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Номер 21.7, страница 121 - 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 21.7, страница 121.
App\Models\Task {#1030 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1129866 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "121" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/21-7" "field_display_title" => "21.7" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #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 {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1071 #items: array:1 [ 0 => App\Models\Branch {#1034 #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 {#1039 #items: array:1 [ 0 => App\Models\Term {#1038 #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 {#1040 #items: array:1 [ 0 => App\Models\Book {#1041 #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 {#1042 #items: array:1 [ 0 => App\Models\Term {#1043 …30} ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1044 #items: array:1 [ 0 => App\Models\Term {#1045 …30} ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:1 [ 0 => App\Models\Term {#1047 …30} ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1048 #items: array:2 [ 0 => App\Models\Term {#1049 …30} 1 => App\Models\Term {#1050 …30} ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1051 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1052 #items: array:1 [ 0 => App\Models\Term {#1053 …30} ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1054 #items: array:1 [ 0 => App\Models\Term {#1055 …30} ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1056 #items: array:1 [ 0 => App\Models\Term {#1057 …30} ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1058 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1059 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1060 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1061 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1062 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1063 #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 {#1064 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1065 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1066 #items: [] #escapeWhenCastingToString: false } "field_url" => "/7-klass/geometrija/smirnova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1067 #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 {#1042} "field_class" => Illuminate\Database\Eloquent\Collection {#1044} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046} "field_author" => Illuminate\Database\Eloquent\Collection {#1048} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1051} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1052} "field_country" => Illuminate\Database\Eloquent\Collection {#1054} "field_city" => Illuminate\Database\Eloquent\Collection {#1056} "field_series" => Illuminate\Database\Eloquent\Collection {#1058} "field_umk" => Illuminate\Database\Eloquent\Collection {#1059} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1060} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1061} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1062} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1063} "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 {#1064} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1065} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1066} "field_url" => "/7-klass/geometrija/smirnova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1067} "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 {#1068 #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 {#1039} "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 {#1040} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1068} ] #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 {#1104 #items: array:2 [ 0 => App\Models\Branch {#1112 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1129379 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Окружность. Геометрические построения" "field_branch_order" => "4" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1116 #items: array:1 [ 0 => App\Models\Term {#1113 #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 {#1144 #items: array:1 [ 0 => App\Models\Book {#1115 #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 {#1117 #items: array:1 [ 0 => App\Models\Term {#1114 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "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 {#1118 #items: array:1 [ 0 => App\Models\Term {#1119 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array: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 {#1120 #items: array:1 [ 0 => App\Models\Term {#1121 #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 {#1122 #items: array:2 [ 0 => App\Models\Term {#1123 