gdz.top
Номер 21.10, страница 122 - 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 21. Геометрические места точек - номер 21.10, страница 122.
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" => 1129869 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "122" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/21-10" "field_display_title" => "21.10" "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" => 1129383 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Геометрические места точек" "field_branch_order" => "21" "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" => "118" "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 {#1069 …30} ] #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 {#1070 #items: array:1 [ 0 => App\Models\Book {#1071 …30} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1098 #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 {#1070} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1098} ] #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" => 1129383 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Геометрические места точек" "field_branch_order" => "21" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1063} "field_page_start" => "118" "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 {#1099 #items: array:3 [ 0 => App\Models\Element {#1100 #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" => 1319998 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1101 #items: array:1 [ 0 => App\Models\Edition {#1102 #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 {#1103 #items: array:1 [ 0 => App\Models\Term {#1104 …30} ] #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 {#1105 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1106 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1107 #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 {#1103} "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 {#1105} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1106} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1107} "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" => "1129869" "type" => "task" ] "text" => "<p><strong>21.10.</strong> На прямой c отметьте точку C, равноудаленную от сторон угла AOB (рис. 21.9).</p><p><strong>a)</strong></p><p><strong>б)</strong></p><p><strong>в)</strong></p><p>Рис. 21.9</p>" "img" => array:1 [ 0 => array:5 [ "name" => "10-1.jpg" "alt" => null "width" => "1784" "height" => 828 "path" => "/media/geometrija_07/Smirnova-u17/0-21/10-1.webp?ts=1753982473" ] ] ] #original: array:7 [ "id" => 1319998 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1101} "task" => array:2 [ "refs" => "1129869" "type" => "task" ] "text" => "<p><strong>21.10.</strong> На прямой c отметьте точку C, равноудаленную от сторон угла AOB (рис. 21.9).</p><p><strong>a)</strong></p><p><strong>б)</strong></p><p><strong>в)</strong></p><p>Рис. 21.9</p>" "img" => array:1 [ 0 => array:5 [ "name" => "10-1.jpg" "alt" => null "width" => "1784" "height" => 828 "path" => "/media/geometrija_07/Smirnova-u17/0-21/10-1.webp?ts=1753982473" ] ] ] #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 {#1108 #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" => 1320434 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1109 #items: array:1 [ 0 => App\Models\Edition {#1110 #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 {#1111 #items: array:1 [ 0 => App\Models\Term {#1112 …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 {#1113 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1114 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1115 #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 {#1111} "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 {#1113} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1114} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1115} "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" => "1129869" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "10-1.jpg" "alt" => null "width" => "1501" "height" => 826 "path" => "/media/geometrija_07/Smirnova-u17/1-21/10-1.webp?ts=1753982588" ] 1 => array:5 [ "name" => "10-2.jpg" "alt" => null "width" => "1501" "height" => 661 "path" => "/media/geometrija_07/Smirnova-u17/1-21/10-2.webp?ts=1753982588" ] ] ] #original: array:6 [ "id" => 1320434 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1109} "task" => array:2 [ "refs" => "1129869" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "10-1.jpg" "alt" => null "width" => "1501" "height" => 826 "path" => "/media/geometrija_07/Smirnova-u17/1-21/10-1.webp?ts=1753982588" ] 1 => array:5 [ "name" => "10-2.jpg" "alt" => null "width" => "1501" "height" => 