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Номер 626, страница 175 - гдз по геометрии 11 класс учебник Солтан, Солтан
Авторы: Солтан Г. Н., Солтан А. Е., Жумадилова А. Ж.
Тип: Учебник
Издательство: Кокшетау
Год издания: 2020 - 2026
ISBN: 978-601-317-528-7
Рекомендовано Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 11 классе
Повторение курса геометрии 10-11 классов - номер 626, страница 175.
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" => 1144846 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "175" "field_page_end" => null "field_url" => "/11-klass/geometrija/kokshetau-soltan-uchebnik/upr-626" "field_display_title" => "626" "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 {#1046 #items: array:1 [ 0 => App\Models\Branch {#1045 #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" => 1144195 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Повторение курса геометрии 10-11 классов" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1047 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "173" "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 {#1078 #items: array:1 [ 0 => App\Models\Book {#1048 #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" => 4313 "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 {#1050 #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: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 {#1051 #items: array:1 [ 0 => App\Models\Term {#1052 #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" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ "field_accusative_case" => "одиннадцатый" "field_creative_case" => "одиннадцатым" "field_dative_case" => "одиннадцатому" "field_genitive_case" => "одинадцатого" "field_nominative_case" => "одинадцатый" "field_prepositional_case" => "одиннадцатом" ] "field_translit" => "odinadcatyj" ] #original: array:6 [ "id" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ "field_accusative_case" => "одиннадцатый" "field_creative_case" => "одиннадцатым" "field_dative_case" => "одиннадцатому" "field_genitive_case" => "одинадцатого" "field_nominative_case" => "одинадцатый" "field_prepositional_case" => "одиннадцатом" ] "field_translit" => "odinadcatyj" ] #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 {#1053 #items: array:1 [ 0 => App\Models\Term {#1054 #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" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "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" => "Kokshetau" ] #original: array:6 [ "id" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "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" => "Kokshetau" ] #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 {#1055 #items: array:3 [ 0 => App\Models\Term {#1056 #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" => 5836 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Г" "field_patronymic" => "Н" "field_surname_rp" => null ] #original: array:12 [ "id" => 5836 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Г" "field_patronymic" => "Н" "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 {#1057 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7004 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алла" "field_patronymic" => "Евгеньевна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7004 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алла" "field_patronymic" => "Евгеньевна" "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 => "*" ] } 2 => App\Models\Term {#1058 #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" => 7005 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумадилова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Аманбала" "field_patronymic" => "Жумадиловна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7005 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумадилова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Аманбала" "field_patronymic" => "Жумадиловна" "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 {#1059 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1060 #items: array:1 [ 0 => App\Models\Term {#1061 #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 {#1062 #items: array:1 [ 0 => App\Models\Term {#1063 #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 {#1064 #items: array:1 [ 0 => App\Models\Term {#1065 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "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 {#1066 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1067 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1068 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1069 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1070 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1071 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2020" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "для учащихся 10 (ч. 1), 11 (ч. 2) классов естественно-математического направления общеобразовательной школы" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "679" "field_priority" => "8" "field_default_folder" => "/geometrija_11/Kokshetau-u20/" "field_isbn" => "978-601-317-528-7" "field_cover" => array:1 [ 0 => "/media/geometrija_11/Kokshetau-u20/covers/cover1.webp?ts=1753338288" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_11/Kokshetau-u20/covers/cover1.webp?ts=1753338288" "alt" => "" "width" => "1632" "height" => "2472" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1072 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1073 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1074 #items: [] #escapeWhenCastingToString: false } "field_url" => "/11-klass/geometrija/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1075 #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" => 382 "subject" => 543 "class_subject" => 393 ] ] #original: array:50 [ "id" => 4313 "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 {#1050} "field_class" => Illuminate\Database\Eloquent\Collection {#1051} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1053} "field_author" => Illuminate\Database\Eloquent\Collection {#1055} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1059} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1060} "field_country" => Illuminate\Database\Eloquent\Collection {#1062} "field_city" => Illuminate\Database\Eloquent\Collection {#1064} "field_series" => Illuminate\Database\Eloquent\Collection {#1066} "field_umk" => Illuminate\Database\Eloquent\Collection {#1067} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1068} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1069} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1070} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1071} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2020" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "для учащихся 10 (ч. 1), 11 (ч. 2) классов естественно-математического направления общеобразовательной школы" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "679" "field_priority" => "8" "field_default_folder" => "/geometrija_11/Kokshetau-u20/" "field_isbn" => "978-601-317-528-7" "field_cover" => array:1 [ 0 => "/media/geometrija_11/Kokshetau-u20/covers/cover1.webp?ts=1753338288" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_11/Kokshetau-u20/covers/cover1.webp?ts=1753338288" "alt" => "" "width" => "1632" "height" => "2472" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1072} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1073} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1074} "field_url" => "/11-klass/geometrija/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1075} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 382 "subject" => 543 "class_subject" => 393 ] ] #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 {#1076 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1144195 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Повторение курса геометрии 10-11 классов" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1047} "field_page_start" => "173" "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 {#1078} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1076} ] #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 {#1038 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1042 #items: array:3 [ 0 => App\Models\Element {#1083 #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" => 1380402 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1093 #items: array:1 [ 0 => App\Models\Edition {#1092 #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" => 5039 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1097 #items: array:1 [ 0 => App\Models\Term {#1094 #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 {#1101 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1095 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1098 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5039 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1097} "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 {#1101} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1095} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1098} "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" => "1144846" "type" => "task" ] "text" => "<p><strong>626.</strong> Основание пирамиды $DABC - \triangle ABC$, в котором $AB = AC = 25$ см, $BC = 40$ см. Грань $BCD$ перпендикулярна основанию. Расстояние от основания высоты $DM$ пирамиды до грани $ACD$ равно $6\sqrt{2}$ см. Найдите площадь боковой поверхности пирамиды.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "626-1.jpg" "alt" => null "width" => "1538" "height" => 308 "path" => "/media/geometrija_11/Kokshetau-u20/0-upr/626-1.webp?ts=1754600635" ] ] ] #original: array:7 [ "id" => 1380402 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1093} "task" => array:2 [ "refs" => "1144846" "type" => "task" ] "text" => "<p><strong>626.</strong> Основание пирамиды $DABC - \triangle ABC$, в котором $AB = AC = 25$ см, $BC = 40$ см. Грань $BCD$ перпендикулярна основанию. Расстояние от основания высоты $DM$ пирамиды до грани $ACD$ равно $6\sqrt{2}$ см. Найдите площадь боковой поверхности пирамиды.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "626-1.jpg" "alt" => null "width" => "1538" "height" => 308 "path" => "/media/geometrija_11/Kokshetau-u20/0-upr/626-1.webp?ts=1754600635" ] ] ] #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 {#1085 #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" => 1380725 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1102 #items: array:1 [ 0 => App\Models\Edition {#1086 #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" => 5040 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1089 #items: array:1 [ 0 => App\Models\Term {#1084 #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 {#1087 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1090 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1088 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5040 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1089} "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 {#1087} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1090} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1088} "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" => "1144846" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "626-1.jpg" "alt" => null "width" => "1459" "height" => 501 "path" => "/media/geometrija_11/Kokshetau-u20/1-upr/626-1.webp?ts=1754600692" ] ] ] #original: array:6 [ "id" => 1380725 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1102} "task" => array:2 [ "refs" => "1144846" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "626-1.jpg" "alt" => null "width" => "1459" "height" => 501 "path" => "/media/geometrija_11/Kokshetau-u20/1-upr/626-1.webp?ts=1754600692" ] ] ] #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 {#1096 #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" => 1562931 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1110 #items: array:1 [ 0 => App\Models\Edition {#1099 #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" => 5685 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1103 #items: array:1 [ 0 => App\Models\Term {#1100 #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 {#1104 #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" => 5685 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1103} "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 {#1104} "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" => "1144846" "type" => "task" ] "text" => "<p><strong>1.