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Номер 17.8, страница 96 - 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 3. Взаимное расположение прямых. Параграф 17. Неравенство треугольника - номер 17.8, страница 96.
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" => 1129782 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "96" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/17-8" "field_display_title" => "17.8" "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" => 1129374 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Взаимное расположение прямых" "field_branch_order" => "3" "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 [ …6] "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 [ …6] "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" => "79" "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 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1063 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_url" => "/7-klass/geometrija/smirnova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1067 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1063 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_url" => "/7-klass/geometrija/smirnova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1067 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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" => 1129374 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Взаимное расположение прямых" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039} "field_page_start" => "79" "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 {#1139 #items: array:2 [ 0 => App\Models\Branch {#1069 #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" => 1129374 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Взаимное расположение прямых" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1072 #items: array:1 [ 0 => App\Models\Term {#1070 #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 [ …6] "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 [ …6] "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" => "79" "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 {#1073 #items: array:1 [ 0 => App\Models\Book {#1074 #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 {#1075 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1077 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1079 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1081 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1094 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1096 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1098 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_url" => "/7-klass/geometrija/smirnova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1100 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 {#1075 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1077 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1079 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1081 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1094 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1096 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1098 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_url" => "/7-klass/geometrija/smirnova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1100 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 {#1101 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1129374 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Взаимное расположение прямых" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1072} "field_page_start" => "79" "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 {#1073} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1101} ] #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 {#1102 #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" => 1129378 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Неравенство треугольника" "field_branch_order" => "17" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1103 #items: array:1 [ 0 => App\Models\Term {#1104 #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 [ …6] "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 [ …6] "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" => "92" "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 {#1105 #items: array:1 [ 0 => App\Models\Book {#1106 #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 {#1107 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1109 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1111 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1113 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1116 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1117 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1125 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1127 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1128 