661 "path" => "/media/geometrija_07/Smirnova-u17/1-21/10-2.webp?ts=1753982588" ] ] ] #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 {#1116 #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" => 1541696 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1117 #items: array:1 [ 0 => App\Models\Edition {#1118 #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 {#1119 #items: array:1 [ 0 => App\Models\Term {#1120 …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 {#1121 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1122 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1123 #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 {#1119} "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 {#1121} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1122} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1123} "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" => "1129869" "type" => "task" ] "text" => "<p>Для решения задачи воспользуемся свойством биссектрисы угла: любая точка, лежащая на биссектрисе угла, равноудалена от его сторон. Следовательно, искомая точка $C$ должна лежать на биссектрисе угла $AOB$. Так как по условию точка $C$ также должна лежать на прямой $c$, то она является точкой пересечения биссектрисы угла $AOB$ и прямой $c$.</p><p><strong>а)</strong> Угол $AOB$ — прямой, его вершина $O$ находится в начале координат, а стороны $OA$ и $OB$ лежат на осях координат. Биссектриса прямого угла делит его на два угла по $45^\circ$, ее уравнение $y=x$. Прямая $c$ проходит через точки с координатами $(4,0)$ и $(1,3)$, ее уравнение $y = -x+4$. Точка пересечения $C$ биссектрисы и прямой $c$ имеет координаты, удовлетворяющие обоим уравнениям. Для системы уравнений $y=x$ и $y=-x+4$ решением является точка $C(2,2)$. Построим биссектрису угла $AOB$ (синяя пунктирная линия) и отметим точку ее пересечения с прямой $c$.</p><svg width="220" height="180" viewBox="0 0 220 180" xmlns="http://www.w3.org/2000/svg"><defs><style> .grid { stroke: #d3d3d3; stroke-width: 1; } .line { stroke: #d9534f; stroke-width: 2; } .bisector { stroke: #428bca; stroke-width: 2; stroke-dasharray: 4 4; } .point { fill: #d9534f; } .label { font-family: sans-serif; font-size: 16px; } </style></defs><path class="grid" d="M 60 20 V 140 M 100 20 V 140 M 140 20 V 140 M 180 20 V 140 M 20 60 H 180 M 20 100 H 180"></path><path class="grid" d="M 20 20 H 180 V 140 H 20 Z" fill="none" stroke="#d3d3d3" stroke-width="1"></path><line class="line" x1="20" y1="140" x2="200" y2="140"></line><line class="line" x1="20" y1="140" x2="20" y2="0"></line><line class="line" x1="180" y1="140" x2="60" y2="20"></line><line class="bisector" x1="20" y1="140" x2="140" y2="20"></line><circle class="point" cx="100" cy="60" r="4"></circle><text class="label" x="5" y="155">O</text><text class="label" x="205" y="155">A</text><text class="label" x="5" y="15">B</text><text class="label" x="125" y="75">c</text><text class="label" x="105" y="55">C</text></svg><p><strong>Ответ:</strong> Искомая точка $C$ является точкой пересечения биссектрисы угла $AOB$ и прямой $c$. Она расположена в узле сетки с координатами $(2,2)$, если принять точку $O$ за начало координат.</p><p><strong>б)</strong> Вершина угла $O$ находится в начале координат. Сторона $OA$ проходит через точку $(4,1)$, а сторона $OB$ — через точку $(1,4)$. Угол $AOB$ симметричен относительно прямой $y=x$, так как координаты точек на его сторонах взаимно обратны. Следовательно, биссектрисой угла $AOB$ является прямая $y=x$. Прямая $c$ — это горизонтальная прямая $y=2$. Точка пересечения $C$ биссектрисы $y=x$ и прямой $c$ ($y=2$) имеет координаты $C(2,2)$. Построим биссектрису угла $AOB$ и найдем ее пересечение с прямой $c$.</p><svg width="220" height="220" viewBox="0 0 220 220" xmlns="http://www.w3.org/2000/svg"><defs><style> .grid { stroke: #d3d3d3; stroke-width: 1; } .line { stroke: #d9534f; stroke-width: 2; } .bisector { stroke: #428bca; stroke-width: 2; stroke-dasharray: 4 4; } .point { fill: #d9534f; } .label { font-family: sans-serif; font-size: 16px; } </style></defs><path class="grid" d="M 60 20 V 180 M 100 20 V 180 M 140 20 V 180 M 180 20 V 180 M 20 60 H 180 M 20 100 H 180 M 20 140 H 180"></path><path class="grid" d="M 20 20 H 180 V 180 H 20 Z" fill="none" stroke="#d3d3d3" stroke-width="1"></path><line class="line" x1="20" y1="180" x2="180" y2="140"></line><line class="line" x1="20" y1="180" x2="60" y2="20"></line><line class="line" x1="0" y1="100" x2="220" y2="100"></line><line class="bisector" x1="20" y1="180" x2="180" y2="20"></line><circle class="point" cx="100" cy="100" r="4"></circle><text class="label" x="5" y="195">O</text><text class="label" x="185" y="135">A</text><text class="label" x="50" y="15">B</text><text class="label" x="125" y="95">c</text><text class="label" x="105" y="90">C</text></svg><p><strong>Ответ:</strong> Искомая точка $C$ является точкой пересечения биссектрисы угла $AOB$ и прямой $c$. Она расположена в узле сетки с координатами $(2,2)$, если принять точку $O$ за начало координат.