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array: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 {#1124 #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 {#1125 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1126 #items: array:1 [ 0 => App\Models\Term {#1127 #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 {#1128 #items: array:1 [ 0 => App\Models\Term {#1129 #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 {#1130 #items: array:1 [ 0 => App\Models\Term {#1131 #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 {#1132 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1133 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1134 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1135 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1136 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1137 #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 {#1138 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1139 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1140 #items: [] #escapeWhenCastingToString: false } "field_url" => "/7-klass/geometrija/smirnova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1141 #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 {#1117} "field_class" => Illuminate\Database\Eloquent\Collection {#1118} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1120} "field_author" => Illuminate\Database\Eloquent\Collection {#1122} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1125} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1126} "field_country" => Illuminate\Database\Eloquent\Collection {#1128} "field_city" => Illuminate\Database\Eloquent\Collection {#1130} "field_series" => Illuminate\Database\Eloquent\Collection {#1132} "field_umk" => Illuminate\Database\Eloquent\Collection {#1133} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1134} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1135} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1136} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1137} "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 {#1138} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1139} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1140} "field_url" => "/7-klass/geometrija/smirnova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1141} "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 {#1142 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1129379 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Окружность. Геометрические построения" "field_branch_order" => "4" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1116} "field_page_start" => "101" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1144} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1142} ] #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 {#1143 #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 {#1148 #items: array:1 [ 0 => App\Models\Term {#1145 #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 {#1147 #items: array:1 [ 0 => App\Models\Book {#1115} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1146 #items: array:1 [ 0 => App\Models\Branch {#1034} ] #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 {#1148} "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 {#1147} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1146} ] #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 {#1101 #items: array:3 [ 0 => App\Models\Element {#1094 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 1319993 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1086 #items: array:1 [ 0 => App\Models\Edition {#1093 #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 {#1091 #items: array:1 [ 0 => App\Models\Term {#1092 …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 {#1089 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1090 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1087 #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 {#1091} "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 {#1089} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1090} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1087} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1129866" "type" => "task" ] "text" => "<p><strong>21.7.</strong> Отметьте точку, равноудаленную от точек $A$, $B$ и $C$ (рис. 21.7).</p><strong>а)</strong><strong>б)</strong><p><strong>в)</strong> Рис. 21.7</p>" "img" => array:1 [ 0 => array:5 [ "name" => "7-1.jpg" "alt" => null "width" => "1763" "height" => 787 "path" => "/media/geometrija_07/Smirnova-u17/0-21/7-1.webp?ts=1753982473" ] ] ] #original: array:7 [ "id" => 1319993 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1086} "task" => array:2 [ "refs" => "1129866" "type" => "task" ] "text" => "<p><strong>21.7.</strong> Отметьте точку, равноудаленную от точек $A$, $B$ и $C$ (рис. 21.7).