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1129 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1131 …2} "field_url" => "/7-klass/geometrija/smirnova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1132 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 {#1107 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1109 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1111 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1113 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1116 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1117 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1125 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1127 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1128 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1129 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1131 …2} "field_url" => "/7-klass/geometrija/smirnova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1132 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 {#1133 #items: array:1 [ 0 => App\Models\Branch {#1134 #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" => 1129374 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Взаимное расположение прямых" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1135 …2} "field_page_start" => "79" "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 {#1137 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1138 …2} ] #original: array:24 [ "id" => 1129374 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Взаимное расположение прямых" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1135 …2} "field_page_start" => "79" "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 {#1137 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1138 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1129378 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Неравенство треугольника" "field_branch_order" => "17" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1103} "field_page_start" => "92" "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 {#1105} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1133} ] #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 {#1142 #items: array:3 [ 0 => App\Models\Element {#1151 #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" => 1319852 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1160 #items: array:1 [ 0 => App\Models\Edition {#1152 #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 {#1154 …2} "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 {#1155 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1156 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1157 …2} "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 {#1154 …2} "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 {#1155 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1156 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1157 …2} "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" => "1129782" "type" => "task" ] "text" => "<p><strong>17.8.</strong> На прямой с укажите точку C, для которой сумма расстояний $AC + CB$ наименьшая (рис. 17.6).</p><p><strong>а)</strong></p><p><strong>б)</strong></p><p><strong>в)</strong></p><p>Рис. 17.6</p>" "img" => array:1 [ 0 => array:5 [ "name" => "8-1.jpg" "alt" => null "width" => "1701" "height" => 892 "path" => "/media/geometrija_07/Smirnova-u17/0-17/8-1.webp?ts=1753982425" ] ] ] #original: array:7 [ "id" => 1319852 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1160} "task" => array:2 [ "refs" => "1129782" "type" => "task" ] "text" => "<p><strong>17.8.</strong> На прямой с укажите точку C, для которой сумма расстояний $AC + CB$ наименьшая (рис. 17.6).</p><p><strong>а)</strong></p><p><strong>б)</strong></p><p><strong>в)</strong></p><p>Рис. 17.6</p>" "img" => array:1 [ 0 => array:5 [ "name" => "8-1.jpg" "alt" => null "width" => "1701" "height" => 892 "path" => "/media/geometrija_07/Smirnova-u17/0-17/8-1.webp?ts=1753982425" ] ] ] #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 {#1158 #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" => 1320356 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1168 #items: array:1 [ 0 => App\Models\Edition {#1159 #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 {#1162 …2} "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 {#1163 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1164 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1165 …2} "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 {#1162 …2} "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 {#1163 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1164 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1165 …2} "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" => "1129782" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "8-1.jpg" "alt" => null "width" => "1439" "height" => 732 "path" => "/media/geometrija_07/Smirnova-u17/1-17/8-1.webp?ts=1753982538" ] ] ] #original: array:6 [ "id" => 1320356 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1168} "task" => array:2 [ "refs" => "1129782" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "8-1.jpg" "alt" => null "width" => "1439" "height" => 732 "path" => "/media/geometrija_07/Smirnova-u17/1-17/8-1.webp?ts=1753982538" ] ] ] #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 {#1166 #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" => 1541618 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1176 #items: array:1 [ 0 => App\Models\Edition {#1167 #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 {#1170 …2} "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 {#1171 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1172 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1173 …2} "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 {#1170 …2} "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 {#1171 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1172 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1173 …2} "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" => "1129782" "type" => "task" ] "text" => "<p>Задача заключается в нахождении на прямой $c$ такой точки $C$, чтобы сумма расстояний $AC + CB$ была наименьшей. Решение этой задачи, известной как задача Герона, зависит от взаимного расположения точек A и B относительно прямой $c$.</p><p><strong>1.</strong> Если точки A и B лежат по одну сторону от прямой $c$, то для нахождения искомой точки C необходимо построить точку B', симметричную точке B относительно прямой $c$. Тогда для любой точки C на прямой $c$ длина отрезка $CB$ будет равна длине отрезка $CB'$, и сумма $AC + CB$ будет равна сумме $AC + CB'$. Эта сумма будет наименьшей, когда точки A, C и B' лежат на одной прямой, так как кратчайшее расстояние между двумя точками — это прямая. Таким образом, искомая точка C является точкой пересечения отрезка AB' с прямой $c$.</p><p><strong>2.</strong> Если точки A и B лежат по разные стороны от прямой $c$, то согласно неравенству треугольника, для любой точки C, не лежащей на отрезке AB, выполняется $AC + CB > AB$. Равенство $AC + CB = AB$ достигается только тогда, когда точка C лежит на отрезке AB. Следовательно, искомая точка C является точкой пересечения отрезка AB с прямой $c$.</p><p><strong>а)</strong> В данном случае точки A и B расположены по одну сторону от прямой c. Для нахождения точки C, минимизирующей сумму $AC+CB$, воспользуемся методом отражения. Отразим точку B относительно прямой c, чтобы получить точку B'. Искомая точка C будет лежать на пересечении отрезка AB' и прямой c. На рисунке ниже показано это построение: зеленой линией показан кратчайший путь, а серой пунктирной линией — построение точки B'.</p><svg width="240" height="200" viewBox="-0.5 -1 7 8" style="vertical-align: middle; margin: 10px 0;"><defs><pattern id="grid_a" width="1" height="1" patternUnits="userSpaceOnUse"><path d="M 1 0 L 0 0 0 1" fill="none" stroke="#a0d8f0" stroke-width="0.05"></path></pattern></defs><rect x="0" y="0" width="6" height="5" fill="url(#grid_a)"></rect><rect x="0" y="0" width="6" height="5" fill="none" stroke="#a0d8f0" stroke-width="0.1"></rect><path d="M 5 1 L 5 7" stroke="gray" stroke-dasharray="0.2 0.2" stroke-width="0.05"></path><line x1="1" y1="1" x2="5" y2="7" stroke="#4caf50" stroke-width="0.08"></line><circle cx="5" cy="7" r="0.1" fill="gray"></circle><text x="5.2" y="7.1" font-size="0.3" fill="gray">B'</text><line x1="0" y1="4" x2="6" y2="4" stroke="red" stroke-width="0.1"></line><text x="0.2" y="3.8" font-size="0.3" fill="black">c</text><circle cx="1" cy="1" r="0.1" fill="red"></circle><text x="0.6" y="1.1" font-size="0.3" fill="black">A</text><circle cx="5" cy="1" r="0.1" fill="red"></circle><text x="5.2" y="1.1" font-size="0.3" fill="black">B</text><circle cx="3" cy="4" r="0.12" fill="blue"></circle><text x="2.9" y="3.8" font-size="0.3" fill="blue">C</text></svg><p>Введем систему координат, приняв левый нижний угол сетки за начало координат (0,0), а сторону клетки за единицу длины. Координаты точек: A(1, 1) и B(5, 1). Уравнение прямой c: $y=4$. Расстояние от точки B до прямой c равно $4-1=3$. Следовательно, отраженная точка B' будет иметь координаты (5, 4+3), то есть B'(5, 7). Найдем уравнение прямой, проходящей через точки A(1, 1) и B'(5, 7): $y - 1 = \frac{7-1}{5-1}(x-1)$, что упрощается до $y - 1 = \frac{3}{2}(x-1)$. Для нахождения точки C подставим в это уравнение $y=4$: $4 - 1 = \frac{3}{2}(x-1) \Rightarrow 3 = \frac{3}{2}(x-1) \Rightarrow 2 = x-1 \Rightarrow x=3$. Таким образом, координаты точки C равны (3, 4). Так как точки A и B в данном случае равноудалены от прямой c, искомая точка C также является основанием перпендикуляра, опущенного из середины отрезка AB на прямую c.