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Номер 21.9, 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 21.9, страница 122.
App\Models\Task {#1003 // 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" => 1129868 "created_at" => "2026-04-10 00:54:11" "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 {#1009 #items: array:1 [ 0 => App\Models\Term {#1008 #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 00:54:11" "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 00:54:11" "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 {#1007 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1018 #items: array:1 [ 0 => App\Models\Branch {#1017 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:23 [ "id" => 1129379 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "field_display_title" => "Окружность. Геометрические построения" "field_branch_order" => "4" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1022 #items: array:1 [ 0 => App\Models\Term {#1019 #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 00:54:11" "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 00:54:11" "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" => [] "book" => Illuminate\Database\Eloquent\Collection {#1050 #items: array:1 [ 0 => App\Models\Book {#1021 #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 00:54:11" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1023 #items: array:1 [ 0 => App\Models\Term {#1020 #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 00:54:11" "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 00:54:11" "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 {#1024 #items: array:1 [ 0 => App\Models\Term {#1025 #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 00:54:11" "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 00:54:11" "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 {#1026 #items: array:1 [ 0 => App\Models\Term {#1027 #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 00:54:11" "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 00:54:11" "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 {#1028 #items: array:2 [ 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:12 [ "id" => 5801 "created_at" => "2026-04-10 00:54:11" "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 00:54:11" "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 {#1030 #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 00:54:11" "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 00:54:11" "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 {#1031 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1032 #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:10 [ "id" => 6671 "created_at" => "2026-04-10 00:54:11" "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 00:54:11" "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 {#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:6 [ "id" => 6994 "created_at" => "2026-04-10 00:54:11" "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 00:54:11" "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 {#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" => 6995 "created_at" => "2026-04-10 00:54:11" "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 00:54:11" "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 {#1038 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1039 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1040 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1041 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1042 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1043 #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" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_07/Smirnova-u17/covers/cover1.webp" "alt" => "" "width" => "1632" "height" => "2472" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1044 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1045 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1046 #items: [] #escapeWhenCastingToString: false } "field_url" => "/7-klass/geometrija/smirnova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1047 #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 00:54:11" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1023} "field_class" => Illuminate\Database\Eloquent\Collection {#1024} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1026} "field_author" => Illuminate\Database\Eloquent\Collection {#1028} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1031} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1032} "field_country" => Illuminate\Database\Eloquent\Collection {#1034} "field_city" => Illuminate\Database\Eloquent\Collection {#1036} "field_series" => Illuminate\Database\Eloquent\Collection {#1038} "field_umk" => Illuminate\Database\Eloquent\Collection {#1039} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1040} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1041} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1042} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1043} "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" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_07/Smirnova-u17/covers/cover1.webp" "alt" => "" "width" => "1632" "height" => "2472" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1044} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1045} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1046} "field_url" => "/7-klass/geometrija/smirnova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1047} "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 {#1048 #items: [] #escapeWhenCastingToString: false } ] #original: array:23 [ "id" => 1129379 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "field_display_title" => "Окружность. Геометрические построения" "field_branch_order" => "4" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1022} "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" => [] "book" => Illuminate\Database\Eloquent\Collection {#1050} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1048} ] #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 {#1077 #items: array:2 [ 0 => App\Models\Branch {#1085 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:23 [ "id" => 1129379 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "field_display_title" => "Окружность. Геометрические построения" "field_branch_order" => "4" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1086 #items: array:1 [ 0 => App\Models\Term {#1019} ] #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" => [] "book" => Illuminate\Database\Eloquent\Collection {#1087 #items: array:1 [ 0 => App\Models\Book {#1021} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1088 #items: [] #escapeWhenCastingToString: false } ] #original: array:23 [ "id" => 1129379 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "field_display_title" => "Окружность. Геометрические построения" "field_branch_order" => "4" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1086} "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" => [] "book" => Illuminate\Database\Eloquent\Collection {#1087} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1088} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Branch {#1089 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:23 [ "id" => 1129383 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "field_display_title" => "Геометрические места точек" "field_branch_order" => "21" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1093 #items: array:1 [ 0 => App\Models\Term {#1090 #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 00:54:11" "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 00:54:11" "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" => [] "book" => Illuminate\Database\Eloquent\Collection {#1092 #items: array:1 [ 0 => App\Models\Book {#1021} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1091 #items: array:1 [ 0 => App\Models\Branch {#1017} ] #escapeWhenCastingToString: false } ] #original: array:23 [ "id" => 1129383 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "field_display_title" => "Геометрические места точек" "field_branch_order" => "21" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1093} "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" => [] "book" => Illuminate\Database\Eloquent\Collection {#1092} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1091} ] #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 {#1074 #items: array:3 [ 0 => App\Models\Element {#1067 #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" => 1319997 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1059 #items: array:1 [ 0 => App\Models\Edition {#1066 #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 00:54:11" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1064 #items: array:1 [ 0 => App\Models\Term {#1065 #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 {#1062 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1063 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1060 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5009 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1064} "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 {#1062} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1063} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1060} "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" => "1129868" "type" => "task" ] "text" => "<p><strong>21.9.</strong> Изобразите геометрическое место внутренних точек угла $AOB$, равноудаленных от его сторон (рис. 21.8).</p><p><strong>а)</strong></p><p><strong>б)</strong></p><p><strong>в)</strong></p><p>Рис. 21.8</p>" "img" => array:1 [ 0 => array:5 [ "name" => "9-1.jpg" "alt" => null "width" => "1711" "height" => 815 "path" => "/media/geometrija_07/Smirnova-u17/0-21/9-1.webp?ts=1753982473" ] ] ] #original: array:7 [ "id" => 1319997 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1059} "task" => array:2 [ "refs" => "1129868" "type" => "task" ] "text" => "<p><strong>21.9.</strong> Изобразите геометрическое место внутренних точек угла $AOB$, равноудаленных от его сторон (рис. 21.8).</p><p><strong>а)</strong></p><p><strong>б)</strong></p><p><strong>в)</strong></p><p>Рис. 21.8</p>" "img" => array:1 [ 0 => array:5 [ "name" => "9-1.jpg" "alt" => null "width" => "1711" "height" => 815 "path" => "/media/geometrija_07/Smirnova-u17/0-21/9-1.