</strong> Анализ основания и геометрии пирамиды.</p><p>Основанием пирамиды является равнобедренный треугольник $ABC$ со сторонами $AB = AC = 25$ см и $BC = 40$ см. Проведем высоту $AK$ к основанию $BC$ в треугольнике $ABC$. Так как треугольник равнобедренный, $AK$ является также медианой, поэтому $K$ – середина $BC$, и $BK = KC = BC/2 = 40/2 = 20$ см. Из прямоугольного треугольника $AKC$ по теореме Пифагора найдем высоту $AK$: $AK = \sqrt{AC^2 - KC^2} = \sqrt{25^2 - 20^2} = \sqrt{625 - 400} = \sqrt{225} = 15$ см.</p><p>По условию, грань $BCD$ перпендикулярна плоскости основания $ABC$. Высота пирамиды $DM$ опущена из вершины $D$ на плоскость основания. Так как $D$ лежит в плоскости $BCD$ и $DM \perp (ABC)$, то основание высоты $M$ должно лежать на линии пересечения плоскостей, то есть на прямой $BC$. Таким образом, высота пирамиды $DM$ является также высотой треугольника $BCD$, проведенной к стороне $BC$. Обозначим высоту пирамиды $H = DM$.</p><p><strong>2.</strong> Нахождение высоты пирамиды.</p><p>Для нахождения расстояния от точки $M$ до плоскости грани $ACD$ используем метод перпендикуляров. Проведем из точки $M$ в плоскости основания перпендикуляр $MP$ к прямой $AC$. Так как $DM \perp (ABC)$, то $DM \perp AC$. Поскольку прямая $AC$ перпендикулярна двум пересекающимся прямым $DM$ и $MP$ в плоскости $DMP$, то $AC \perp (DMP)$. Следовательно, плоскость $ACD$ перпендикулярна плоскости $DMP$. Расстояние от точки $M$ до плоскости $ACD$ – это длина перпендикуляра $MQ$, опущенного из $M$ на линию пересечения этих плоскостей, то есть на прямую $DP$. Таким образом, $MQ \perp DP$, и по условию $MQ = 6\sqrt{2}$ см.</p><p>Рассмотрим прямоугольный треугольник $DMP$ (угол $DMP = 90^\circ$). $DM = H$ – высота пирамиды, $MP$ – расстояние от $M$ до $AC$. $MQ$ – высота, проведенная к гипотенузе $DP$. Для такого треугольника справедливо соотношение: $ \frac{1}{MQ^2} = \frac{1}{DM^2} + \frac{1}{MP^2} $</p><p>Найдем длину $MP$. В плоскости основания рассмотрим треугольники $MPC$ и $AKC$. Они оба прямоугольные ($\angle MPC = \angle AKC = 90^\circ$) и имеют общий угол $C$. Следовательно, $\triangle MPC \sim \triangle AKC$. Из подобия следует: $\frac{MP}{AK} = \frac{MC}{AC}$. Отсюда $MP = \frac{AK \cdot MC}{AC} = \frac{15 \cdot MC}{25} = \frac{3}{5}MC$.</p><p>Подставим все в формулу для высоты: $ \frac{1}{(6\sqrt{2})^2} = \frac{1}{H^2} + \frac{1}{(\frac{3}{5}MC)^2} \implies \frac{1}{72} = \frac{1}{H^2} + \frac{25}{9 \cdot MC^2} $</p><p>Так как в условии не дано положение точки $M$ на $BC$, а задача должна иметь единственное решение, следует предположить симметричность конструкции относительно плоскости $ADK$ (где $K$ - середина $BC$), что обусловлено равнобедренным треугольником в основании. В этом случае точка $M$ совпадает с точкой $K$. Тогда $MC = KC = 20$ см. Подставим это значение: $ \frac{1}{72} = \frac{1}{H^2} + \frac{25}{9 \cdot 20^2} = \frac{1}{H^2} + \frac{25}{9 \cdot 400} = \frac{1}{H^2} + \frac{1}{9 \cdot 16} = \frac{1}{H^2} + \frac{1}{144} $ $ \frac{1}{H^2} = \frac{1}{72} - \frac{1}{144} = \frac{2}{144} - \frac{1}{144} = \frac{1}{144} $ $ H^2 = 144 \implies H = 12 $ см.</p><p><strong>3.</strong> Вычисление площади боковой поверхности.</p><p>Площадь боковой поверхности $S_{бок}$ равна сумме площадей граней $BCD$, $ACD$ и $ABD$. $S_{бок} = S_{BCD} + S_{ACD} + S_{ABD}$.</p><p><strong>Площадь грани BCD:</strong> Эта грань является треугольником с основанием $BC=40$ см и высотой $DM=DK=H=12$ см. $ S_{BCD} = \frac{1}{2} \cdot BC \cdot H = \frac{1}{2} \cdot 40 \cdot 12 = 240 $ см<sup>2</sup>.