точки $A_{min}$ равно $O_1A_{min} = R_1$. Расстояние от центра $O_1$ до точки $B_{min}$ равно $O_1B_{min} = O_1O_2 + O_2B_{min} = d + R_2$.
Поскольку обе точки лежат на одном луче из $O_1$, расстояние между ними равно разности их расстояний от $O_1$: $A_{min}B_{min} = |O_1A_{min} - O_1B_{min}| = |R_1 - (d + R_2)|$.
Из условия задачи $d < R_1 - R_2$, что эквивалентно $d + R_2 < R_1$. Поэтому выражение под знаком модуля положительно. Таким образом, наименьшее расстояние равно $R_1 - (d + R_2) = R_1 - d - R_2$.
Ответ: $R_1 - R_2 - d$.
Наибольшее расстояние
Наибольшее расстояние между точками двух окружностей будет между точками $A_{max}$ и $B_{max}$, которые лежат на линии центров и расположены так, чтобы максимизировать длину отрезка между ними. Это происходит, когда точки находятся на линии центров по разные стороны от отрезка $O_1O_2$.
Пусть точка $A_{max}$ лежит на первой окружности, а $B_{max}$ — на второй. Тогда расстояние $A_{max}B_{max}$ равно сумме расстояния от $A_{max}$ до $O_1$, расстояния между центрами $O_1O_2$ и расстояния от $O_2$ до $B_{max}$: $A_{max}B_{max} = A_{max}O_1 + O_1O_2 + O_2B_{max}$.
Поскольку $A_{max}$ лежит на первой окружности, $A_{max}O_1 = R_1$. Поскольку $B_{max}$ лежит на второй окружности, $O_2B_{max} = R_2$. Расстояние между центрами равно $d$. Следовательно, наибольшее расстояние равно $R_1 + d + R_2$.
Ответ: $R_1 + R_2 + d$.
Другие задания:
Помогло решение? Оставьте отзыв в комментариях ниже.
Присоединяйтесь к Телеграм-группе @top_gdz
ПрисоединитьсяМы подготовили для вас ответ c подробным объяснением домашего задания по геометрии за 7 класс, для упражнения номер 20.9 расположенного на странице 117 к учебнику 2017 года издания для учащихся школ и гимназий.
Теперь на нашем сайте ГДЗ.ТОП вы всегда легко и бесплатно найдёте условие с правильным ответом на вопрос «Как решить ДЗ» и «Как сделать» задание по геометрии к упражнению №20.9 (с. 117), авторов: Смирнов (Владимир Алексеевич), Туяков (Есенкельды Алыбаевич), учебного пособия издательства Мектеп.