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 {#1061 #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" => 1320433 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1051 #items: array:1 [ 0 => App\Models\Edition {#1058 #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 00:54:11" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1054 #items: array:1 [ 0 => App\Models\Term {#1057 #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 {#1056 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1055 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1052 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5010 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1054} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1056} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1055} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1052} "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" => "1129868" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "9-1.jpg" "alt" => null "width" => "1501" "height" => 1217 "path" => "/media/geometrija_07/Smirnova-u17/1-21/9-1.webp?ts=1753982588" ] ] ] #original: array:6 [ "id" => 1320433 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1051} "task" => array:2 [ "refs" => "1129868" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "9-1.jpg" "alt" => null "width" => "1501" "height" => 1217 "path" => "/media/geometrija_07/Smirnova-u17/1-21/9-1.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 {#1053 #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" => 1541695 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1080 #items: array:1 [ 0 => App\Models\Edition {#1049 #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 00:54:11" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1013 #items: array:1 [ 0 => App\Models\Term {#1015 #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 {#1006 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1011 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1076 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5631 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1013} "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 {#1006} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1011} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1076} "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" => "1129868" "type" => "task" ] "text" => "<p>Геометрическое место внутренних точек угла, равноудаленных от его сторон, — это биссектриса этого угла. Биссектриса — это луч, который исходит из вершины угла и делит его на два равных угла. Для каждого из представленных углов необходимо построить его биссектрису.</p><p><strong>а)</strong>Угол $AOB$ — прямой, так как его стороны лежат на перпендикулярных линиях сетки. Его градусная мера равна $90^\circ$. Биссектриса прямого угла делит его на два угла по $45^\circ$. В данной системе координат, если принять точку $O$ за начало, это будет луч, проходящий по диагоналям клеток сетки. На рисунке искомое геометрическое место точек изображено синим лучом.</p><svg width="180" height="180" viewBox="0 0 110 110"><style> .grid { stroke: #d3e5f7; stroke-width: 1; } .angle { stroke: #e53935; stroke-width: 2; fill: none; } .bisector { stroke: #1e88e5; stroke-width: 2; fill: none; } .label { font-family: sans-serif; font-size: 14px; fill: black; } </style><path class="grid" d="M 10 10 H 90 M 10 30 H 90 M 10 50 H 90 M 10 70 H 90 M 10 90 H 90 M 10 10 V 90 M 30 10 V 90 M 50 10 V 90 M 70 10 V 90 M 90 10 V 90"></path><path class="angle" d="M 90 10 L 10 10 L 10 90"></path><path class="bisector" d="M 10 10 L 90 90"></path><text x="5" y="10" class="label">O</text><text x="95" y="10" class="label">A</text><text x="5" y="98" class="label">B</text></svg><p>Ответ: Искомое геометрическое место точек — биссектриса угла $AOB$, изображенная на рисунке синим лучом.</p><p><strong>б)</strong>Чтобы построить биссектрису неразвернутого угла $AOB$, можно найти луч, исходящий из вершины $O$, все точки которого находятся на равном расстоянии от сторон $OA$ и $OB$. Примем точку $O$ за начало координат (0,0). Луч $OA$ проходит через точку с координатами (4,1). Луч $OB$ проходит через точку с координатами (2,4). Можно аналитически или с помощью геометрических построений показать, что биссектриса этого угла будет проходить через узел сетки с координатами (5,4) относительно точки $O$. На рисунке искомое геометрическое место точек изображено синим лучом.</p><svg width="180" height="180" viewBox="0 0 110 110"><style> .grid { stroke: #d3e5f7; stroke-width: 1; } .angle { stroke: #e53935; stroke-width: 2; fill: none; } .bisector { stroke: #1e88e5; stroke-width: 2; fill: none; } .label { font-family: sans-serif; font-size: 14px; fill: black; } </style><path class="grid" d="M 10 10 H 90 M 10 30 H 90 M 10 50 H 90 M 10 70 H 90 M 10 90 H 90 M 10 10 V 90 M 30 10 V 90 M 50 10 V 90 M 70 10 V 90 M 90 10 V 90"></path><path class="angle" d="M 90 30 L 10 10 L 50 90"></path><path class="bisector" d="M 10 10 L 90 74"></path><text x="5" y="10" class="label">O</text><text x="95" y="35" class="label">A</text><text x="50" y="100" class="label">B</text></svg><p>Ответ: Искомое геометрическое место точек — биссектриса угла $AOB$, изображенная на рисунке синим лучом.