</p><strong>а)</strong><strong>б)</strong><p><strong>в)</strong> Рис. 21.7</p>" "img" => array:1 [ 0 => array:5 [ "name" => "7-1.jpg" "alt" => null "width" => "1763" "height" => 787 "path" => "/media/geometrija_07/Smirnova-u17/0-21/7-1.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 {#1088 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1320431 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1078 #items: array:1 [ 0 => App\Models\Edition {#1085 #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 {#1081 #items: 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 {#1083 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1082 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1079 #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 {#1081} "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 {#1083} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1082} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1079} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1129866" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "7-1.jpg" "alt" => null "width" => "1501" "height" => 1138 "path" => "/media/geometrija_07/Smirnova-u17/1-21/7-1.webp?ts=1753982588" ] ] ] #original: array:6 [ "id" => 1320431 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1078} "task" => array:2 [ "refs" => "1129866" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "7-1.jpg" "alt" => null "width" => "1501" "height" => 1138 "path" => "/media/geometrija_07/Smirnova-u17/1-21/7-1.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 {#1080 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1541693 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1107 #items: array:1 [ 0 => App\Models\Edition {#1077 #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 {#1076 #items: 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 {#1072 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1070 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1103 #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 {#1076} "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 {#1072} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1070} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1103} "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" => "1129866" "type" => "task" ] "text" => "<p>Для нахождения точки, равноудаленной от трех заданных точек А, В и С, необходимо найти центр окружности, описанной около треугольника АВС. Центр описанной окружности находится в точке пересечения серединных перпендикуляров к сторонам треугольника.</p><p><strong>а)</strong> Введем систему координат, приняв за единицу измерения сторону клетки сетки, а за начало координат — левый нижний узел сетки. В этой системе точки имеют координаты: A(0, 2), B(3, 1), C(3, 3). Найдем серединный перпендикуляр к стороне BC. Так как отрезок BC вертикален, его серединный перпендикуляр будет горизонтальной прямой. Середина отрезка BC имеет координаты $M_{BC}(\frac{3+3}{2}, \frac{1+3}{2}) = (3, 2)$. Уравнение серединного перпендикуляра к BC: $y = 2$. Найдем серединный перпендикуляр к стороне AB. Середина отрезка AB имеет координаты $M_{AB}(\frac{0+3}{2}, \frac{2+1}{2}) = (1.5, 1.5)$. Угловой коэффициент прямой AB равен $k_{AB} = \frac{1-2}{3-0} = -\frac{1}{3}$. Угловой коэффициент серединного перпендикуляра равен $k_{\perp} = -\frac{1}{k_{AB}} = 3$. Уравнение перпендикуляра, проходящего через точку $M_{AB}(1.5, 1.5)$ с угловым коэффициентом 3, имеет вид: $y - 1.5 = 3(x - 1.5)$, что упрощается до $y = 3x - 3$. Для нахождения точки пересечения (обозначим ее O) решим систему уравнений: $$ \begin{cases} y = 2 \\ y = 3x - 3 \end{cases} $$ Подставляя $y=2$ во второе уравнение, получаем $2 = 3x - 3$, откуда $3x = 5$ и $x = \frac{5}{3}$. Таким образом, искомая точка O имеет координаты O($\frac{5}{3}, 2$). На рисунке эта точка отмечена синим крестиком.</p><strong>Ответ:</strong><svg width="180" height="150" viewBox="0 0 180 150" xmlns="http://www.w3.org/2000/svg"><!-- Grid --><defs><pattern id="grid_a" width="30" height="30" patternUnits="userSpaceOnUse"><path d="M 30 0 L 0 0 0 30" fill="none" stroke="lightgray" stroke-width="1"/></pattern></defs><rect x="15" y="15" width="150" height="120" fill="url(#grid_a)"/><!-- Perpendicular Bisectors --><line x1="15" y1="75" x2="165" y2="75" stroke="green" stroke-width="1.5" stroke-dasharray="4 2"/><line x1="45" y1="135" x2="85" y2="15" stroke="green" stroke-width="1.5" stroke-dasharray="4 2"/><!-- Points --><circle cx="15" cy="75" r="3.5" fill="red"/><text x="5" y="79" font-family="sans-serif" font-size="14">A</text><circle cx="105" cy="105" r="3.5" fill="red"/><text x="110" y="115" font-family="sans-serif" font-size="14">B</text><circle cx="105" cy="45" r="3.5" fill="red"/><text x="110" y="49" font-family="sans-serif" font-size="14">C</text><!