</p><p><strong>Ответ:</strong> Искомая точка C находится на прямой c в точке с координатами (3, 4), если принять левый нижний узел сетки за начало координат. Эта точка является пересечением прямой c и перпендикуляра к ней, проходящего через середину отрезка AB.</p><p><strong>б)</strong> В этом случае точки A и B также расположены по одну сторону от прямой c, поэтому применяем тот же метод отражения. Отражаем точку B относительно прямой c, получаем точку B', и искомая точка C будет являться пересечением отрезка AB' с прямой c.</p><svg width="240" height="200" viewBox="-0.5 -1 7 8" style="vertical-align: middle; margin: 10px 0;"><defs><pattern id="grid_b" width="1" height="1" patternUnits="userSpaceOnUse"><path d="M 1 0 L 0 0 0 1" fill="none" stroke="#a0d8f0" stroke-width="0.05"></path></pattern></defs><rect x="0" y="0" width="6" height="5" fill="url(#grid_b)"></rect><rect x="0" y="0" width="6" height="5" fill="none" stroke="#a0d8f0" stroke-width="0.1"></rect><path d="M 5 1 L 5 7" stroke="gray" stroke-dasharray="0.2 0.2" stroke-width="0.05"></path><line x1="1" y1="2" x2="5" y2="7" stroke="#4caf50" stroke-width="0.08"></line><circle cx="5" cy="7" r="0.1" fill="gray"></circle><text x="5.2" y="7.1" font-size="0.3" fill="gray">B'</text><line x1="0" y1="4" x2="6" y2="4" stroke="red" stroke-width="0.1"></line><text x="0.2" y="3.8" font-size="0.3" fill="black">c</text><circle cx="1" cy="2" r="0.1" fill="red"></circle><text x="0.6" y="2.1" font-size="0.3" fill="black">A</text><circle cx="5" cy="1" r="0.1" fill="red"></circle><text x="5.2" y="1.1" font-size="0.3" fill="black">B</text><circle cx="2.6" cy="4" r="0.12" fill="blue"></circle><text x="2.5" y="3.8" font-size="0.3" fill="blue">C</text></svg><p>Введем аналогичную систему координат. Координаты точек: A(1, 2) и B(5, 1). Уравнение прямой c: $y=4$. Отразим точку B(5, 1) относительно прямой $y=4$. Расстояние от B до прямой равно $4-1=3$. Координаты отраженной точки B' будут (5, 4+3), то есть B'(5, 7). Найдем уравнение прямой, проходящей через точки A(1, 2) и B'(5, 7): $y - 2 = \frac{7-2}{5-1}(x-1)$, что равносильно $y - 2 = \frac{5}{4}(x-1)$. Чтобы найти координаты точки C, подставим $y=4$: $4 - 2 = \frac{5}{4}(x-1) \Rightarrow 2 = \frac{5}{4}(x-1) \Rightarrow 8 = 5(x-1) \Rightarrow 8 = 5x-5 \Rightarrow 5x=13 \Rightarrow x=2.6$. Координаты точки C: (2.6, 4).</p><p><strong>Ответ:</strong> Искомая точка C находится на прямой c в точке с координатами (2.6, 4).</p><p><strong>в)</strong> Здесь точки A и B лежат по разные стороны от прямой c. Кратчайшее расстояние между двумя точками — это длина отрезка, их соединяющего. Сумма $AC+CB$ будет минимальной и равной длине отрезка $AB$, если точка C лежит на этом отрезке. Следовательно, искомая точка C — это точка пересечения отрезка AB и прямой c.</p><svg width="240" height="200" viewBox="-0.5 -1 6.5 6" style="vertical-align: middle; margin: 10px 0;"><defs><pattern id="grid_c" width="1" height="1" patternUnits="userSpaceOnUse"><path d="M 1 0 L 0 0 0 1" fill="none" stroke="#a0d8f0" stroke-width="0.05"></path></pattern></defs><rect x="0" y="0" width="6" height="5" fill="url(#grid_c)"></rect><rect x="0" y="0" width="6" height="5" fill="none" stroke="#a0d8f0" stroke-width="0.1"></rect><line x1="1" y1="3" x2="4" y2="1" stroke="#4caf50" stroke-width="0.08"></line><line x1="0" y1="5" x2="5" y2="0" stroke="red" stroke-width="0.1"></line><text x="0.5" y="4.2" font-size="0.3" fill="black">c</text><circle cx="1" cy="3" r="0.1" fill="red"></circle><text x="0.6" y="3.1" font-size="0.3" fill="black">A</text><circle cx="4" cy="1" r="0.1" fill="red"></circle><text x="4.2" y="1.1" font-size="0.3" fill="black">B</text><circle cx="4" cy="1" r="0.15" fill="none" stroke="blue" stroke-width="0.08"></circle><text x="3.9" y="0.8" font-size="0.3" fill="blue">C</text></svg><p>Введем систему координат. Координаты точек: A(1, 3) и B(4, 1). Прямая c проходит через точки (0, 5) и (5, 0), ее уравнение: $y = -x+5$. Проверим, лежит ли точка B на прямой c, подставив ее координаты в уравнение: $1 = -4+5$, что является верным равенством. Так как точка B уже лежит на прямой c, то отрезок AB пересекает прямую c в самой точке B. Таким образом, искомая точка C совпадает с точкой B, а минимальная сумма расстояний равна длине отрезка AB.</p><p><strong>Ответ:</strong> Искомая точка C совпадает с точкой B.</p>" ] #original: array:6 [ "id" => 1541618 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1176} "task" => array:2 [ "refs" => "1129782" "type" => "task" ] "text" => "<p>Задача заключается в нахождении на прямой $c$ такой точки $C$, чтобы сумма расстояний $AC + CB$ была наименьшей. Решение этой задачи, известной как задача Герона, зависит от взаимного расположения точек A и B относительно прямой $c$.</p><p><strong>1.