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Номер 1.4, страница 8 - 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 1. Начальные геометрические сведения. Параграф 1. Основные понятия геометрии - номер 1.4, страница 8.
App\Models\Task {#1024 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1129391 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "8" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/01-4" "field_display_title" => "1.4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#331 #items: array:1 [ 0 => App\Models\Term {#1029 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1026 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#342 #items: array:1 [ 0 => App\Models\Branch {#1023 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1129359 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Начальные геометрические сведения" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333 #items: array:1 [ 0 => App\Models\Term {#1033 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "field_page_check_not_needed" => null "field_short_name" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1030 #items: array:1 [ 0 => App\Models\Book {#1032 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 2907 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1034 #items: array:1 [ 0 => App\Models\Term {#1035 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6477 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Геометрия" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "геометрию" "field_creative_case" => "геометрией" "field_dative_case" => "геометрии" "field_genitive_case" => "геометрии" "field_nominative_case" => "геометрия" "field_prepositional_case" => "геометрии" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "geometrija" ] #original: array:10 [ "id" => 6477 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Геометрия" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "геометрию" "field_creative_case" => "геометрией" "field_dative_case" => "геометрии" "field_genitive_case" => "геометрии" "field_nominative_case" => "геометрия" "field_prepositional_case" => "геометрии" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "geometrija" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1036 #items: array:1 [ 0 => App\Models\Term {#1037 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5456 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "7" "field_cases" => array:6 [ "field_accusative_case" => "седьмой" "field_creative_case" => "седьмым" "field_dative_case" => "седьмому" "field_genitive_case" => "седьмого" "field_nominative_case" => "седьмой" "field_prepositional_case" => "седьмом" ] "field_translit" => "sedmoj" ] #original: array:6 [ "id" => 5456 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "7" "field_cases" => array:6 [ "field_accusative_case" => "седьмой" "field_creative_case" => "седьмым" "field_dative_case" => "седьмому" "field_genitive_case" => "седьмого" "field_nominative_case" => "седьмой" "field_prepositional_case" => "седьмом" ] "field_translit" => "sedmoj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1038 #items: array:1 [ 0 => App\Models\Term {#1039 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #original: array:6 [ "id" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1040 #items: array:2 [ 0 => App\Models\Term {#1041 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 5801 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Смирнов" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Владимир" "field_patronymic" => "Алексеевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 5801 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Смирнов" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Владимир" "field_patronymic" => "Алексеевич" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1042 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7017 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Туяков" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Есенкельды" "field_patronymic" => "Алыбаевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 7017 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Туяков" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Есенкельды" "field_patronymic" => "Алыбаевич" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1043 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1044 #items: array:1 [ 0 => App\Models\Term {#1045 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:1 [ 0 => App\Models\Term {#1047 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "kazakhstan" ] #original: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "kazakhstan" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1048 #items: array:1 [ 0 => App\Models\Term {#1049 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almaty" ] #original: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almaty" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1050 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1051 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1052 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1053 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1054 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1055 