</p><p><strong>в)</strong> Вершина угла $O$ находится в начале координат. Сторона $OA$ проходит через точку $(4,2)$, а сторона $OB$ — через точку $(2,4)$. Угловые коэффициенты сторон равны $k_{OA} = 2/4 = 1/2$ и $k_{OB} = 4/2 = 2$. Так как $k_{OB} = 1/k_{OA}$, угол $AOB$ симметричен относительно прямой $y=x$, которая и является его биссектрисой. Прямая $c$ — это горизонтальная прямая $y=3$. Точка пересечения $C$ биссектрисы $y=x$ и прямой $c$ ($y=3$) имеет координаты $C(3,3)$. Построим биссектрису угла $AOB$ и найдем ее пересечение с прямой $c$.</p><svg width="220" height="220" viewBox="0 0 220 220" xmlns="http://www.w3.org/2000/svg"><defs><style> .grid { stroke: #d3d3d3; stroke-width: 1; } .line { stroke: #d9534f; stroke-width: 2; } .bisector { stroke: #428bca; stroke-width: 2; stroke-dasharray: 4 4; } .point { fill: #d9534f; } .label { font-family: sans-serif; font-size: 16px; } </style></defs><path class="grid" d="M 60 20 V 180 M 100 20 V 180 M 140 20 V 180 M 180 20 V 180 M 20 60 H 180 M 20 100 H 180 M 20 140 H 180"></path><path class="grid" d="M 20 20 H 180 V 180 H 20 Z" fill="none" stroke="#d3d3d3" stroke-width="1"></path><line class="line" x1="20" y1="180" x2="180" y2="100"></line><line class="line" x1="20" y1="180" x2="100" y2="20"></line><line class="line" x1="0" y1="60" x2="220" y2="60"></line><line class="bisector" x1="20" y1="180" x2="180" y2="20"></line><circle class="point" cx="140" cy="60" r="4"></circle><text class="label" x="5" y="195">O</text><text class="label" x="185" y="95">A</text><text class="label" x="90" y="15">B</text><text class="label" x="165" y="55">c</text><text class="label" x="145" y="50">C</text></svg><p><strong>Ответ:</strong> Искомая точка $C$ является точкой пересечения биссектрисы угла $AOB$ и прямой $c$. Она расположена в узле сетки с координатами $(3,3)$, если принять точку $O$ за начало координат.</p>" ] #original: array:6 [ "id" => 1541696 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1117} "task" => array:2 [ "refs" => "1129869" "type" => "task" ] "text" => "<p>Для решения задачи воспользуемся свойством биссектрисы угла: любая точка, лежащая на биссектрисе угла, равноудалена от его сторон. Следовательно, искомая точка $C$ должна лежать на биссектрисе угла $AOB$. Так как по условию точка $C$ также должна лежать на прямой $c$, то она является точкой пересечения биссектрисы угла $AOB$ и прямой $c$.</p><p><strong>а)</strong> Угол $AOB$ — прямой, его вершина $O$ находится в начале координат, а стороны $OA$ и $OB$ лежат на осях координат. Биссектриса прямого угла делит его на два угла по $45^\circ$, ее уравнение $y=x$. Прямая $c$ проходит через точки с координатами $(4,0)$ и $(1,3)$, ее уравнение $y = -x+4$. Точка пересечения $C$ биссектрисы и прямой $c$ имеет координаты, удовлетворяющие обоим уравнениям. Для системы уравнений $y=x$ и $y=-x+4$ решением является точка $C(2,2)$. Построим биссектрису угла $AOB$ (синяя пунктирная линия) и отметим точку ее пересечения с прямой $c$.</p><svg width="220" height="180" viewBox="0 0 220 180" xmlns="http://www.w3.org/2000/svg"><defs><style> .grid { stroke: #d3d3d3; stroke-width: 1; } .line { stroke: #d9534f; stroke-width: 2; } .bisector { stroke: #428bca; stroke-width: 2; stroke-dasharray: 4 4; } .point { fill: #d9534f; } .label { font-family: sans-serif; font-size: 16px; } </style></defs><path class="grid" d="M 60 20 V 140 M 100 20 V 140 M 140 20 V 140 M 180 20 V 140 M 20 60 H 180 M 20 100 H 180"></path><path class="grid" d="M 20 20 H 180 V 140 H 20 Z" fill="none" stroke="#d3d3d3" stroke-width="1"></path><line class="line" x1="20" y1="140" x2="200" y2="140"></line><line class="line" x1="20" y1="140" x2="20" y2="0"></line><line class="line" x1="180" y1="140" x2="60" y2="20"></line><line class="bisector" x1="20" y1="140" x2="140" y2="20"></line><circle class="point" cx="100" cy="60" r="4"></circle><text class="label" x="5" y="155">O</text><text class="label" x="205" y="155">A</text><text class="label" x="5" y="15">B</text><text class="label" x="125" y="75">c</text><text class="label" x="105" y="55">C</text></svg><p><strong>Ответ:</strong> Искомая точка $C$ является точкой пересечения биссектрисы угла $AOB$ и прямой $c$. Она расположена в узле сетки с координатами $(2,2)$, если принять точку $O$ за начало координат.