-- Solution Point O --><line x1="61" y1="71" x2="69" y2="79" stroke="blue" stroke-width="2"/><line x1="61" y1="79" x2="69" y2="71" stroke="blue" stroke-width="2"/></svg><p><strong>б)</strong> Аналогично, введем систему координат с началом в левом нижнем углу. Координаты точек: A(0, 3), B(2, 1), C(3, 3). Найдем серединный перпендикуляр к стороне AC. Отрезок AC горизонтален. Его середина $M_{AC}(\frac{0+3}{2}, \frac{3+3}{2}) = (1.5, 3)$. Серединный перпендикуляр — вертикальная прямая $x = 1.5$. Найдем серединный перпендикуляр к стороне AB. Середина отрезка AB имеет координаты $M_{AB}(\frac{0+2}{2}, \frac{3+1}{2}) = (1, 2)$. Угловой коэффициент прямой AB равен $k_{AB} = \frac{1-3}{2-0} = -1$. Угловой коэффициент серединного перпендикуляра равен $k_{\perp} = -\frac{1}{k_{AB}} = 1$. Уравнение перпендикуляра: $y - 2 = 1(x - 1)$, что упрощается до $y = x + 1$. Найдем точку пересечения O, решив систему: $$ \begin{cases} x = 1.5 \\ y = x + 1 \end{cases} $$ Подставляя $x=1.5$ во второе уравнение, получаем $y = 1.5 + 1 = 2.5$. Искомая точка O имеет координаты O(1.5, 2.5). На рисунке эта точка отмечена синим крестиком.</p><strong>Ответ:</strong><svg width="180" height="150" viewBox="0 0 180 150" xmlns="http://www.w3.org/2000/svg"><!-- Grid --><defs><pattern id="grid_b" width="30" height="30" patternUnits="userSpaceOnUse"><path d="M 30 0 L 0 0 0 30" fill="none" stroke="lightgray" stroke-width="1"/></pattern></defs><rect x="15" y="15" width="150" height="120" fill="url(#grid_b)"/><!-- Perpendicular Bisectors --><line x1="60" y1="15" x2="60" y2="135" stroke="green" stroke-width="1.5" stroke-dasharray="4 2"/><line x1="15" y1="105" x2="105" y2="15" stroke="green" stroke-width="1.5" stroke-dasharray="4 2"/><!-- Points --><circle cx="15" cy="45" r="3.5" fill="red"/><text x="5" y="49" font-family="sans-serif" font-size="14">A</text><circle cx="75" cy="105" r="3.5" fill="red"/><text x="65" y="115" font-family="sans-serif" font-size="14">B</text><circle cx="105" cy="45" r="3.5" fill="red"/><text x="110" y="49" font-family="sans-serif" font-size="14">C</text><!-- Solution Point O --><line x1="56" y1="56" x2="64" y2="64" stroke="blue" stroke-width="2"/><line x1="56" y1="64" x2="64" y2="56" stroke="blue" stroke-width="2"/></svg><p><strong>в)</strong> Снова используем метод серединных перпендикуляров. В системе координат с началом в левом нижнем углу точки имеют координаты: A(0, 1), B(2, 1), C(4, 2). Найдем серединный перпендикуляр к стороне AB. Отрезок AB горизонтален. Его середина $M_{AB}(\frac{0+2}{2}, \frac{1+1}{2}) = (1, 1)$. Серединный перпендикуляр — вертикальная прямая $x = 1$. Найдем серединный перпендикуляр к стороне BC. Середина отрезка BC имеет координаты $M_{BC}(\frac{2+4}{2}, \frac{1+2}{2}) = (3, 1.5)$. Угловой коэффициент прямой BC равен $k_{BC} = \frac{2-1}{4-2} = \frac{1}{2}$. Угловой коэффициент серединного перпендикуляра равен $k_{\perp} = -\frac{1}{k_{BC}} = -2$. Уравнение перпендикуляра: $y - 1.5 = -2(x - 3)$, что упрощается до $y = -2x + 7.5$. Найдем точку пересечения O, решив систему: $$ \begin{cases} x = 1 \\ y = -2x + 7.5 \end{cases} $$ Подставляя $x=1$ во второе уравнение, получаем $y = -2(1) + 7.5 = 5.5$. Искомая точка O имеет координаты O(1, 5.5). Эта точка лежит за пределами исходной сетки, что является допустимым результатом. На рисунке сетка продлена вверх для отображения этой точки.</p><strong>Ответ:</strong><svg width="180" height="210" viewBox="0 0 180 210" xmlns="http://www.w3.org/2000/svg"><!-- Grid --><path d="M 15 15 H 165 M 15 45 H 165 M 15 75 H 165 M 15 105 H 165 M 15 135 H 165 M 15 165 H 165 M 15 195 H 165 M 15 15 V 195 M 45 15 V 195 M 75 15 V 195 M 105 15 V 195 M 135 15 V 195 M 165 15 V 195" fill="none" stroke="lightgray" stroke-width="1"/><!-- Extended Grid --><path d="M 15 15 V -15 M 45 15 V -15 M 75 15 V -15 M 105 15 V -15 M 135 15 V -15 M 165 15 V -15" fill="none" stroke="lightgray" stroke-width="1" stroke-dasharray="2 2"/><!-- Perpendicular Bisectors --><line x1="45" y1="-15" x2="45" y2="195" stroke="green" stroke-width="1.5" stroke-dasharray="4 2"/><line x1="22.5" y1="75" x2="112.5" y2="-105" transform="translate(0, 75)" stroke="green" stroke-width="1.5" stroke-dasharray="4 2"/><line x1="45" y1="30" x2="105" y2="150" stroke="green" stroke-width="1.5" stroke-dasharray="4 2" visibility="hidden"/><!-- Calculation guide line --><!-- Points --><circle cx="15" cy="165" r="3.5" fill="red"/><text x="5" y="169" font-family="sans-serif" font-size="14">A</text><circle cx="75" cy="165" r="3.5" fill="red"/><text x="65" y="175" font-family="sans-serif" font-size="14">B</text><circle cx="135" cy="135" r="3.5" fill="red"/><text x="140" y="139" font-family="sans-serif" font-size="14">C</text><!