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2017" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 7 класса общеобразовательной школы" "field_allowed" => "Утверждено Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "528" "field_priority" => "2" "field_default_folder" => "/geometrija_07/Smirnova-u17/" "field_isbn" => "978-601-07-0873-0" "field_cover" => array:1 [ 0 => "/media/geometrija_07/Smirnova-u17/covers/cover1.webp?ts=1753259592" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_07/Smirnova-u17/covers/cover1.webp?ts=1753259592" "alt" => "" "width" => "1632" "height" => "2472" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1056 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1057 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1058 #items: [] #escapeWhenCastingToString: false } "field_url" => "/7-klass/geometrija/smirnova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1059 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 378 "subject" => 543 "class_subject" => 389 ] ] #original: array:50 [ "id" => 2907 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1034} "field_class" => Illuminate\Database\Eloquent\Collection {#1036} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1038} "field_author" => Illuminate\Database\Eloquent\Collection {#1040} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1043} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1044} "field_country" => Illuminate\Database\Eloquent\Collection {#1046} "field_city" => Illuminate\Database\Eloquent\Collection {#1048} "field_series" => Illuminate\Database\Eloquent\Collection {#1050} "field_umk" => Illuminate\Database\Eloquent\Collection {#1051} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1052} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1053} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1054} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1055} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2017" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 7 класса общеобразовательной школы" "field_allowed" => "Утверждено Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "528" "field_priority" => "2" "field_default_folder" => "/geometrija_07/Smirnova-u17/" "field_isbn" => "978-601-07-0873-0" "field_cover" => array:1 [ 0 => "/media/geometrija_07/Smirnova-u17/covers/cover1.webp?ts=1753259592" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_07/Smirnova-u17/covers/cover1.webp?ts=1753259592" "alt" => "" "width" => "1632" "height" => "2472" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1056} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1057} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1058} "field_url" => "/7-klass/geometrija/smirnova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1059} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 378 "subject" => 543 "class_subject" => 389 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1060 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1129359 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Начальные геометрические сведения" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333} "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1030} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1060} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1061 #items: array:2 [ 0 => App\Models\Branch {#1023} 1 => App\Models\Branch {#1062 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1129360 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Основные понятия геометрии" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1063 #items: array:1 [ 0 => App\Models\Term {#1064 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ "field_accusative_case" => "Параграф" "field_creative_case" => "Параграфом" "field_dative_case" => "Параграфу" "field_genitive_case" => "Параграфа" "field_nominative_case" => "Параграф" "field_prepositional_case" => "Параграфе" ] "field_page_check_not_needed" => null "field_short_name" => "§" ] #original: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ "field_accusative_case" => "Параграф" "field_creative_case" => "Параграфом" "field_dative_case" => "Параграфу" "field_genitive_case" => "Параграфа" "field_nominative_case" => "Параграф" "field_prepositional_case" => "Параграфе" ] "field_page_check_not_needed" => null "field_short_name" => "§" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1065 #items: array:1 [ 0 => App\Models\Book {#1032} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1066 #items: array:1 [ 0 => App\Models\Branch {#1067 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1129359 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Начальные геометрические сведения" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1068 #items: array:1 [ 0 => App\Models\Term {#1069 #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 [ …7] #original: array:7 [ …7] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1070 #items: array:1 [ 0 => App\Models\Book {#1071 #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 [ …50] #original: array:50 [ …50] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1098 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1129359 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Начальные