</strong> Если точки A и B лежат по одну сторону от прямой $c$, то для нахождения искомой точки C необходимо построить точку B', симметричную точке B относительно прямой $c$. Тогда для любой точки C на прямой $c$ длина отрезка $CB$ будет равна длине отрезка $CB'$, и сумма $AC + CB$ будет равна сумме $AC + CB'$. Эта сумма будет наименьшей, когда точки A, C и B' лежат на одной прямой, так как кратчайшее расстояние между двумя точками — это прямая. Таким образом, искомая точка C является точкой пересечения отрезка AB' с прямой $c$.</p><p><strong>2.</strong> Если точки A и B лежат по разные стороны от прямой $c$, то согласно неравенству треугольника, для любой точки C, не лежащей на отрезке AB, выполняется $AC + CB > AB$. Равенство $AC + CB = AB$ достигается только тогда, когда точка C лежит на отрезке AB. Следовательно, искомая точка C является точкой пересечения отрезка AB с прямой $c$.</p><p><strong>а)</strong> В данном случае точки A и B расположены по одну сторону от прямой c. Для нахождения точки C, минимизирующей сумму $AC+CB$, воспользуемся методом отражения. Отразим точку B относительно прямой c, чтобы получить точку B'. Искомая точка C будет лежать на пересечении отрезка AB' и прямой c. На рисунке ниже показано это построение: зеленой линией показан кратчайший путь, а серой пунктирной линией — построение точки B'.</p><svg width="240" height="200" viewBox="-0.5 -1 7 8" style="vertical-align: middle; margin: 10px 0;"><defs><pattern id="grid_a" width="1" height="1" patternUnits="userSpaceOnUse"><path d="M 1 0 L 0 0 0 1" fill="none" stroke="#a0d8f0" stroke-width="0.05"></path></pattern></defs><rect x="0" y="0" width="6" height="5" fill="url(#grid_a)"></rect><rect x="0" y="0" width="6" height="5" fill="none" stroke="#a0d8f0" stroke-width="0.1"></rect><path d="M 5 1 L 5 7" stroke="gray" stroke-dasharray="0.2 0.2" stroke-width="0.05"></path><line x1="1" y1="1" x2="5" y2="7" stroke="#4caf50" stroke-width="0.08"></line><circle cx="5" cy="7" r="0.1" fill="gray"></circle><text x="5.2" y="7.1" font-size="0.3" fill="gray">B'</text><line x1="0" y1="4" x2="6" y2="4" stroke="red" stroke-width="0.1"></line><text x="0.2" y="3.8" font-size="0.3" fill="black">c</text><circle cx="1" cy="1" r="0.1" fill="red"></circle><text x="0.6" y="1.1" font-size="0.3" fill="black">A</text><circle cx="5" cy="1" r="0.1" fill="red"></circle><text x="5.2" y="1.1" font-size="0.3" fill="black">B</text><circle cx="3" cy="4" r="0.12" fill="blue"></circle><text x="2.9" y="3.8" font-size="0.3" fill="blue">C</text></svg><p>Введем систему координат, приняв левый нижний угол сетки за начало координат (0,0), а сторону клетки за единицу длины. Координаты точек: A(1, 1) и B(5, 1). Уравнение прямой c: $y=4$. Расстояние от точки B до прямой c равно $4-1=3$. Следовательно, отраженная точка B' будет иметь координаты (5, 4+3), то есть B'(5, 7). Найдем уравнение прямой, проходящей через точки A(1, 1) и B'(5, 7): $y - 1 = \frac{7-1}{5-1}(x-1)$, что упрощается до $y - 1 = \frac{3}{2}(x-1)$. Для нахождения точки C подставим в это уравнение $y=4$: $4 - 1 = \frac{3}{2}(x-1) \Rightarrow 3 = \frac{3}{2}(x-1) \Rightarrow 2 = x-1 \Rightarrow x=3$. Таким образом, координаты точки C равны (3, 4). Так как точки A и B в данном случае равноудалены от прямой c, искомая точка C также является основанием перпендикуляра, опущенного из середины отрезка AB на прямую c.</p><p><strong>Ответ:</strong> Искомая точка C находится на прямой c в точке с координатами (3, 4), если принять левый нижний узел сетки за начало координат. Эта точка является пересечением прямой c и перпендикуляра к ней, проходящего через середину отрезка AB.</p><p><strong>б)</strong> В этом случае точки A и B также расположены по одну сторону от прямой c, поэтому применяем тот же метод отражения. Отражаем точку B относительно прямой c, получаем точку B', и искомая точка C будет являться пересечением отрезка AB' с прямой c.</p><svg width="240" height="200" viewBox="-0.5 -1 7 8" style="vertical-align: middle; margin: 10px 0;"><defs><pattern id="grid_b" width="1" height="1" patternUnits="userSpaceOnUse"><path d="M 1 0 L 0 0 0 1" fill="none" stroke="#a0d8f0" stroke-width="0.05"></path></pattern></defs><rect x="0" y="0" width="6" height="5" fill="url(#grid_b)"></rect><rect x="0" y="0" width="6" height="5" fill="none" stroke="#a0d8f0" stroke-width="0.1"></rect><path d="M 5 1 L 5 7" stroke="gray" stroke-dasharray="0.2 0.2" stroke-width="0.05"></path><line x1="1" y1="2" x2="5" y2="7" stroke="#4caf50" stroke-width="0.08"></line><circle cx="5" cy="7" r="0.1" fill="gray"></circle><text x="5.2" y="7.1" font-size="0.3" fill="gray">B'</text><line x1="0" y1="4" x2="6" y2="4" stroke="red" stroke-width="0.1"></line><text x="0.2" y="3.8" font-size="0.3" fill="black">c</text><circle cx="1" cy="2" r="0.1" fill="red"></circle><text x="0.6" y="2.1" font-size="0.3" fill="black">A</text><circle cx="5" cy="1" r="0.1" fill="red"></circle><text x="5.2" y="1.1" font-size="0.3" fill="black">B</text><circle cx="2.6" cy="4" r="0.12" fill="blue"></circle><text x="2.5" y="3.8" font-size="0.3" fill="blue">C</text></svg><p>Введем аналогичную систему координат. Координаты точек: A(1, 2) и B(5, 1). Уравнение прямой c: $y=4$. Отразим точку B(5, 1) относительно прямой $y=4$. Расстояние от B до прямой равно $4-1=3$. Координаты отраженной точки B' будут (5, 4+3), то есть B'(5, 7). Найдем уравнение прямой, проходящей через точки A(1, 2) и B'(5, 7): $y - 2 = \frac{7-2}{5-1}(x-1)$, что равносильно $y - 2 = \frac{5}{4}(x-1)$. Чтобы найти координаты точки C, подставим $y=4$: $4 - 2 = \frac{5}{4}(x-1) \Rightarrow 2 = \frac{5}{4}(x-1) \Rightarrow 8 = 5(x-1) \Rightarrow 8 = 5x-5 \Rightarrow 5x=13 \Rightarrow x=2.6$. Координаты точки C: (2.6, 4).