-- Solution Point O --><line x1="41" y1="26" x2="49" y2="34" stroke="blue" stroke-width="2"/><line x1="41" y1="34" x2="49" y2="26" stroke="blue" stroke-width="2"/></svg>" ] #original: array:6 [ "id" => 1541693 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1107} "task" => array:2 [ "refs" => "1129866" "type" => "task" ] "text" => "<p>Для нахождения точки, равноудаленной от трех заданных точек А, В и С, необходимо найти центр окружности, описанной около треугольника АВС. Центр описанной окружности находится в точке пересечения серединных перпендикуляров к сторонам треугольника.</p><p><strong>а)</strong> Введем систему координат, приняв за единицу измерения сторону клетки сетки, а за начало координат — левый нижний узел сетки. В этой системе точки имеют координаты: A(0, 2), B(3, 1), C(3, 3). Найдем серединный перпендикуляр к стороне BC. Так как отрезок BC вертикален, его серединный перпендикуляр будет горизонтальной прямой. Середина отрезка BC имеет координаты $M_{BC}(\frac{3+3}{2}, \frac{1+3}{2}) = (3, 2)$. Уравнение серединного перпендикуляра к BC: $y = 2$. Найдем серединный перпендикуляр к стороне AB. Середина отрезка AB имеет координаты $M_{AB}(\frac{0+3}{2}, \frac{2+1}{2}) = (1.5, 1.5)$. Угловой коэффициент прямой AB равен $k_{AB} = \frac{1-2}{3-0} = -\frac{1}{3}$. Угловой коэффициент серединного перпендикуляра равен $k_{\perp} = -\frac{1}{k_{AB}} = 3$. Уравнение перпендикуляра, проходящего через точку $M_{AB}(1.5, 1.5)$ с угловым коэффициентом 3, имеет вид: $y - 1.5 = 3(x - 1.5)$, что упрощается до $y = 3x - 3$. Для нахождения точки пересечения (обозначим ее O) решим систему уравнений: $$ \begin{cases} y = 2 \\ y = 3x - 3 \end{cases} $$ Подставляя $y=2$ во второе уравнение, получаем $2 = 3x - 3$, откуда $3x = 5$ и $x = \frac{5}{3}$. Таким образом, искомая точка O имеет координаты O($\frac{5}{3}, 2$). На рисунке эта точка отмечена синим крестиком.</p><strong>Ответ:</strong><svg width="180" height="150" viewBox="0 0 180 150" xmlns="http://www.w3.org/2000/svg"><!-- Grid --><defs><pattern id="grid_a" width="30" height="30" patternUnits="userSpaceOnUse"><path d="M 30 0 L 0 0 0 30" fill="none" stroke="lightgray" stroke-width="1"/></pattern></defs><rect x="15" y="15" width="150" height="120" fill="url(#grid_a)"/><!-- Perpendicular Bisectors --><line x1="15" y1="75" x2="165" y2="75" stroke="green" stroke-width="1.5" stroke-dasharray="4 2"/><line x1="45" y1="135" x2="85" y2="15" stroke="green" stroke-width="1.5" stroke-dasharray="4 2"/><!-- Points --><circle cx="15" cy="75" r="3.5" fill="red"/><text x="5" y="79" font-family="sans-serif" font-size="14">A</text><circle cx="105" cy="105" r="3.5" fill="red"/><text x="110" y="115" font-family="sans-serif" font-size="14">B</text><circle cx="105" cy="45" r="3.5" fill="red"/><text x="110" y="49" font-family="sans-serif" font-size="14">C</text><!-- Solution Point O --><line x1="61" y1="71" x2="69" y2="79" stroke="blue" stroke-width="2"/><line x1="61" y1="79" x2="69" y2="71" stroke="blue" stroke-width="2"/></svg><p><strong>б)</strong> Аналогично, введем систему координат с началом в левом нижнем углу. Координаты точек: A(0, 3), B(2, 1), C(3, 3). Найдем серединный перпендикуляр к стороне AC. Отрезок AC горизонтален. Его середина $M_{AC}(\frac{0+3}{2}, \frac{3+3}{2}) = (1.5, 3)$. Серединный перпендикуляр — вертикальная прямая $x = 1.5$. Найдем серединный перпендикуляр к стороне AB. Середина отрезка AB имеет координаты $M_{AB}(\frac{0+2}{2}, \frac{3+1}{2}) = (1, 2)$. Угловой коэффициент прямой AB равен $k_{AB} = \frac{1-3}{2-0} = -1$. Угловой коэффициент серединного перпендикуляра равен $k_{\perp} = -\frac{1}{k_{AB}} = 1$. Уравнение перпендикуляра: $y - 2 = 1(x - 1)$, что упрощается до $y = x + 1$. Найдем точку пересечения O, решив систему: $$ \begin{cases} x = 1.5 \\ y = x + 1 \end{cases} $$ Подставляя $x=1.5$ во второе уравнение, получаем $y = 1.5 + 1 = 2.5$. Искомая точка O имеет координаты O(1.5, 2.5). На рисунке эта точка отмечена синим крестиком.</p><strong>Ответ:</strong><svg width="180" height="150" viewBox="0 0 180 150" xmlns="http://www.w3.org/2000/svg"><!-- Grid --><defs><pattern id="grid_b" width="30" height="30" patternUnits="userSpaceOnUse"><path d="M 30 0 L 0 0 0 30" fill="none" stroke="lightgray" stroke-width="1"/></pattern></defs><rect x="15" y="15" width="150" height="120" fill="url(#grid_b)"/><!-- Perpendicular Bisectors --><line x1="60" y1="15" x2="60" y2="135" stroke="green" stroke-width="1.5" stroke-dasharray="4 2"/><line x1="15" y1="105" x2="105" y2="15" stroke="green" stroke-width="1.5" stroke-dasharray="4 2"/><!-- Points --><circle cx="15" cy="45" r="3.5" fill="red"/><text x="5" y="49" font-family="sans-serif" font-size="14">A</text><circle cx="75" cy="105" r="3.5" fill="red"/><text x="65" y="115" font-family="sans-serif" font-size="14">B</text><circle cx="105" cy="45" r="3.5" fill="red"/><text x="110" y="49" font-family="sans-serif" font-size="14">C</text><!