геометрические сведения" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1068} "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1070} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1098} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1129360 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Основные понятия геометрии" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1063} "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1065} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1066} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1099 #items: array:3 [ 0 => App\Models\Element {#1100 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 1319471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1101 #items: array:1 [ 0 => App\Models\Edition {#1102 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5009 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1103 #items: array:1 [ 0 => App\Models\Term {#1104 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1105 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1106 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1107 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5009 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1103} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1105} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1106} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1107} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1129391" "type" => "task" ] "text" => "<p><strong>1.4.</strong> Изобразите: а) четыре точки; б) пять точек; в) шесть точек, никакие три из которых не принадлежат одной прямой (рис. 1.4). Проведите прямые, проходящие через различные пары из этих точек. Сколько всего таких прямых?</p><p>Рис. 1.4</p>" "img" => array:1 [ 0 => array:5 [ "name" => "4-1.jpg" "alt" => null "width" => "1774" "height" => 925 "path" => "/media/geometrija_07/Smirnova-u17/0-01/4-1.webp?ts=1753982184" ] ] ] #original: array:7 [ "id" => 1319471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1101} "task" => array:2 [ "refs" => "1129391" "type" => "task" ] "text" => "<p><strong>1.4.</strong> Изобразите: а) четыре точки; б) пять точек; в) шесть точек, никакие три из которых не принадлежат одной прямой (рис. 1.4). Проведите прямые, проходящие через различные пары из этих точек. Сколько всего таких прямых?</p><p>Рис. 1.4</p>" "img" => array:1 [ 0 => array:5 [ "name" => "4-1.jpg" "alt" => null "width" => "1774" "height" => 925 "path" => "/media/geometrija_07/Smirnova-u17/0-01/4-1.webp?ts=1753982184" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1108 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1319815 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1109 #items: array:1 [ 0 => App\Models\Edition {#1110 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5010 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1111 #items: array:1 [ 0 => App\Models\Term {#1112 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1113 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1114 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1115 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5010 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1111} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1113} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1114} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1115} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1129391" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "4-1.jpg" "alt" => null "width" => "1438" "height" => 781 "path" => "/media/geometrija_07/Smirnova-u17/1-01/4-1.webp?ts=1753982292" ] 1 => array:5 [ "name" => "4-2.jpg" "alt" => null "width" => "1438" "height" => 781 "path" => "/media/geometrija_07/Smirnova-u17/1-01/4-2.webp?ts=1753982292" ] ] ] #original: array:6 [ "id" => 1319815 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1109} "task" => array:2 [ "refs" => "1129391" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "4-1.jpg" "alt" => null "width" => "1438" "height" => 781 "path" => "/media/geometrija_07/Smirnova-u17/1-01/4-1.webp?ts=1753982292" ] 1 => array:5 [ "name" => "4-2.jpg" "alt" => null "width" => "1438" "height" => 781 "path" => "/media/geometrija_07/Smirnova-u17/1-01/4-2.webp?ts=1753982292" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Element {#1116 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1541261 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1117 #items: array:1 [ 0 => App\Models\Edition {#1118 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5631 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1119 #items: array:1 [ 0 => App\Models\Term {#1120 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1121 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1122 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1123 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5631 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1119} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1121} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1122} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1123} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1129391" "type" => "task" ] "text" => "<p><strong>а)</strong> Изобразим четыре точки A, B, C, D, никакие три из которых не лежат на одной прямой. Чтобы найти количество прямых, которые можно провести через различные пары этих точек, можно либо перечислить все возможные пары точек, либо использовать комбинаторный подход.