</p><p><strong>Площади граней ACD и ABD:</strong> Поскольку $M=K$, пирамида симметрична, и грани $ACD$ и $ABD$ равны, их площади также равны. Найдем площадь $S_{ACD}$. $S_{ACD} = \frac{1}{2} \cdot AC \cdot DP$, где $DP$ - апофема (высота грани $ACD$, проведенная к стороне $AC$). $DP$ является гипотенузой в прямоугольном треугольнике $DKP$, где $DK = H = 12$ см. Найдем катет $KP$. Так как $M=K$, то $MP=KP$. Из подобия $\triangle KPC \sim \triangle AKC$: $ KP = \frac{AK \cdot KC}{AC} = \frac{15 \cdot 20}{25} = 12 $ см. Теперь по теореме Пифагора для $\triangle DKP$: $ DP = \sqrt{DK^2 + KP^2} = \sqrt{12^2 + 12^2} = \sqrt{144 + 144} = \sqrt{2 \cdot 144} = 12\sqrt{2} $ см. Теперь найдем площадь грани $ACD$: $ S_{ACD} = \frac{1}{2} \cdot AC \cdot DP = \frac{1}{2} \cdot 25 \cdot 12\sqrt{2} = 150\sqrt{2} $ см<sup>2</sup>. Так как $S_{ABD} = S_{ACD}$, то $S_{ABD} = 150\sqrt{2}$ см<sup>2</sup>.</p><p><strong>Общая площадь боковой поверхности:</strong> $ S_{бок} = S_{BCD} + S_{ACD} + S_{ABD} = 240 + 150\sqrt{2} + 150\sqrt{2} = 240 + 300\sqrt{2} $ см<sup>2</sup>.</p><p><strong>Ответ:</strong> $240 + 300\sqrt{2}$ см<sup>2</sup>.</p>" ] #original: array:6 [ "id" => 1562931 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1110} "task" => array:2 [ "refs" => "1144846" "type" => "task" ] "text" => "<p><strong>1.</strong> Анализ основания и геометрии пирамиды.</p><p>Основанием пирамиды является равнобедренный треугольник $ABC$ со сторонами $AB = AC = 25$ см и $BC = 40$ см. Проведем высоту $AK$ к основанию $BC$ в треугольнике $ABC$. Так как треугольник равнобедренный, $AK$ является также медианой, поэтому $K$ – середина $BC$, и $BK = KC = BC/2 = 40/2 = 20$ см. Из прямоугольного треугольника $AKC$ по теореме Пифагора найдем высоту $AK$: $AK = \sqrt{AC^2 - KC^2} = \sqrt{25^2 - 20^2} = \sqrt{625 - 400} = \sqrt{225} = 15$ см.</p><p>По условию, грань $BCD$ перпендикулярна плоскости основания $ABC$. Высота пирамиды $DM$ опущена из вершины $D$ на плоскость основания. Так как $D$ лежит в плоскости $BCD$ и $DM \perp (ABC)$, то основание высоты $M$ должно лежать на линии пересечения плоскостей, то есть на прямой $BC$. Таким образом, высота пирамиды $DM$ является также высотой треугольника $BCD$, проведенной к стороне $BC$. Обозначим высоту пирамиды $H = DM$.</p><p><strong>2.</strong> Нахождение высоты пирамиды.</p><p>Для нахождения расстояния от точки $M$ до плоскости грани $ACD$ используем метод перпендикуляров. Проведем из точки $M$ в плоскости основания перпендикуляр $MP$ к прямой $AC$. Так как $DM \perp (ABC)$, то $DM \perp AC$. Поскольку прямая $AC$ перпендикулярна двум пересекающимся прямым $DM$ и $MP$ в плоскости $DMP$, то $AC \perp (DMP)$. Следовательно, плоскость $ACD$ перпендикулярна плоскости $DMP$. Расстояние от точки $M$ до плоскости $ACD$ – это длина перпендикуляра $MQ$, опущенного из $M$ на линию пересечения этих плоскостей, то есть на прямую $DP$. Таким образом, $MQ \perp DP$, и по условию $MQ = 6\sqrt{2}$ см.</p><p>Рассмотрим прямоугольный треугольник $DMP$ (угол $DMP = 90^\circ$). $DM = H$ – высота пирамиды, $MP$ – расстояние от $M$ до $AC$. $MQ$ – высота, проведенная к гипотенузе $DP$. Для такого треугольника справедливо соотношение: $ \frac{1}{MQ^2} = \frac{1}{DM^2} + \frac{1}{MP^2} $</p><p>Найдем длину $MP$. В плоскости основания рассмотрим треугольники $MPC$ и $AKC$. Они оба прямоугольные ($\angle MPC = \angle AKC = 90^\circ$) и имеют общий угол $C$. Следовательно, $\triangle MPC \sim \triangle AKC$. Из подобия следует: $\frac{MP}{AK} = \frac{MC}{AC}$. Отсюда $MP = \frac{AK \cdot MC}{AC} = \frac{15 \cdot MC}{25} = \frac{3}{5}MC$.</p><p>Подставим все в формулу для высоты: $ \frac{1}{(6\sqrt{2})^2} = \frac{1}{H^2} + \frac{1}{(\frac{3}{5}MC)^2} \implies \frac{1}{72} = \frac{1}{H^2} + \frac{25}{9 \cdot MC^2} $</p><p>Так как в условии не дано положение точки $M$ на $BC$, а задача должна иметь единственное решение, следует предположить симметричность конструкции относительно плоскости $ADK$ (где $K$ - середина $BC$), что обусловлено равнобедренным треугольником в основании. В этом случае точка $M$ совпадает с точкой $K$. Тогда $MC = KC = 20$ см. Подставим это значение: $ \frac{1}{72} = \frac{1}{H^2} + \frac{25}{9 \cdot 20^2} = \frac{1}{H^2} + \frac{25}{9 \cdot 400} = \frac{1}{H^2} + \frac{1}{9 \cdot 16} = \frac{1}{H^2} + \frac{1}{144} $ $ \frac{1}{H^2} = \frac{1}{72} - \frac{1}{144} = \frac{2}{144} - \frac{1}{144} = \frac{1}{144} $ $ H^2 = 144 \implies H = 12 $ см.