</p><p><strong>Ответ:</strong> Искомая точка C находится на прямой c в точке с координатами (2.6, 4).</p><p><strong>в)</strong> Здесь точки A и B лежат по разные стороны от прямой c. Кратчайшее расстояние между двумя точками — это длина отрезка, их соединяющего. Сумма $AC+CB$ будет минимальной и равной длине отрезка $AB$, если точка C лежит на этом отрезке. Следовательно, искомая точка C — это точка пересечения отрезка AB и прямой c.</p><svg width="240" height="200" viewBox="-0.5 -1 6.5 6" style="vertical-align: middle; margin: 10px 0;"><defs><pattern id="grid_c" width="1" height="1" patternUnits="userSpaceOnUse"><path d="M 1 0 L 0 0 0 1" fill="none" stroke="#a0d8f0" stroke-width="0.05"></path></pattern></defs><rect x="0" y="0" width="6" height="5" fill="url(#grid_c)"></rect><rect x="0" y="0" width="6" height="5" fill="none" stroke="#a0d8f0" stroke-width="0.1"></rect><line x1="1" y1="3" x2="4" y2="1" stroke="#4caf50" stroke-width="0.08"></line><line x1="0" y1="5" x2="5" y2="0" stroke="red" stroke-width="0.1"></line><text x="0.5" y="4.2" font-size="0.3" fill="black">c</text><circle cx="1" cy="3" r="0.1" fill="red"></circle><text x="0.6" y="3.1" font-size="0.3" fill="black">A</text><circle cx="4" cy="1" r="0.1" fill="red"></circle><text x="4.2" y="1.1" font-size="0.3" fill="black">B</text><circle cx="4" cy="1" r="0.15" fill="none" stroke="blue" stroke-width="0.08"></circle><text x="3.9" y="0.8" font-size="0.3" fill="blue">C</text></svg><p>Введем систему координат. Координаты точек: A(1, 3) и B(4, 1). Прямая c проходит через точки (0, 5) и (5, 0), ее уравнение: $y = -x+5$. Проверим, лежит ли точка B на прямой c, подставив ее координаты в уравнение: $1 = -4+5$, что является верным равенством. Так как точка B уже лежит на прямой c, то отрезок AB пересекает прямую c в самой точке B. Таким образом, искомая точка C совпадает с точкой B, а минимальная сумма расстояний равна длине отрезка AB.</p><p><strong>Ответ:</strong> Искомая точка C совпадает с точкой B.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1129783" "type" => "task" ] "previous" => array:2 [ "refs" => "1129781" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1195 #items: array:1 [ 0 => App\Models\Book {#1148 #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 {#1141 #items: array:1 [ 0 => App\Models\Term {#1140 #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 [ …6] "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 [ …6] "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 {#1147 #items: array:1 [ 0 => App\Models\Term {#1149 #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 [ …6] "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 [ …6] "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 {#1145 #items: array:1 [ 0 => App\Models\Term {#1143 #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 [ …6] "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 [ …6] "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 {#1146 #items: array:2 [ 0 => App\Models\Term {#1144 #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 {#1174 #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 {#1175 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1177 #items: array:1 [ 0 => App\Models\Term {#1178 #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 [ …6] "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 [ …6] "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 {#1179 #items: array:1 [ 0 => App\Models\Term {#1180 #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 [ …6] "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 [ …6] "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 {#1181 #items: array:1 [ 0 => App\Models\Term {#1182 #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 [ …6] "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 [ …6] "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 {#1183 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1184 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1185 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1186 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1187 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1188 #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 {#1189 