</p><p>При перечислении получаем следующие прямые: AB, AC, AD, BC, BD, CD. Всего получается 6 прямых.</p><p>С точки зрения комбинаторики, количество прямых равно числу сочетаний из 4 элементов по 2, поскольку каждая прямая однозначно определяется двумя точками. Формула для числа сочетаний:</p><p>$C_n^k = \frac{n!}{k!(n-k)!}$</p><p>В данном случае, количество точек $n=4$, и для проведения прямой мы выбираем $k=2$ точки:</p><p>$C_4^2 = \frac{4!}{2!(4-2)!} = \frac{4!}{2! \cdot 2!} = \frac{4 \cdot 3}{2 \cdot 1} = 6$</p><p><svg width="200" height="150" viewBox="0 0 120 110"><title>Четыре точки и шесть прямых</title><style> .line { stroke: #666; stroke-width: 1; } .point { fill: red; } .label { font-family: sans-serif; font-size: 12px; } </style><!-- Линии --><line class="line" x1="20" y1="80" x2="70" y2="85"></line><line class="line" x1="20" y1="80" x2="80" y2="40"></line><line class="line" x1="20" y1="80" x2="30" y2="30"></line><line class="line" x1="70" y1="85" x2="80" y2="40"></line><line class="line" x1="70" y1="85" x2="30" y2="30"></line><line class="line" x1="80" y1="40" x2="30" y2="30"></line><!-- Точки --><circle class="point" cx="20" cy="80" r="3"></circle><circle class="point" cx="70" cy="85" r="3"></circle><circle class="point" cx="80" cy="40" r="3"></circle><circle class="point" cx="30" cy="30" r="3"></circle><!-- Метки --><text class="label" x="10" y="80">A</text><text class="label" x="75" y="95">B</text><text class="label" x="90" y="40">C</text><text class="label" x="35" y="25">D</text></svg></p><p>Ответ: 6 прямых.</p><p><strong>б)</strong> Для пяти точек, расположенных так, что никакие три из них не лежат на одной прямой, количество прямых, которые можно провести через них, также вычисляется с помощью формулы числа сочетаний. В этом случае $n=5$ и $k=2$.</p><p>$C_5^2 = \frac{5!}{2!(5-2)!} = \frac{5!}{2! \cdot 3!} = \frac{5 \cdot 4}{2 \cdot 1} = 10$</p><p>Можно провести 10 различных прямых. На рисунке ниже изображены 5 точек и все 10 прямых, проходящих через них.</p><p><svg width="200" height="150" viewBox="0 0 120 110"><title>Пять точек и десять прямых</title><style> .line { stroke: #666; stroke-width: 1; } .point { fill: red; } .label { font-family: sans-serif; font-size: 12px; } </style><!-- Координаты точек --><defs><g id="points-b"><circle id="pA" class="point" cx="30" cy="85" r="3"></circle><circle id="pB" class="point" cx="80" cy="85" r="3"></circle><circle id="pC" class="point" cx="75" cy="45" r="3"></circle><circle id="pD" class="point" cx="90" cy="20" r="3"></circle><circle id="pE" class="point" cx="40" cy="30" r="3"></circle></g></defs><!-- Линии --><line class="line" x1="30" y1="85" x2="80" y2="85"></line><!-- AB --><line class="line" x1="30" y1="85" x2="75" y2="45"></line><!-- AC --><line class="line" x1="30" y1="85" x2="90" y2="20"></line><!-- AD --><line class="line" x1="30" y1="85" x2="40" y2="30"></line><!-- AE --><line class="line" x1="80" y1="85" x2="75" y2="45"></line><!-- BC --><line class="line" x1="80" y1="85" x2="90" y2="20"></line><!-- BD --><line class="line" x1="80" y1="85" x2="40" y2="30"></line><!-- BE --><line class="line" x1="75" y1="45" x2="90" y2="20"></line><!-- CD --><line class="line" x1="75" y1="45" x2="40" y2="30"></line><!-- CE --><line class="line" x1="90" y1="20" x2="40" y2="30"></line><!-- DE --><!-- Точки --><use href="#points-b"></use><!-- Метки --><text class="label" x="20" y="85">A</text><text class="label" x="85" y="95">B</text><text class="label" x="85" y="50">C</text><text class="label" x="100" y="20">D</text><text class="label" x="45" y="25">E</text></svg></p><p>Ответ: 10 прямых.</p><p><strong>в)</strong> Для шести точек, никакие три из которых не являются коллинеарными, количество прямых, которые можно провести через них, вычисляется аналогично. Здесь $n=6$ и $k=2$.</p><p>$C_6^2 = \frac{6!}{2!(6-2)!} = \frac{6!}{2! \cdot 4!} = \frac{6 \cdot 5}{2 \cdot 1} = 15$</p><p>Следовательно, можно провести 15 различных прямых. На рисунке показаны 6 точек и 15 прямых.</p><p><svg width="200" height="150" viewBox="0 0 120 110"><title>Шесть точек и пятнадцать прямых</title><style> .line { stroke: #666; stroke-width: 1; } .point { fill: red; } .label { font-family: sans-serif; font-size: 12px; } </style><!-- Координаты точек --><defs><g id="points-c"><circle id="pA2" class="point" cx="80" cy="90" r="3"></circle><circle id="pB2" class="point" cx="95" cy="60" r="3"></circle><circle id="pC2" class="point" cx="75" cy="20" r="3"></circle><circle id="pD2" class="point" cx="10" cy="40" r="3"></circle><circle id="pE2" class="point" cx="25" cy="15" r="3"></circle><circle id="pF2" class="point" cx="45" cy="55" r="3"></circle></g></defs><!-- Линии --><line class="line" x1="80" y1="90" x2="95" y2="60"></line><!-- AB --><line class="line" x1="80" y1="90" x2="75" y2="20"></line><!-- AC --><line class="line" x1="80" y1="90" x2="10" y2="40"></line><!-- AD --><line class="line" x1="80" y1="90" x2="25" y2="15"></line><!-- AE --><line class="line" x1="80" y1="90" x2="45" y2="55"></line><!