</p><p><strong>б)</strong> Вершина угла $O$ находится в начале координат. Сторона $OA$ проходит через точку $(4,1)$, а сторона $OB$ — через точку $(1,4)$. Угол $AOB$ симметричен относительно прямой $y=x$, так как координаты точек на его сторонах взаимно обратны. Следовательно, биссектрисой угла $AOB$ является прямая $y=x$. Прямая $c$ — это горизонтальная прямая $y=2$. Точка пересечения $C$ биссектрисы $y=x$ и прямой $c$ ($y=2$) имеет координаты $C(2,2)$. Построим биссектрису угла $AOB$ и найдем ее пересечение с прямой $c$.</p><svg width="220" height="220" viewBox="0 0 220 220" xmlns="http://www.w3.org/2000/svg"><defs><style> .grid { stroke: #d3d3d3; stroke-width: 1; } .line { stroke: #d9534f; stroke-width: 2; } .bisector { stroke: #428bca; stroke-width: 2; stroke-dasharray: 4 4; } .point { fill: #d9534f; } .label { font-family: sans-serif; font-size: 16px; } </style></defs><path class="grid" d="M 60 20 V 180 M 100 20 V 180 M 140 20 V 180 M 180 20 V 180 M 20 60 H 180 M 20 100 H 180 M 20 140 H 180"></path><path class="grid" d="M 20 20 H 180 V 180 H 20 Z" fill="none" stroke="#d3d3d3" stroke-width="1"></path><line class="line" x1="20" y1="180" x2="180" y2="140"></line><line class="line" x1="20" y1="180" x2="60" y2="20"></line><line class="line" x1="0" y1="100" x2="220" y2="100"></line><line class="bisector" x1="20" y1="180" x2="180" y2="20"></line><circle class="point" cx="100" cy="100" r="4"></circle><text class="label" x="5" y="195">O</text><text class="label" x="185" y="135">A</text><text class="label" x="50" y="15">B</text><text class="label" x="125" y="95">c</text><text class="label" x="105" y="90">C</text></svg><p><strong>Ответ:</strong> Искомая точка $C$ является точкой пересечения биссектрисы угла $AOB$ и прямой $c$. Она расположена в узле сетки с координатами $(2,2)$, если принять точку $O$ за начало координат.</p><p><strong>в)</strong> Вершина угла $O$ находится в начале координат. Сторона $OA$ проходит через точку $(4,2)$, а сторона $OB$ — через точку $(2,4)$. Угловые коэффициенты сторон равны $k_{OA} = 2/4 = 1/2$ и $k_{OB} = 4/2 = 2$. Так как $k_{OB} = 1/k_{OA}$, угол $AOB$ симметричен относительно прямой $y=x$, которая и является его биссектрисой. Прямая $c$ — это горизонтальная прямая $y=3$. Точка пересечения $C$ биссектрисы $y=x$ и прямой $c$ ($y=3$) имеет координаты $C(3,3)$. Построим биссектрису угла $AOB$ и найдем ее пересечение с прямой $c$.</p><svg width="220" height="220" viewBox="0 0 220 220" xmlns="http://www.w3.org/2000/svg"><defs><style> .grid { stroke: #d3d3d3; stroke-width: 1; } .line { stroke: #d9534f; stroke-width: 2; } .bisector { stroke: #428bca; stroke-width: 2; stroke-dasharray: 4 4; } .point { fill: #d9534f; } .label { font-family: sans-serif; font-size: 16px; } </style></defs><path class="grid" d="M 60 20 V 180 M 100 20 V 180 M 140 20 V 180 M 180 20 V 180 M 20 60 H 180 M 20 100 H 180 M 20 140 H 180"></path><path class="grid" d="M 20 20 H 180 V 180 H 20 Z" fill="none" stroke="#d3d3d3" stroke-width="1"></path><line class="line" x1="20" y1="180" x2="180" y2="100"></line><line class="line" x1="20" y1="180" x2="100" y2="20"></line><line class="line" x1="0" y1="60" x2="220" y2="60"></line><line class="bisector" x1="20" y1="180" x2="180" y2="20"></line><circle class="point" cx="140" cy="60" r="4"></circle><text class="label" x="5" y="195">O</text><text class="label" x="185" y="95">A</text><text class="label" x="90" y="15">B</text><text class="label" x="165" y="55">c</text><text class="label" x="145" y="50">C</text></svg><p><strong>Ответ:</strong> Искомая точка $C$ является точкой пересечения биссектрисы угла $AOB$ и прямой $c$. Она расположена в узле сетки с координатами $(3,3)$, если принять точку $O$ за начало координат.</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" => "1129870" "type" => "task" ] "previous" => array:2 [ "refs" => "1129868" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1124 #items: array:1 [ 0 => App\Models\Book {#1032} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1139 #items: array:1 [ 0 => App\Models\BookPage {#1138 #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" => 1114439 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "122" "field_url" => "/7-klass/geometrija/smirnova-uchebnik/page-122" "field_display_title" => "122" "field_folder" => "1" "field_image_name" => "122" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1140 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1170 #items: array:1 [ 0 => App\Models\Book {#1141 #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 {#1143 #items: array:1 [ 0 => App\Models\Term {#1142 …30} ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1144 #items: array:1 [ 0 => App\Models\Term {#1145 …30} ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1146 #items: array:1 [ 0 => App\Models\Term {#1147 …30} ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1148 #items: array:2 [ 0 => App\Models\Term {#1149 …30} 1 => App\Models\Term {#1150 …30} ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1151 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1152 #items: array:1 [ 0 => App\Models\Term {#1153 …30} ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1154 #items: array:1 [ 0 => App\Models\Term {#1155 …30} ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1156 #items: array:1 [ 0 => App\Models\Term {#1157 …30} ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1158 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1159 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1160 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1161 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1162 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1163 #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 {#1164 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1165 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1166 #items: [] #escapeWhenCastingToString: false } "field_url" => "/7-klass/geometrija/smirnova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1167 #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 {#1143} "field_class" => Illuminate\Database\Eloquent\Collection {#1144} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1146} "field_author" => Illuminate\Database\Eloquent\Collection {#1148} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1151} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1152} "field_country" => Illuminate\Database\Eloquent\Collection {#1154} "field_city" => Illuminate\Database\Eloquent\Collection {#1156} "field_series" => Illuminate\Database\Eloquent\Collection {#1158} "field_umk" => Illuminate\Database\Eloquent\Collection {#1159} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1160} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1161} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1162} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1163} "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 {#1164} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1165} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1166} "field_url" => "/7-klass/geometrija/smirnova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1167} "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 {#1168 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1179 #items: array:1 [ 0 => App\Models\Element {#1178 #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" => 1290626 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1188 #items: array:1 [ 0 => App\Models\Edition {#1180 …30} ] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1114439" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "122-1.jpg" "alt" => null "width" => "1908" "height" => 2818 "path" => "/media/geometrija_07/Smirnova-u17/0-1/122-1.webp?ts=1753260190" ] ] ] #original: array:6 [ "id" => 1290626 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1188} "book_page" => array:2 [ "refs" => "1114439" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "122-1.jpg" "alt" => null "width" => "1908" "height" => 2818 "path" => "/media/geometrija_07/Smirnova-u17/0-1/122-1.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" => "1114440" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1114438" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1686 #items: array:5 [ 0 => App\Models\Task {#1171 #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" => 1129868 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "122" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/21-9" "field_display_title" => "21.9" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1169 #items: array:1 [ 0 => App\Models\Term {#1172 …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 {#1175 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1173 #items: array:1 [ 0 => App\Models\Branch {#1176 …30} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1217 #items: array:2 [ 0 => App\Models\Branch {#1176 …30} 1 => App\Models\Branch {#1218 …30} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1255 #items: array:3 [ 0 => App\Models\Element {#1256 …30} 1 => App\Models\Element {#1264 …30} 2 => App\Models\Element {#1272 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1129869" "type" => "task" ] "previous" => array:2 [ "refs" => "1129867" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1280 #items: array:1 [ 0 => App\Models\Book {#1189 …30} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1114439" "type" => "book_page" ] ] #original: array:24 [ "id" => 1129868 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "122" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/21-9" "field_display_title" => "21.9" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1169} "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 {#1175} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1173} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1217} "content" => Illuminate\Database\Eloquent\Collection {#1255} "next" => array:2 [ "refs" => "1129869" "type" => "task" ] "previous" => array:2 [ "refs" => "1129867" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1280} "page" => array:2 [ "refs" => "1114439" "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 {#1281 #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" => 1129869 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "122" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/21-10" "field_display_title" => "21.10" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1282 #items: array:1 [ 0 => App\Models\Term {#1283 …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 {#1284 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1285 #items: array:1 [ 0 => App\Models\Branch {#1286 …30} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1318 #items: array:2 [ 0 => App\Models\Branch {#1286 …30} 1 => App\Models\Branch {#1319 …30} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1356 #items: array:3 [ 0 => App\Models\Element {#1357 …30} 1 => App\Models\Element {#1365 …30} 2 => App\Models\Element {#1373 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1129870" "type" => "task" ] "previous" => array:2 [ "refs" => "1129868" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1381 #items: array:1 [ 0 => App\Models\Book {#1290 …30} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1114439" "type" => "book_page" ] ] #original: array:24 [ "id" => 1129869 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "122" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/21-10" "field_display_title" => "21.10" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1282} "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 {#1284} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1285} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1318} "content" => Illuminate\Database\Eloquent\Collection {#1356} "next" => array:2 [ "refs" => "1129870" "type" => "task" ] "previous" => array:2 [ "refs" => "1129868" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1381} "page" => array:2 [ "refs" => "1114439" "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 {#1382 #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" => 1129870 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "122" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/21-11" "field_display_title" => "21.11" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1383 #items: array:1 [ 0 => App\Models\Term {#1384 …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 {#1385 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1386 #items: array:1 [ 0 => App\Models\Branch {#1387 …30} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1419 #items: array:2 [ 0 => App\Models\Branch {#1387 …30} 1 => App\Models\Branch {#1420 …30} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1457 #items: array:3 [ 0 => App\Models\Element {#1458 …30} 1 => App\Models\Element {#1466 …30} 2 => App\Models\Element {#1474 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1129871" "type" => "task" ] "previous" => array:2 [ "refs" => "1129869" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1482 #items: array:1 [ 0 => App\Models\Book {#1391 …30} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1114439" "type" => "book_page" ] ] #original: array:24 [ "id" => 1129870 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "122" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/21-11" "field_display_title" => "21.11" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1383} "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 {#1385} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1386} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1419} "content" => Illuminate\Database\Eloquent\Collection {#1457} "next" => array:2 [ "refs" => "1129871" "type" => "task" ] "previous" => array:2 [ "refs" => "1129869" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1482} "page" => array:2 [ "refs" => "1114439" "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 {#1483 #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" => 1129871 