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1190 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1191 #items: [] #escapeWhenCastingToString: false } "field_url" => "/7-klass/geometrija/smirnova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1192 #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 {#1141} "field_class" => Illuminate\Database\Eloquent\Collection {#1147} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1145} "field_author" => Illuminate\Database\Eloquent\Collection {#1146} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1175} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1177} "field_country" => Illuminate\Database\Eloquent\Collection {#1179} "field_city" => Illuminate\Database\Eloquent\Collection {#1181} "field_series" => Illuminate\Database\Eloquent\Collection {#1183} "field_umk" => Illuminate\Database\Eloquent\Collection {#1184} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1185} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1186} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1187} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1188} "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 {#1189} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1190} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1191} "field_url" => "/7-klass/geometrija/smirnova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1192} "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 } "page" => Illuminate\Database\Eloquent\Collection {#1200 #items: array:1 [ 0 => App\Models\BookPage {#1199 #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" => 1114413 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "96" "field_url" => "/7-klass/geometrija/smirnova-uchebnik/page-96" "field_display_title" => "96" "field_folder" => "1" "field_image_name" => "96" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1201 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1202 #items: array:1 [ 0 => App\Models\Book {#1148} ] #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 {#1203 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1213 #items: array:1 [ 0 => App\Models\Element {#1212 #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" => 1290600 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1214 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1290600 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1214 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #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" => "1114414" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1114412" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1369 #items: array:7 [ 0 => App\Models\Task {#1383 #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" => 1129781 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "96" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/17-7" "field_display_title" => "17.7" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1384 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1385 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1386 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1387 …2} "content" => Illuminate\Database\Eloquent\Collection {#1390 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1396 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1129781 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "96" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/17-7" "field_display_title" => "17.7" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1384 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1385 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1386 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1387 …2} "content" => Illuminate\Database\Eloquent\Collection {#1390 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1396 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] 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№17.8 (с. 96)
Условие. №17.8 (с. 96)
скриншот условия
17.8. На прямой с укажите точку C, для которой сумма расстояний $AC + CB$ наименьшая (рис. 17.6).
а)
б)
в)
Рис. 17.6
Решение. №17.8 (с. 96)
Решение 2. №17.8 (с. 96)
Задача заключается в нахождении на прямой $c$ такой точки $C$, чтобы сумма расстояний $AC + CB$ была наименьшей. Решение этой задачи, известной как задача Герона, зависит от взаимного расположения точек A и B относительно прямой $c$.
1. Если точки A и B лежат по одну сторону от прямой $c$, то для нахождения искомой точки C необходимо построить точку B', симметричную точке B относительно прямой $c$. Тогда для любой точки C на прямой $c$ длина отрезка $CB$ будет равна длине отрезка $CB'$, и сумма $AC + CB$ будет равна сумме $AC + CB'$. Эта сумма будет наименьшей, когда точки A, C и B' лежат на одной прямой, так как кратчайшее расстояние между двумя точками — это прямая. Таким образом, искомая точка C является точкой пересечения отрезка AB' с прямой $c$.