-- AF --><line class="line" x1="95" y1="60" x2="75" y2="20"></line><!-- BC --><line class="line" x1="95" y1="60" x2="10" y2="40"></line><!-- BD --><line class="line" x1="95" y1="60" x2="25" y2="15"></line><!-- BE --><line class="line" x1="95" y1="60" x2="45" y2="55"></line><!-- BF --><line class="line" x1="75" y1="20" x2="10" y2="40"></line><!-- CD --><line class="line" x1="75" y1="20" x2="25" y2="15"></line><!-- CE --><line class="line" x1="75" y1="20" x2="45" y2="55"></line><!-- CF --><line class="line" x1="10" y1="40" x2="25" y2="15"></line><!-- DE --><line class="line" x1="10" y1="40" x2="45" y2="55"></line><!-- DF --><line class="line" x1="25" y1="15" x2="45" y2="55"></line><!-- EF --><!-- Точки --><use href="#points-c"></use><!-- Метки --><text class="label" x="75" y="102">A</text><text class="label" x="105" y="60">B</text><text class="label" x="85" y="20">C</text><text class="label" x="0" y="40">D</text><text class="label" x="20" y="8">E</text><text class="label" x="40" y="70">F</text></svg></p><p>Ответ: 15 прямых.</p>" ] #original: array:6 [ "id" => 1541261 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1117} "task" => array:2 [ "refs" => "1129391" "type" => "task" ] "text" => "<p><strong>а)</strong> Изобразим четыре точки A, B, C, D, никакие три из которых не лежат на одной прямой. Чтобы найти количество прямых, которые можно провести через различные пары этих точек, можно либо перечислить все возможные пары точек, либо использовать комбинаторный подход.</p><p>При перечислении получаем следующие прямые: AB, AC, AD, BC, BD, CD. Всего получается 6 прямых.</p><p>С точки зрения комбинаторики, количество прямых равно числу сочетаний из 4 элементов по 2, поскольку каждая прямая однозначно определяется двумя точками. Формула для числа сочетаний:</p><p>$C_n^k = \frac{n!}{k!(n-k)!}$</p><p>В данном случае, количество точек $n=4$, и для проведения прямой мы выбираем $k=2$ точки:</p><p>$C_4^2 = \frac{4!}{2!(4-2)!} = \frac{4!}{2! \cdot 2!} = \frac{4 \cdot 3}{2 \cdot 1} = 6$</p><p><svg width="200" height="150" viewBox="0 0 120 110"><title>Четыре точки и шесть прямых</title><style> .line { stroke: #666; stroke-width: 1; } .point { fill: red; } .label { font-family: sans-serif; font-size: 12px; } </style><!-- Линии --><line class="line" x1="20" y1="80" x2="70" y2="85"></line><line class="line" x1="20" y1="80" x2="80" y2="40"></line><line class="line" x1="20" y1="80" x2="30" y2="30"></line><line class="line" x1="70" y1="85" x2="80" y2="40"></line><line class="line" x1="70" y1="85" x2="30" y2="30"></line><line class="line" x1="80" y1="40" x2="30" y2="30"></line><!-- Точки --><circle class="point" cx="20" cy="80" r="3"></circle><circle class="point" cx="70" cy="85" r="3"></circle><circle class="point" cx="80" cy="40" r="3"></circle><circle class="point" cx="30" cy="30" r="3"></circle><!-- Метки --><text class="label" x="10" y="80">A</text><text class="label" x="75" y="95">B</text><text class="label" x="90" y="40">C</text><text class="label" x="35" y="25">D</text></svg></p><p>Ответ: 6 прямых.</p><p><strong>б)</strong> Для пяти точек, расположенных так, что никакие три из них не лежат на одной прямой, количество прямых, которые можно провести через них, также вычисляется с помощью формулы числа сочетаний. В этом случае $n=5$ и $k=2$.</p><p>$C_5^2 = \frac{5!}{2!(5-2)!} = \frac{5!}{2! \cdot 3!} = \frac{5 \cdot 4}{2 \cdot 1} = 10$</p><p>Можно провести 10 различных прямых. На рисунке ниже изображены 5 точек и все 10 прямых, проходящих через них.</p><p><svg width="200" height="150" viewBox="0 0 120 110"><title>Пять точек и десять прямых</title><style> .line { stroke: #666; stroke-width: 1; } .point { fill: red; } .label { font-family: sans-serif; font-size: 12px; } </style><!-- Координаты точек --><defs><g id="points-b"><circle id="pA" class="point" cx="30" cy="85" r="3"></circle><circle id="pB" class="point" cx="80" cy="85" r="3"></circle><circle id="pC" class="point" cx="75" cy="45" r="3"></circle><circle id="pD" class="point" cx="90" cy="20" r="3"></circle><circle id="pE" class="point" cx="40" cy="30" r="3"></circle></g></defs><!-- Линии --><line class="line" x1="30" y1="85" x2="80" y2="85"></line><!-- AB --><line class="line" x1="30" y1="85" x2="75" y2="45"></line><!-- AC --><line class="line" x1="30" y1="85" x2="90" y2="20"></line><!-- AD --><line class="line" x1="30" y1="85" x2="40" y2="30"></line><!-- AE --><line class="line" x1="80" y1="85" x2="75" y2="45"></line><!-- BC --><line class="line" x1="80" y1="85" x2="90" y2="20"></line><!-- BD --><line class="line" x1="80" y1="85" x2="40" y2="30"></line><!-- BE --><line class="line" x1="75" y1="45" x2="90" y2="20"></line><!-- CD --><line class="line" x1="75" y1="45" x2="40" y2="30"></line><!-- CE --><line class="line" x1="90" y1="20" x2="40" y2="30"></line><!-- DE --><!-- Точки --><use href="#points-b"></use><!-- Метки --><text class="label" x="20" y="85">A</text><text class="label" x="85" y="95">B</text><text class="label" x="85" y="50">C</text><text class="label" x="100" y="20">D</text><text class="label" x="45" y="25">E</text></svg></p><p>Ответ: 10 прямых.</p><p><strong>в)</strong> Для шести точек, никакие три из которых не являются коллинеарными, количество прямых, которые можно провести через них, вычисляется аналогично. Здесь $n=6$ и $k=2$.