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "122" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/21-12" "field_display_title" => "21.12" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1484 #items: array:1 [ 0 => App\Models\Term {#1485 …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 {#1486 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1487 #items: array:1 [ 0 => App\Models\Branch {#1488 …30} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1520 #items: array:2 [ 0 => App\Models\Branch {#1488 …30} 1 => App\Models\Branch {#1521 …30} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1558 #items: array:3 [ 0 => App\Models\Element {#1559 …30} 1 => App\Models\Element {#1567 …30} 2 => App\Models\Element {#1575 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1129872" "type" => "task" ] "previous" => array:2 [ "refs" => "1129870" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1583 #items: array:1 [ 0 => App\Models\Book {#1492 …30} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1114439" "type" => "book_page" ] ] #original: array:24 [ "id" => 1129871 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "122" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/21-12" "field_display_title" => "21.12" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1484} "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 {#1486} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1487} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1520} "content" => Illuminate\Database\Eloquent\Collection {#1558} "next" => array:2 [ "refs" => "1129872" "type" => "task" ] "previous" => array:2 [ "refs" => "1129870" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1583} "page" => array:2 [ "refs" => "1114439" "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 {#1584 #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" => 1129872 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "122" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/21-13" "field_display_title" => "21.13" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1585 #items: array:1 [ …1] #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 {#1587 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1588 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1621 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1659 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1129873" "type" => "task" ] "previous" => array:2 [ "refs" => "1129871" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1684 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1114439" "type" => "book_page" ] ] #original: array:24 [ "id" => 1129872 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "122" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/21-13" "field_display_title" => "21.13" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1585} "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 {#1587} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1588} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1621} "content" => Illuminate\Database\Eloquent\Collection {#1659} "next" => array:2 [ "refs" => "1129873" "type" => "task" ] "previous" => array:2 [ "refs" => "1129871" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1684} "page" => array:2 [ "refs" => "1114439" "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" => 1114439 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "122" "field_url" => "/7-klass/geometrija/smirnova-uchebnik/page-122" "field_display_title" => "122" "field_folder" => "1" "field_image_name" => "122" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1140} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1170} "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 {#1168} "content" => Illuminate\Database\Eloquent\Collection {#1179} "next" => array:2 [ "refs" => "1114440" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1114438" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1686} ] #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" => 1129869 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "122" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/21-10" "field_display_title" => "21.10" "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 {#1099} "next" => array:2 [ "refs" => "1129870" "type" => "task" ] "previous" => array:2 [ "refs" => "1129868" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1124} "page" => array:2 [ "refs" => "1114439" "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 => "*" ] }
№21.10 (с. 122)
Условие. №21.10 (с. 122)
скриншот условия
21.10. На прямой c отметьте точку C, равноудаленную от сторон угла AOB (рис. 21.9).