</p><p><strong>в)</strong>Примем точку $O$ за начало локальной системы координат. Луч $OA$ проходит через точку (3,1), его угловой коэффициент $k_{OA} = 1/3$. Луч $OB$ проходит через точку (-1,3), его угловой коэффициент $k_{OB} = -3$. Поскольку произведение угловых коэффициентов $k_{OA} \cdot k_{OB} = (1/3) \cdot (-3) = -1$, то стороны угла $OA$ и $OB$ перпендикулярны, и угол $AOB$ является прямым ($90^\circ$). Биссектриса делит этот угол на два равных угла по $45^\circ$. Биссектриса данного прямого угла будет иметь угловой коэффициент $k=2$. Этот луч выходит из точки $O$ и проходит через узел сетки с координатами (1,2) относительно $O$. На рисунке искомое геометрическое место точек изображено синим лучом.</p><svg width="180" height="150" viewBox="0 0 120 100"><style> .grid { stroke: #d3e5f7; stroke-width: 1; } .angle { stroke: #e53935; stroke-width: 2; fill: none; } .bisector { stroke: #1e88e5; stroke-width: 2; fill: none; } .label { font-family: sans-serif; font-size: 14px; fill: black; } </style><path class="grid" d="M 10 10 H 110 M 10 30 H 110 M 10 50 H 110 M 10 70 H 110 M 10 90 H 110 M 10 10 V 90 M 30 10 V 90 M 50 10 V 90 M 70 10 V 90 M 90 10 V 90 M 110 10 V 90"></path><path class="angle" d="M 110 30 L 50 10 L 30 70"></path><path class="bisector" d="M 50 10 L 90 90"></path><text x="48" y="5" class="label">O</text><text x="112" y="35" class="label">A</text><text x="25" y="80" class="label">B</text></svg><p>Ответ: Искомое геометрическое место точек — биссектриса угла $AOB$, изображенная на рисунке синим лучом.</p>" ] #original: array:6 [ "id" => 1541695 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1080} "task" => array:2 [ "refs" => "1129868" "type" => "task" ] "text" => "<p>Геометрическое место внутренних точек угла, равноудаленных от его сторон, — это биссектриса этого угла. Биссектриса — это луч, который исходит из вершины угла и делит его на два равных угла. Для каждого из представленных углов необходимо построить его биссектрису.</p><p><strong>а)</strong>Угол $AOB$ — прямой, так как его стороны лежат на перпендикулярных линиях сетки. Его градусная мера равна $90^\circ$. Биссектриса прямого угла делит его на два угла по $45^\circ$. В данной системе координат, если принять точку $O$ за начало, это будет луч, проходящий по диагоналям клеток сетки. На рисунке искомое геометрическое место точек изображено синим лучом.</p><svg width="180" height="180" viewBox="0 0 110 110"><style> .grid { stroke: #d3e5f7; stroke-width: 1; } .angle { stroke: #e53935; stroke-width: 2; fill: none; } .bisector { stroke: #1e88e5; stroke-width: 2; fill: none; } .label { font-family: sans-serif; font-size: 14px; fill: black; } </style><path class="grid" d="M 10 10 H 90 M 10 30 H 90 M 10 50 H 90 M 10 70 H 90 M 10 90 H 90 M 10 10 V 90 M 30 10 V 90 M 50 10 V 90 M 70 10 V 90 M 90 10 V 90"></path><path class="angle" d="M 90 10 L 10 10 L 10 90"></path><path class="bisector" d="M 10 10 L 90 90"></path><text x="5" y="10" class="label">O</text><text x="95" y="10" class="label">A</text><text x="5" y="98" class="label">B</text></svg><p>Ответ: Искомое геометрическое место точек — биссектриса угла $AOB$, изображенная на рисунке синим лучом.</p><p><strong>б)</strong>Чтобы построить биссектрису неразвернутого угла $AOB$, можно найти луч, исходящий из вершины $O$, все точки которого находятся на равном расстоянии от сторон $OA$ и $OB$. Примем точку $O$ за начало координат (0,0). Луч $OA$ проходит через точку с координатами (4,1). Луч $OB$ проходит через точку с координатами (2,4). Можно аналитически или с помощью геометрических построений показать, что биссектриса этого угла будет проходить через узел сетки с координатами (5,4) относительно точки $O$. На рисунке искомое геометрическое место точек изображено синим лучом.</p><svg width="180" height="180" viewBox="0 0 110 110"><style> .grid { stroke: #d3e5f7; stroke-width: 1; } .angle { stroke: #e53935; stroke-width: 2; fill: none; } .bisector { stroke: #1e88e5; stroke-width: 2; fill: none; } .label { font-family: sans-serif; font-size: 14px; fill: black; } </style><path class="grid" d="M 10 10 H 90 M 10 30 H 90 M 10 50 H 90 M 10 70 H 90 M 10 90 H 90 M 10 10 V 90 M 30 10 V 90 M 50 10 V 90 M 70 10 V 90 M 90 10 V 90"></path><path class="angle" d="M 90 30 L 10 10 L 50 90"></path><path class="bisector" d="M 10 10 L 90 74"></path><text x="5" y="10" class="label">O</text><text x="95" y="35" class="label">A</text><text x="50" y="100" class="label">B</text></svg><p>Ответ: Искомое геометрическое место точек — биссектриса угла $AOB$, изображенная на рисунке синим лучом.</p><p><strong>в)</strong>Примем точку $O$ за начало локальной системы координат. Луч $OA$ проходит через точку (3,1), его угловой коэффициент $k_{OA} = 1/3$. Луч $OB$ проходит через точку (-1,3), его угловой коэффициент $k_{OB} = -3$. Поскольку произведение угловых коэффициентов $k_{OA} \cdot k_{OB} = (1/3) \cdot (-3) = -1$, то стороны угла $OA$ и $OB$ перпендикулярны, и угол $AOB$ является прямым ($90^\circ$). Биссектриса делит этот угол на два равных угла по $45^\circ$. Биссектриса данного прямого угла будет иметь угловой коэффициент $k=2$. Этот луч выходит из точки $O$ и проходит через узел сетки с координатами (1,2) относительно $O$. На рисунке искомое геометрическое место точек изображено синим лучом.