</p><p>$C_6^2 = \frac{6!}{2!(6-2)!} = \frac{6!}{2! \cdot 4!} = \frac{6 \cdot 5}{2 \cdot 1} = 15$</p><p>Следовательно, можно провести 15 различных прямых. На рисунке показаны 6 точек и 15 прямых.</p><p><svg width="200" height="150" viewBox="0 0 120 110"><title>Шесть точек и пятнадцать прямых</title><style> .line { stroke: #666; stroke-width: 1; } .point { fill: red; } .label { font-family: sans-serif; font-size: 12px; } </style><!-- Координаты точек --><defs><g id="points-c"><circle id="pA2" class="point" cx="80" cy="90" r="3"></circle><circle id="pB2" class="point" cx="95" cy="60" r="3"></circle><circle id="pC2" class="point" cx="75" cy="20" r="3"></circle><circle id="pD2" class="point" cx="10" cy="40" r="3"></circle><circle id="pE2" class="point" cx="25" cy="15" r="3"></circle><circle id="pF2" class="point" cx="45" cy="55" r="3"></circle></g></defs><!-- Линии --><line class="line" x1="80" y1="90" x2="95" y2="60"></line><!-- AB --><line class="line" x1="80" y1="90" x2="75" y2="20"></line><!-- AC --><line class="line" x1="80" y1="90" x2="10" y2="40"></line><!-- AD --><line class="line" x1="80" y1="90" x2="25" y2="15"></line><!-- AE --><line class="line" x1="80" y1="90" x2="45" y2="55"></line><!-- AF --><line class="line" x1="95" y1="60" x2="75" y2="20"></line><!-- BC --><line class="line" x1="95" y1="60" x2="10" y2="40"></line><!-- BD --><line class="line" x1="95" y1="60" x2="25" y2="15"></line><!-- BE --><line class="line" x1="95" y1="60" x2="45" y2="55"></line><!-- BF --><line class="line" x1="75" y1="20" x2="10" y2="40"></line><!-- CD --><line class="line" x1="75" y1="20" x2="25" y2="15"></line><!-- CE --><line class="line" x1="75" y1="20" x2="45" y2="55"></line><!-- CF --><line class="line" x1="10" y1="40" x2="25" y2="15"></line><!-- DE --><line class="line" x1="10" y1="40" x2="45" y2="55"></line><!-- DF --><line class="line" x1="25" y1="15" x2="45" y2="55"></line><!-- EF --><!-- Точки --><use href="#points-c"></use><!-- Метки --><text class="label" x="75" y="102">A</text><text class="label" x="105" y="60">B</text><text class="label" x="85" y="20">C</text><text class="label" x="0" y="40">D</text><text class="label" x="20" y="8">E</text><text class="label" x="40" y="70">F</text></svg></p><p>Ответ: 15 прямых.</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" => "1129392" "type" => "task" ] "previous" => array:2 [ "refs" => "1129390" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1124 #items: array:1 [ 0 => App\Models\Book {#1032} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1139 #items: array:1 [ 0 => App\Models\BookPage {#1138 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 1114325 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "8" "field_url" => "/7-klass/geometrija/smirnova-uchebnik/page-8" "field_display_title" => "8" "field_folder" => "1" "field_image_name" => "8" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1140 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1170 #items: array:1 [ 0 => App\Models\Book {#1141 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 2907 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1143 #items: array:1 [ 0 => App\Models\Term {#1142 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1144 #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:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1146 #items: array:1 [ 0 => App\Models\Term {#1147 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] …24 } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1148 #items: array:2 [ 0 => App\Models\Term {#1149 …30} 1 => App\Models\Term {#1150 …30} ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1151 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1152 #items: array:1 [ 0 => App\Models\Term {#1153 …30} ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1154 #items: array:1 [ 0 => App\Models\Term {#1155 …30} ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1156 #items: array:1 [ 0 => App\Models\Term {#1157 …30} ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1158 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1159 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1160 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1161 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1162 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1163 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2017" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 7 класса общеобразовательной школы" "field_allowed" => "Утверждено Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "528" "field_priority" => "2" "field_default_folder" => "/geometrija_07/Smirnova-u17/" "field_isbn" => "978-601-07-0873-0" "field_cover" => array:1 [ 0 => "/media/geometrija_07/Smirnova-u17/covers/cover1.webp?ts=1753259592" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_07/Smirnova-u17/covers/cover1.webp?ts=1753259592" "alt" => "" "width" => "1632" "height" => "2472" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1164 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1165 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1166 #items: [] #escapeWhenCastingToString: false } "field_url" => "/7-klass/geometrija/smirnova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1167 