-- Solution Point O --><line x1="56" y1="56" x2="64" y2="64" stroke="blue" stroke-width="2"/><line x1="56" y1="64" x2="64" y2="56" stroke="blue" stroke-width="2"/></svg><p><strong>в)</strong> Снова используем метод серединных перпендикуляров. В системе координат с началом в левом нижнем углу точки имеют координаты: A(0, 1), B(2, 1), C(4, 2). Найдем серединный перпендикуляр к стороне AB. Отрезок AB горизонтален. Его середина $M_{AB}(\frac{0+2}{2}, \frac{1+1}{2}) = (1, 1)$. Серединный перпендикуляр — вертикальная прямая $x = 1$. Найдем серединный перпендикуляр к стороне BC. Середина отрезка BC имеет координаты $M_{BC}(\frac{2+4}{2}, \frac{1+2}{2}) = (3, 1.5)$. Угловой коэффициент прямой BC равен $k_{BC} = \frac{2-1}{4-2} = \frac{1}{2}$. Угловой коэффициент серединного перпендикуляра равен $k_{\perp} = -\frac{1}{k_{BC}} = -2$. Уравнение перпендикуляра: $y - 1.5 = -2(x - 3)$, что упрощается до $y = -2x + 7.5$. Найдем точку пересечения O, решив систему: $$ \begin{cases} x = 1 \\ y = -2x + 7.5 \end{cases} $$ Подставляя $x=1$ во второе уравнение, получаем $y = -2(1) + 7.5 = 5.5$. Искомая точка O имеет координаты O(1, 5.5). Эта точка лежит за пределами исходной сетки, что является допустимым результатом. На рисунке сетка продлена вверх для отображения этой точки.</p><strong>Ответ:</strong><svg width="180" height="210" viewBox="0 0 180 210" xmlns="http://www.w3.org/2000/svg"><!-- Grid --><path d="M 15 15 H 165 M 15 45 H 165 M 15 75 H 165 M 15 105 H 165 M 15 135 H 165 M 15 165 H 165 M 15 195 H 165 M 15 15 V 195 M 45 15 V 195 M 75 15 V 195 M 105 15 V 195 M 135 15 V 195 M 165 15 V 195" fill="none" stroke="lightgray" stroke-width="1"/><!-- Extended Grid --><path d="M 15 15 V -15 M 45 15 V -15 M 75 15 V -15 M 105 15 V -15 M 135 15 V -15 M 165 15 V -15" fill="none" stroke="lightgray" stroke-width="1" stroke-dasharray="2 2"/><!-- Perpendicular Bisectors --><line x1="45" y1="-15" x2="45" y2="195" stroke="green" stroke-width="1.5" stroke-dasharray="4 2"/><line x1="22.5" y1="75" x2="112.5" y2="-105" transform="translate(0, 75)" stroke="green" stroke-width="1.5" stroke-dasharray="4 2"/><line x1="45" y1="30" x2="105" y2="150" stroke="green" stroke-width="1.5" stroke-dasharray="4 2" visibility="hidden"/><!-- Calculation guide line --><!-- Points --><circle cx="15" cy="165" r="3.5" fill="red"/><text x="5" y="169" font-family="sans-serif" font-size="14">A</text><circle cx="75" cy="165" r="3.5" fill="red"/><text x="65" y="175" font-family="sans-serif" font-size="14">B</text><circle cx="135" cy="135" r="3.5" fill="red"/><text x="140" y="139" font-family="sans-serif" font-size="14">C</text><!-- Solution Point O --><line x1="41" y1="26" x2="49" y2="34" stroke="blue" stroke-width="2"/><line x1="41" y1="34" x2="49" y2="26" stroke="blue" stroke-width="2"/></svg>" ] #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" => "1129867" "type" => "task" ] "previous" => array:2 [ "refs" => "1129865" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1097 #items: array:1 [ 0 => App\Models\Book {#1115} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1099 #items: array:1 [ 0 => App\Models\BookPage {#1073 #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" => 1114438 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "121" "field_url" => "/7-klass/geometrija/smirnova-uchebnik/page-121" "field_display_title" => "121" "field_folder" => "1" "field_image_name" => "121" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1074 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1102 #items: array:1 [ 0 => App\Models\Book {#1115} ] #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 {#1111 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1155 #items: array:1 [ 0 => App\Models\Element {#1154 #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" => 1290625 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1156 #items: array:1 [ …1] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1114438" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ …5] ] ] #original: array:6 [ "id" => 1290625 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1156} "book_page" => array:2 [ "refs" => "1114438" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ …5] ] ] #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" => "1114439" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1114437" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1252 #items: array:5 [ 0 => App\Models\Task {#1264 #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" => 1129863 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "121" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/21-4" "field_display_title" => "21.4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1265 #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 {#1266 