</p><p><strong>3.</strong> Вычисление площади боковой поверхности.</p><p>Площадь боковой поверхности $S_{бок}$ равна сумме площадей граней $BCD$, $ACD$ и $ABD$. $S_{бок} = S_{BCD} + S_{ACD} + S_{ABD}$.</p><p><strong>Площадь грани BCD:</strong> Эта грань является треугольником с основанием $BC=40$ см и высотой $DM=DK=H=12$ см. $ S_{BCD} = \frac{1}{2} \cdot BC \cdot H = \frac{1}{2} \cdot 40 \cdot 12 = 240 $ см<sup>2</sup>.</p><p><strong>Площади граней ACD и ABD:</strong> Поскольку $M=K$, пирамида симметрична, и грани $ACD$ и $ABD$ равны, их площади также равны. Найдем площадь $S_{ACD}$. $S_{ACD} = \frac{1}{2} \cdot AC \cdot DP$, где $DP$ - апофема (высота грани $ACD$, проведенная к стороне $AC$). $DP$ является гипотенузой в прямоугольном треугольнике $DKP$, где $DK = H = 12$ см. Найдем катет $KP$. Так как $M=K$, то $MP=KP$. Из подобия $\triangle KPC \sim \triangle AKC$: $ KP = \frac{AK \cdot KC}{AC} = \frac{15 \cdot 20}{25} = 12 $ см. Теперь по теореме Пифагора для $\triangle DKP$: $ DP = \sqrt{DK^2 + KP^2} = \sqrt{12^2 + 12^2} = \sqrt{144 + 144} = \sqrt{2 \cdot 144} = 12\sqrt{2} $ см. Теперь найдем площадь грани $ACD$: $ S_{ACD} = \frac{1}{2} \cdot AC \cdot DP = \frac{1}{2} \cdot 25 \cdot 12\sqrt{2} = 150\sqrt{2} $ см<sup>2</sup>. Так как $S_{ABD} = S_{ACD}$, то $S_{ABD} = 150\sqrt{2}$ см<sup>2</sup>.</p><p><strong>Общая площадь боковой поверхности:</strong> $ S_{бок} = S_{BCD} + S_{ACD} + S_{ABD} = 240 + 150\sqrt{2} + 150\sqrt{2} = 240 + 300\sqrt{2} $ см<sup>2</sup>.</p><p><strong>Ответ:</strong> $240 + 300\sqrt{2}$ см<sup>2</sup>.</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" => "1144847" "type" => "task" ] "previous" => array:2 [ "refs" => "1144845" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1080 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1077 #items: array:1 [ 0 => App\Models\BookPage 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№626 (с. 175)
Условие. №626 (с. 175)
скриншот условия
626. Основание пирамиды $DABC - \triangle ABC$, в котором $AB = AC = 25$ см, $BC = 40$ см. Грань $BCD$ перпендикулярна основанию. Расстояние от основания высоты $DM$ пирамиды до грани $ACD$ равно $6\sqrt{2}$ см. Найдите площадь боковой поверхности пирамиды.
Решение. №626 (с. 175)
Решение 2 (rus). №626 (с. 175)
1. Анализ основания и геометрии пирамиды.
Основанием пирамиды является равнобедренный треугольник $ABC$ со сторонами $AB = AC = 25$ см и $BC = 40$ см. Проведем высоту $AK$ к основанию $BC$ в треугольнике $ABC$. Так как треугольник равнобедренный, $AK$ является также медианой, поэтому $K$ – середина $BC$, и $BK = KC = BC/2 = 40/2 = 20$ см. Из прямоугольного треугольника $AKC$ по теореме Пифагора найдем высоту $AK$: $AK = \sqrt{AC^2 - KC^2} = \sqrt{25^2 - 20^2} = \sqrt{625 - 400} = \sqrt{225} = 15$ см.
По условию, грань $BCD$ перпендикулярна плоскости основания $ABC$. Высота пирамиды $DM$ опущена из вершины $D$ на плоскость основания. Так как $D$ лежит в плоскости $BCD$ и $DM \perp (ABC)$, то основание высоты $M$ должно лежать на линии пересечения плоскостей, то есть на прямой $BC$. Таким образом, высота пирамиды $DM$ является также высотой треугольника $BCD$, проведенной к стороне $BC$. Обозначим высоту пирамиды $H = DM$.
2. Нахождение высоты пирамиды.
Для нахождения расстояния от точки $M$ до плоскости грани $ACD$ используем метод перпендикуляров. Проведем из точки $M$ в плоскости основания перпендикуляр $MP$ к прямой $AC$. Так как $DM \perp (ABC)$, то $DM \perp AC$. Поскольку прямая $AC$ перпендикулярна двум пересекающимся прямым $DM$ и $MP$ в плоскости $DMP$, то $AC \perp (DMP)$. Следовательно, плоскость $ACD$ перпендикулярна плоскости $DMP$. Расстояние от точки $M$ до плоскости $ACD$ – это длина перпендикуляра $MQ$, опущенного из $M$ на линию пересечения этих плоскостей, то есть на прямую $DP$. Таким образом, $MQ \perp DP$, и по условию $MQ = 6\sqrt{2}$ см.