2. Если точки A и B лежат по разные стороны от прямой $c$, то согласно неравенству треугольника, для любой точки C, не лежащей на отрезке AB, выполняется $AC + CB > AB$. Равенство $AC + CB = AB$ достигается только тогда, когда точка C лежит на отрезке AB. Следовательно, искомая точка C является точкой пересечения отрезка AB с прямой $c$.
а) В данном случае точки A и B расположены по одну сторону от прямой c. Для нахождения точки C, минимизирующей сумму $AC+CB$, воспользуемся методом отражения. Отразим точку B относительно прямой c, чтобы получить точку B'. Искомая точка C будет лежать на пересечении отрезка AB' и прямой c. На рисунке ниже показано это построение: зеленой линией показан кратчайший путь, а серой пунктирной линией — построение точки B'.
Введем систему координат, приняв левый нижний угол сетки за начало координат (0,0), а сторону клетки за единицу длины. Координаты точек: A(1, 1) и B(5, 1). Уравнение прямой c: $y=4$. Расстояние от точки B до прямой c равно $4-1=3$. Следовательно, отраженная точка B' будет иметь координаты (5, 4+3), то есть B'(5, 7). Найдем уравнение прямой, проходящей через точки A(1, 1) и B'(5, 7): $y - 1 = \frac{7-1}{5-1}(x-1)$, что упрощается до $y - 1 = \frac{3}{2}(x-1)$. Для нахождения точки C подставим в это уравнение $y=4$: $4 - 1 = \frac{3}{2}(x-1) \Rightarrow 3 = \frac{3}{2}(x-1) \Rightarrow 2 = x-1 \Rightarrow x=3$. Таким образом, координаты точки C равны (3, 4). Так как точки A и B в данном случае равноудалены от прямой c, искомая точка C также является основанием перпендикуляра, опущенного из середины отрезка AB на прямую c.
Ответ: Искомая точка C находится на прямой c в точке с координатами (3, 4), если принять левый нижний узел сетки за начало координат. Эта точка является пересечением прямой c и перпендикуляра к ней, проходящего через середину отрезка AB.
б) В этом случае точки A и B также расположены по одну сторону от прямой c, поэтому применяем тот же метод отражения. Отражаем точку B относительно прямой c, получаем точку B', и искомая точка C будет являться пересечением отрезка AB' с прямой c.
Введем аналогичную систему координат. Координаты точек: A(1, 2) и B(5, 1). Уравнение прямой c: $y=4$. Отразим точку B(5, 1) относительно прямой $y=4$. Расстояние от B до прямой равно $4-1=3$. Координаты отраженной точки B' будут (5, 4+3), то есть B'(5, 7). Найдем уравнение прямой, проходящей через точки A(1, 2) и B'(5, 7): $y - 2 = \frac{7-2}{5-1}(x-1)$, что равносильно $y - 2 = \frac{5}{4}(x-1)$. Чтобы найти координаты точки C, подставим $y=4$: $4 - 2 = \frac{5}{4}(x-1) \Rightarrow 2 = \frac{5}{4}(x-1) \Rightarrow 8 = 5(x-1) \Rightarrow 8 = 5x-5 \Rightarrow 5x=13 \Rightarrow x=2.6$. Координаты точки C: (2.6, 4).
Ответ: Искомая точка C находится на прямой c в точке с координатами (2.6, 4).
в) Здесь точки A и B лежат по разные стороны от прямой c. Кратчайшее расстояние между двумя точками — это длина отрезка, их соединяющего. Сумма $AC+CB$ будет минимальной и равной длине отрезка $AB$, если точка C лежит на этом отрезке. Следовательно, искомая точка C — это точка пересечения отрезка AB и прямой c.
Введем систему координат. Координаты точек: A(1, 3) и B(4, 1). Прямая c проходит через точки (0, 5) и (5, 0), ее уравнение: $y = -x+5$. Проверим, лежит ли точка B на прямой c, подставив ее координаты в уравнение: $1 = -4+5$, что является верным равенством. Так как точка B уже лежит на прямой c, то отрезок AB пересекает прямую c в самой точке B. Таким образом, искомая точка C совпадает с точкой B, а минимальная сумма расстояний равна длине отрезка AB.
Ответ: Искомая точка C совпадает с точкой B.
Другие задания:
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