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1267 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1268 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1271 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1129864" "type" => "task" ] "previous" => array:2 [ "refs" => "1129862" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1277 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1114438" "type" => "book_page" ] ] #original: array:24 [ "id" => 1129863 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "121" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/21-4" "field_display_title" => "21.4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1265} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1266} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1267} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1268} "content" => Illuminate\Database\Eloquent\Collection {#1271} "next" => array:2 [ "refs" => "1129864" "type" => "task" ] "previous" => array:2 [ "refs" => "1129862" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1277} "page" => array:2 [ "refs" => "1114438" "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 {#1270 #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" => 1129864 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "121" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/21-5" "field_display_title" => "21.5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1269 #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 {#1276 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1278 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1274 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1273 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1129865" "type" => "task" ] "previous" => array:2 [ "refs" => "1129863" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1291 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1114438" "type" => "book_page" ] ] #original: array:24 [ "id" => 1129864 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "121" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/21-5" "field_display_title" => "21.5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1269} "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 {#1276} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1278} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1274} "content" => Illuminate\Database\Eloquent\Collection {#1273} "next" => array:2 [ "refs" => "1129865" "type" => "task" ] "previous" => array:2 [ "refs" => "1129863" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1291} "page" => array:2 [ "refs" => "1114438" "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 {#1275 #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" => 1129865 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "121" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/21-6" "field_display_title" => "21.6" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1272 #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 {#1290 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1292 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1288 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1287 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1129866" "type" => "task" ] "previous" => array:2 [ "refs" => "1129864" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1305 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1114438" "type" => "book_page" ] ] #original: array:24 [ "id" => 1129865 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "121" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/21-6" "field_display_title" => "21.6" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1272} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] 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+wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1129866 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "121" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/21-7" "field_display_title" => "21.7" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1286 #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 {#1304 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1306 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1302 #items: array:2 [ …2] 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№21.7 (с. 121)
Условие. №21.7 (с. 121)
скриншот условия
21.7. Отметьте точку, равноудаленную от точек $A$, $B$ и $C$ (рис. 21.7).
а)б)в) Рис. 21.7
Решение. №21.7 (с. 121)
Решение 2. №21.7 (с. 121)
Для нахождения точки, равноудаленной от трех заданных точек А, В и С, необходимо найти центр окружности, описанной около треугольника АВС. Центр описанной окружности находится в точке пересечения серединных перпендикуляров к сторонам треугольника.