Рассмотрим прямоугольный треугольник $DMP$ (угол $DMP = 90^\circ$). $DM = H$ – высота пирамиды, $MP$ – расстояние от $M$ до $AC$. $MQ$ – высота, проведенная к гипотенузе $DP$. Для такого треугольника справедливо соотношение: $ \frac{1}{MQ^2} = \frac{1}{DM^2} + \frac{1}{MP^2} $
Найдем длину $MP$. В плоскости основания рассмотрим треугольники $MPC$ и $AKC$. Они оба прямоугольные ($\angle MPC = \angle AKC = 90^\circ$) и имеют общий угол $C$. Следовательно, $\triangle MPC \sim \triangle AKC$. Из подобия следует: $\frac{MP}{AK} = \frac{MC}{AC}$. Отсюда $MP = \frac{AK \cdot MC}{AC} = \frac{15 \cdot MC}{25} = \frac{3}{5}MC$.
Подставим все в формулу для высоты: $ \frac{1}{(6\sqrt{2})^2} = \frac{1}{H^2} + \frac{1}{(\frac{3}{5}MC)^2} \implies \frac{1}{72} = \frac{1}{H^2} + \frac{25}{9 \cdot MC^2} $
Так как в условии не дано положение точки $M$ на $BC$, а задача должна иметь единственное решение, следует предположить симметричность конструкции относительно плоскости $ADK$ (где $K$ - середина $BC$), что обусловлено равнобедренным треугольником в основании. В этом случае точка $M$ совпадает с точкой $K$. Тогда $MC = KC = 20$ см. Подставим это значение: $ \frac{1}{72} = \frac{1}{H^2} + \frac{25}{9 \cdot 20^2} = \frac{1}{H^2} + \frac{25}{9 \cdot 400} = \frac{1}{H^2} + \frac{1}{9 \cdot 16} = \frac{1}{H^2} + \frac{1}{144} $ $ \frac{1}{H^2} = \frac{1}{72} - \frac{1}{144} = \frac{2}{144} - \frac{1}{144} = \frac{1}{144} $ $ H^2 = 144 \implies H = 12 $ см.
3. Вычисление площади боковой поверхности.
Площадь боковой поверхности $S_{бок}$ равна сумме площадей граней $BCD$, $ACD$ и $ABD$. $S_{бок} = S_{BCD} + S_{ACD} + S_{ABD}$.
Площадь грани BCD: Эта грань является треугольником с основанием $BC=40$ см и высотой $DM=DK=H=12$ см. $ S_{BCD} = \frac{1}{2} \cdot BC \cdot H = \frac{1}{2} \cdot 40 \cdot 12 = 240 $ см2.
Площади граней ACD и ABD: Поскольку $M=K$, пирамида симметрична, и грани $ACD$ и $ABD$ равны, их площади также равны. Найдем площадь $S_{ACD}$. $S_{ACD} = \frac{1}{2} \cdot AC \cdot DP$, где $DP$ - апофема (высота грани $ACD$, проведенная к стороне $AC$). $DP$ является гипотенузой в прямоугольном треугольнике $DKP$, где $DK = H = 12$ см. Найдем катет $KP$. Так как $M=K$, то $MP=KP$. Из подобия $\triangle KPC \sim \triangle AKC$: $ KP = \frac{AK \cdot KC}{AC} = \frac{15 \cdot 20}{25} = 12 $ см. Теперь по теореме Пифагора для $\triangle DKP$: $ DP = \sqrt{DK^2 + KP^2} = \sqrt{12^2 + 12^2} = \sqrt{144 + 144} = \sqrt{2 \cdot 144} = 12\sqrt{2} $ см. Теперь найдем площадь грани $ACD$: $ S_{ACD} = \frac{1}{2} \cdot AC \cdot DP = \frac{1}{2} \cdot 25 \cdot 12\sqrt{2} = 150\sqrt{2} $ см2. Так как $S_{ABD} = S_{ACD}$, то $S_{ABD} = 150\sqrt{2}$ см2.
Общая площадь боковой поверхности: $ S_{бок} = S_{BCD} + S_{ACD} + S_{ABD} = 240 + 150\sqrt{2} + 150\sqrt{2} = 240 + 300\sqrt{2} $ см2.
Ответ: $240 + 300\sqrt{2}$ см2.
Другие задания:
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