</p><svg width="180" height="150" viewBox="0 0 120 100"><style> .grid { stroke: #d3e5f7; stroke-width: 1; } .angle { stroke: #e53935; stroke-width: 2; fill: none; } .bisector { stroke: #1e88e5; stroke-width: 2; fill: none; } .label { font-family: sans-serif; font-size: 14px; fill: black; } </style><path class="grid" d="M 10 10 H 110 M 10 30 H 110 M 10 50 H 110 M 10 70 H 110 M 10 90 H 110 M 10 10 V 90 M 30 10 V 90 M 50 10 V 90 M 70 10 V 90 M 90 10 V 90 M 110 10 V 90"></path><path class="angle" d="M 110 30 L 50 10 L 30 70"></path><path class="bisector" d="M 50 10 L 90 90"></path><text x="48" y="5" class="label">O</text><text x="112" y="35" class="label">A</text><text x="25" y="80" class="label">B</text></svg><p>Ответ: Искомое геометрическое место точек — биссектриса угла $AOB$, изображенная на рисунке синим лучом.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] 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false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1290626 "created_at" => "2026-04-10 00:54:11" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1101 #items: array:1 [ 0 => App\Models\Edition {#1066} ] #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 00:54:11" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1101} "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: [] 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1129869 "created_at" => "2026-04-10 00:54:11" "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 {#1189 #items: array:1 [ 0 => App\Models\Term {#1008} ] #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 {#1190 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1191 #items: array:1 [ 0 => App\Models\Branch {#1085} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1192 #items: array:2 [ 0 => App\Models\Branch {#1085} 1 => App\Models\Branch {#1089} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1195 #items: array:3 [ 0 => App\Models\Element {#1204 #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 [ …7] #original: array:7 [ …7] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Element {#1206 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false 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{#1008} ] #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 {#1214 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1216 #items: array:1 [ 0 => App\Models\Branch {#1085} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1212 #items: array:2 [ 0 => App\Models\Branch {#1085} 1 => App\Models\Branch {#1089} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1211 #items: array:3 [ 0 => App\Models\Element {#1232 …30} 1 => App\Models\Element {#1234 …30} 2 => App\Models\Element {#1236 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1129872" "type" => "task" ] "previous" => 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"refs" => "1129867" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1070} "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 => "*" ] }
№21.9 (с. 122)
Условие. №21.9 (с. 122)
скриншот условия
21.9. Изобразите геометрическое место внутренних точек угла $AOB$, равноудаленных от его сторон (рис. 21.8).
а)
б)
в)
Рис. 21.8
Решение. №21.9 (с. 122)
Решение 2. №21.9 (с. 122)
Геометрическое место внутренних точек угла, равноудаленных от его сторон, — это биссектриса этого угла. Биссектриса — это луч, который исходит из вершины угла и делит его на два равных угла. Для каждого из представленных углов необходимо построить его биссектрису.
а)Угол $AOB$ — прямой, так как его стороны лежат на перпендикулярных линиях сетки. Его градусная мера равна $90^\circ$. Биссектриса прямого угла делит его на два угла по $45^\circ$. В данной системе координат, если принять точку $O$ за начало, это будет луч, проходящий по диагоналям клеток сетки. На рисунке искомое геометрическое место точек изображено синим лучом.
Ответ: Искомое геометрическое место точек — биссектриса угла $AOB$, изображенная на рисунке синим лучом.
б)Чтобы построить биссектрису неразвернутого угла $AOB$, можно найти луч, исходящий из вершины $O$, все точки которого находятся на равном расстоянии от сторон $OA$ и $OB$. Примем точку $O$ за начало координат (0,0). Луч $OA$ проходит через точку с координатами (4,1). Луч $OB$ проходит через точку с координатами (2,4). Можно аналитически или с помощью геометрических построений показать, что биссектриса этого угла будет проходить через узел сетки с координатами (5,4) относительно точки $O$. На рисунке искомое геометрическое место точек изображено синим лучом.
Ответ: Искомое геометрическое место точек — биссектриса угла $AOB$, изображенная на рисунке синим лучом.
в)Примем точку $O$ за начало локальной системы координат. Луч $OA$ проходит через точку (3,1), его угловой коэффициент $k_{OA} = 1/3$. Луч $OB$ проходит через точку (-1,3), его угловой коэффициент $k_{OB} = -3$. Поскольку произведение угловых коэффициентов $k_{OA} \cdot k_{OB} = (1/3) \cdot (-3) = -1$, то стороны угла $OA$ и $OB$ перпендикулярны, и угол $AOB$ является прямым ($90^\circ$). Биссектриса делит этот угол на два равных угла по $45^\circ$. Биссектриса данного прямого угла будет иметь угловой коэффициент $k=2$. Этот луч выходит из точки $O$ и проходит через узел сетки с координатами (1,2) относительно $O$. На рисунке искомое геометрическое место точек изображено синим лучом.
Ответ: Искомое геометрическое место точек — биссектриса угла $AOB$, изображенная на рисунке синим лучом.
Другие задания:
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