а) Введем систему координат, приняв за единицу измерения сторону клетки сетки, а за начало координат — левый нижний узел сетки. В этой системе точки имеют координаты: A(0, 2), B(3, 1), C(3, 3). Найдем серединный перпендикуляр к стороне BC. Так как отрезок BC вертикален, его серединный перпендикуляр будет горизонтальной прямой. Середина отрезка BC имеет координаты $M_{BC}(\frac{3+3}{2}, \frac{1+3}{2}) = (3, 2)$. Уравнение серединного перпендикуляра к BC: $y = 2$. Найдем серединный перпендикуляр к стороне AB. Середина отрезка AB имеет координаты $M_{AB}(\frac{0+3}{2}, \frac{2+1}{2}) = (1.5, 1.5)$. Угловой коэффициент прямой AB равен $k_{AB} = \frac{1-2}{3-0} = -\frac{1}{3}$. Угловой коэффициент серединного перпендикуляра равен $k_{\perp} = -\frac{1}{k_{AB}} = 3$. Уравнение перпендикуляра, проходящего через точку $M_{AB}(1.5, 1.5)$ с угловым коэффициентом 3, имеет вид: $y - 1.5 = 3(x - 1.5)$, что упрощается до $y = 3x - 3$. Для нахождения точки пересечения (обозначим ее O) решим систему уравнений: $$ \begin{cases} y = 2 \\ y = 3x - 3 \end{cases} $$ Подставляя $y=2$ во второе уравнение, получаем $2 = 3x - 3$, откуда $3x = 5$ и $x = \frac{5}{3}$. Таким образом, искомая точка O имеет координаты O($\frac{5}{3}, 2$). На рисунке эта точка отмечена синим крестиком.
Ответ:б) Аналогично, введем систему координат с началом в левом нижнем углу. Координаты точек: A(0, 3), B(2, 1), C(3, 3). Найдем серединный перпендикуляр к стороне AC. Отрезок AC горизонтален. Его середина $M_{AC}(\frac{0+3}{2}, \frac{3+3}{2}) = (1.5, 3)$. Серединный перпендикуляр — вертикальная прямая $x = 1.5$. Найдем серединный перпендикуляр к стороне AB. Середина отрезка AB имеет координаты $M_{AB}(\frac{0+2}{2}, \frac{3+1}{2}) = (1, 2)$. Угловой коэффициент прямой AB равен $k_{AB} = \frac{1-3}{2-0} = -1$. Угловой коэффициент серединного перпендикуляра равен $k_{\perp} = -\frac{1}{k_{AB}} = 1$. Уравнение перпендикуляра: $y - 2 = 1(x - 1)$, что упрощается до $y = x + 1$. Найдем точку пересечения O, решив систему: $$ \begin{cases} x = 1.5 \\ y = x + 1 \end{cases} $$ Подставляя $x=1.5$ во второе уравнение, получаем $y = 1.5 + 1 = 2.5$. Искомая точка O имеет координаты O(1.5, 2.5). На рисунке эта точка отмечена синим крестиком.
Ответ:в) Снова используем метод серединных перпендикуляров. В системе координат с началом в левом нижнем углу точки имеют координаты: A(0, 1), B(2, 1), C(4, 2). Найдем серединный перпендикуляр к стороне AB. Отрезок AB горизонтален. Его середина $M_{AB}(\frac{0+2}{2}, \frac{1+1}{2}) = (1, 1)$. Серединный перпендикуляр — вертикальная прямая $x = 1$. Найдем серединный перпендикуляр к стороне BC. Середина отрезка BC имеет координаты $M_{BC}(\frac{2+4}{2}, \frac{1+2}{2}) = (3, 1.5)$. Угловой коэффициент прямой BC равен $k_{BC} = \frac{2-1}{4-2} = \frac{1}{2}$. Угловой коэффициент серединного перпендикуляра равен $k_{\perp} = -\frac{1}{k_{BC}} = -2$. Уравнение перпендикуляра: $y - 1.5 = -2(x - 3)$, что упрощается до $y = -2x + 7.5$. Найдем точку пересечения O, решив систему: $$ \begin{cases} x = 1 \\ y = -2x + 7.5 \end{cases} $$ Подставляя $x=1$ во второе уравнение, получаем $y = -2(1) + 7.5 = 5.5$. Искомая точка O имеет координаты O(1, 5.5). Эта точка лежит за пределами исходной сетки, что является допустимым результатом. На рисунке сетка продлена вверх для отображения этой точки.
Ответ:Другие задания:
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