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 378 "subject" => 543 "class_subject" => 389 ] ] #original: array:50 [ "id" => 2907 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1143} "field_class" => Illuminate\Database\Eloquent\Collection {#1144} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1146} "field_author" => Illuminate\Database\Eloquent\Collection {#1148} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1151} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1152} "field_country" => Illuminate\Database\Eloquent\Collection {#1154} "field_city" => Illuminate\Database\Eloquent\Collection {#1156} "field_series" => Illuminate\Database\Eloquent\Collection {#1158} "field_umk" => Illuminate\Database\Eloquent\Collection {#1159} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1160} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1161} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1162} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1163} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2017" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 7 класса общеобразовательной школы" "field_allowed" => "Утверждено Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "528" "field_priority" => "2" "field_default_folder" => "/geometrija_07/Smirnova-u17/" "field_isbn" => "978-601-07-0873-0" "field_cover" => array:1 [ 0 => "/media/geometrija_07/Smirnova-u17/covers/cover1.webp?ts=1753259592" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_07/Smirnova-u17/covers/cover1.webp?ts=1753259592" "alt" => "" "width" => "1632" "height" => "2472" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1164} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1165} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1166} "field_url" => "/7-klass/geometrija/smirnova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1167} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 378 "subject" => 543 "class_subject" => 389 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1168 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1179 #items: array:1 [ 0 => App\Models\Element {#1178 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1290512 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1188 #items: array:1 [ 0 => App\Models\Edition {#1180 …30} ] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1114325" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "8-1.jpg" "alt" => null "width" => "1906" "height" => 2818 "path" => "/media/geometrija_07/Smirnova-u17/0-1/8-1.webp?ts=1753260117" ] ] ] #original: array:6 [ "id" => 1290512 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1188} "book_page" => array:2 [ "refs" => "1114325" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "8-1.jpg" "alt" => null "width" => "1906" "height" => 2818 "path" => "/media/geometrija_07/Smirnova-u17/0-1/8-1.webp?ts=1753260117" ] ] ] #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" => "1114326" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1114324" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1484 #items: array:3 [ 0 => App\Models\Task {#1171 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1129391 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "8" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/01-4" "field_display_title" => "1.4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1169 #items: array:1 [ 0 => App\Models\Term {#1172 …30} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1175 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1173 #items: array:1 [ 0 => App\Models\Branch {#1176 …30} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1217 #items: array:2 [ 0 => App\Models\Branch {#1176 …30} 1 => App\Models\Branch {#1218 …30} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1255 #items: array:3 [ 0 => App\Models\Element {#1256 …30} 1 => App\Models\Element {#1264 …30} 2 => App\Models\Element {#1272 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1129392" "type" => "task" ] "previous" => array:2 [ "refs" => "1129390" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1280 #items: array:1 [ 0 => App\Models\Book {#1189 …30} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1114325" "type" => "book_page" ] ] #original: array:24 [ "id" => 1129391 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "8" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/01-4" "field_display_title" => "1.4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1169} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1175} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1173} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1217} "content" => Illuminate\Database\Eloquent\Collection {#1255} "next" => array:2 [ "refs" => "1129392" "type" => "task" ] "previous" => array:2 [ "refs" => "1129390" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1280} "page" => array:2 [ "refs" => "1114325" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Task {#1281 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1129392 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "8" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/01-5" "field_display_title" => "1.5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1282 #items: array:1 [ 0 => App\Models\Term {#1283 …30} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1284 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1285 #items: array:1 [ 0 => App\Models\Branch {#1286 …30} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1318 #items: array:2 [ 0 => App\Models\Branch {#1286 …30} 1 => App\Models\Branch {#1319 …30} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1356 #items: array:3 [ 0 => App\Models\Element {#1357 …30} 1 => App\Models\Element {#1365 …30} 2 => App\Models\Element {#1373 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1129393" "type" => "task" ] "previous" => array:2 [ "refs" => "1129391" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1381 #items: array:1 [ 0 => App\Models\Book {#1290 …30} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1114325" "type" => "book_page" ] ] #original: array:24 [ "id" => 1129392 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "8" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/01-5" "field_display_title" => "1.5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1282} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1284} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1285} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1318} "content" => Illuminate\Database\Eloquent\Collection {#1356} "next" => array:2 [ "refs" => "1129393" "type" => "task" ] "previous" => array:2 [ "refs" => "1129391" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1381} "page" => array:2 [ "refs" => "1114325" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Task {#1382 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1129393 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "8" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/01-6" "field_display_title" => "1.6" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1383 #items: array:1 [ 0 => App\Models\Term {#1384 …30} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1385 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1386 #items: array:1 [ 0 => App\Models\Branch {#1387 …30} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1419 #items: array:2 [ 0 => App\Models\Branch {#1387 …30} 1 => App\Models\Branch {#1420 …30} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1457 #items: array:3 [ 0 => App\Models\Element {#1458 …30} 1 => App\Models\Element {#1466 …30} 2 => App\Models\Element {#1474 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1129394" "type" => "task" ] "previous" => array:2 [ "refs" => "1129392" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1482 #items: array:1 [ 0 => App\Models\Book {#1391 …30} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1114325" "type" => "book_page" ] ] #original: array:24 [ "id" => 1129393 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "8" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/01-6" "field_display_title" => "1.6" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1383} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => 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"task" ] "book" => Illuminate\Database\Eloquent\Collection {#1124} "page" => array:2 [ "refs" => "1114325" "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 => "*" ] }
№1.4 (с. 8)
Условие. №1.4 (с. 8)
скриншот условия
1.4. Изобразите: а) четыре точки; б) пять точек; в) шесть точек, никакие три из которых не принадлежат одной прямой (рис. 1.4). Проведите прямые, проходящие через различные пары из этих точек. Сколько всего таких прямых?
Рис. 1.4
Решение. №1.4 (с. 8)
Решение 2. №1.4 (с. 8)
а) Изобразим четыре точки A, B, C, D, никакие три из которых не лежат на одной прямой. Чтобы найти количество прямых, которые можно провести через различные пары этих точек, можно либо перечислить все возможные пары точек, либо использовать комбинаторный подход.
При перечислении получаем следующие прямые: AB, AC, AD, BC, BD, CD. Всего получается 6 прямых.
С точки зрения комбинаторики, количество прямых равно числу сочетаний из 4 элементов по 2, поскольку каждая прямая однозначно определяется двумя точками. Формула для числа сочетаний:
$C_n^k = \frac{n!}{k!(n-k)!}$
В данном случае, количество точек $n=4$, и для проведения прямой мы выбираем $k=2$ точки:
$C_4^2 = \frac{4!}{2!(4-2)!} = \frac{4!}{2! \cdot 2!} = \frac{4 \cdot 3}{2 \cdot 1} = 6$
Ответ: 6 прямых.
б) Для пяти точек, расположенных так, что никакие три из них не лежат на одной прямой, количество прямых, которые можно провести через них, также вычисляется с помощью формулы числа сочетаний. В этом случае $n=5$ и $k=2$.
$C_5^2 = \frac{5!}{2!(5-2)!} = \frac{5!}{2! \cdot 3!} = \frac{5 \cdot 4}{2 \cdot 1} = 10$
Можно провести 10 различных прямых. На рисунке ниже изображены 5 точек и все 10 прямых, проходящих через них.
Ответ: 10 прямых.
в) Для шести точек, никакие три из которых не являются коллинеарными, количество прямых, которые можно провести через них, вычисляется аналогично. Здесь $n=6$ и $k=2$.
$C_6^2 = \frac{6!}{2!(6-2)!} = \frac{6!}{2! \cdot 4!} = \frac{6 \cdot 5}{2 \cdot 1} = 15$
Следовательно, можно провести 15 различных прямых. На рисунке показаны 6 точек и 15 прямых.
Ответ: 15 прямых.
Другие задания:
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