a)
б)
в)
Рис. 21.9
Решение. №21.10 (с. 122)
Решение 2. №21.10 (с. 122)
Для решения задачи воспользуемся свойством биссектрисы угла: любая точка, лежащая на биссектрисе угла, равноудалена от его сторон. Следовательно, искомая точка $C$ должна лежать на биссектрисе угла $AOB$. Так как по условию точка $C$ также должна лежать на прямой $c$, то она является точкой пересечения биссектрисы угла $AOB$ и прямой $c$.
а) Угол $AOB$ — прямой, его вершина $O$ находится в начале координат, а стороны $OA$ и $OB$ лежат на осях координат. Биссектриса прямого угла делит его на два угла по $45^\circ$, ее уравнение $y=x$. Прямая $c$ проходит через точки с координатами $(4,0)$ и $(1,3)$, ее уравнение $y = -x+4$. Точка пересечения $C$ биссектрисы и прямой $c$ имеет координаты, удовлетворяющие обоим уравнениям. Для системы уравнений $y=x$ и $y=-x+4$ решением является точка $C(2,2)$. Построим биссектрису угла $AOB$ (синяя пунктирная линия) и отметим точку ее пересечения с прямой $c$.
Ответ: Искомая точка $C$ является точкой пересечения биссектрисы угла $AOB$ и прямой $c$. Она расположена в узле сетки с координатами $(2,2)$, если принять точку $O$ за начало координат.
б) Вершина угла $O$ находится в начале координат. Сторона $OA$ проходит через точку $(4,1)$, а сторона $OB$ — через точку $(1,4)$. Угол $AOB$ симметричен относительно прямой $y=x$, так как координаты точек на его сторонах взаимно обратны. Следовательно, биссектрисой угла $AOB$ является прямая $y=x$. Прямая $c$ — это горизонтальная прямая $y=2$. Точка пересечения $C$ биссектрисы $y=x$ и прямой $c$ ($y=2$) имеет координаты $C(2,2)$. Построим биссектрису угла $AOB$ и найдем ее пересечение с прямой $c$.
Ответ: Искомая точка $C$ является точкой пересечения биссектрисы угла $AOB$ и прямой $c$. Она расположена в узле сетки с координатами $(2,2)$, если принять точку $O$ за начало координат.
в) Вершина угла $O$ находится в начале координат. Сторона $OA$ проходит через точку $(4,2)$, а сторона $OB$ — через точку $(2,4)$. Угловые коэффициенты сторон равны $k_{OA} = 2/4 = 1/2$ и $k_{OB} = 4/2 = 2$. Так как $k_{OB} = 1/k_{OA}$, угол $AOB$ симметричен относительно прямой $y=x$, которая и является его биссектрисой. Прямая $c$ — это горизонтальная прямая $y=3$. Точка пересечения $C$ биссектрисы $y=x$ и прямой $c$ ($y=3$) имеет координаты $C(3,3)$. Построим биссектрису угла $AOB$ и найдем ее пересечение с прямой $c$.
Ответ: Искомая точка $C$ является точкой пересечения биссектрисы угла $AOB$ и прямой $c$. Она расположена в узле сетки с координатами $(3,3)$, если принять точку $O$ за начало координат.
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 21.10 расположенного на странице 122 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 №21.10 (с. 122), авторов: Смирнов (Владимир Алексеевич), Туяков (Есенкельды Алыбаевич), учебного пособия издательства Мектеп.