MIME-Version: 1.0 Content-Type: multipart/related; boundary="----=_NextPart_01C9DADA.21B4EA00" Данный документ является веб-страницей в одном файле, также называемой файлом веб-архива. Если вы видите это сообщение, значит данный обозреватель или редактор не поддерживает файлы веб-архива. Загрузите обозреватель, поддерживающий веб-архивы, например Microsoft Internet Explorer. ------=_NextPart_01C9DADA.21B4EA00 Content-Location: file:///C:/17254952/mhogogr.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Многогранl= 5;ики

              =            Н= . = В. Макарова

 

 

 

 

МНОГО = 43;РАННИКИ<= /p>

 

 (моног= ;рафия)

 

 

В монографии на уровне школьной программы рассказываk= 7;тся о многограннl= 0;ках всех видов. Теоретичесl= 2;ий материал сопровождаk= 7;тся большим количествоl= 4; самых разны = 93; задач о многограннl= 0;ках с решением. Есть оригин = 72;льные выводы, подходы к решению нек = 86;торых задач.

Книга адресована школьникам, учащимся ра = 79;ных учебных заведений, а также всем, кто интерес = 91;ется стереометрl= 0;ей.

 

 

СОДЕРЖh= 0;НИЕ

 

Глава 1. Общие понятия

Глава 2. Призма

Глава 2.1 Задачи о призме

Глава 2.2 Приложения=

Глава 3. Параллелепl= 0;пед

Глава 3.1 Задачи о параллелепl= 0;педе

Глав= а 4. Куб

Глав= а 4.1 Задачи о кубе

Глав= а 4.2 Приложения<= o:p>

Глава 5. Пирамида<= /b>

Глава 5.1 Задачи о правильном тетраэдре<= /span>

Глава 5.2 Правильная пирамида<= /b>

Глава 5.3 Задачи о правильной пирамиде =

Глава 6. Задачи на комбинации многограннl= 0;ков

Глава 7. Вписанные и описанные многограннl= 0;ки

Глава 8. Задачи о многограннl= 0;ках, вписанных в шар

 

Примеча = 85;ие: выделен приведённыl= 1; здесь фрагмент книги.

 

 

Глава 4. Куб

 

Кто из вас не играл в детстве с кубиками?! Кубик такой “правильныl= 1;”, как хочешь его поверни, а на гранях – буквы (азбук= 072; на кубиках) или картинк = 80;; прикладываn= 3; кубики друг = 082; другу, можно сложить слово или большую картину из отдельных е = 77; частей, изображеннm= 9;х на гранях отдельных кубиков. Был еще игральн = 99;й кубик, на гранях которого изображалоl= 9;ь число очков; помните, как кубик хорош = 86; катится при бросании? Словом, куб знаком каждому с самого детства. Какой же многограннl= 0;к является кубом?

 

Определ = 77;ние 1. Кубом называется прямоугольl= 5;ый параллелепl= 0;пед, все измерен = 80;я которого равны.

 

Таким образом, все грани куба – квадраты; граней у куб= 072; шесть, поэтому его = 085;азывают также гексаэдром, что означае = 90; в переводе с греческого R= 11; шестиграннl= 0;к. На рис. 4.1 вы видите изображениk= 7; куба и его развертки.=

 

 

 

Не забудьте, чт= 086; куб, как частный случай параллелепl= 0;педа, тоже является призмой. На языке множеств эт = 86; можно выразить та = 82;: из множеств = 72; всех выпукл = 99;х многограннl= 0;ков выделяем подмножестk= 4;о призм (см. главу 2, определениk= 7; 1), в этом подмн= ;ожестве выделяем подмножестk= 4;о четырехугоl= 3;ьных призм - это все параллелепl= 0;педы (см. главу 3, определениk= 7; 1); далее из всех паралл = 77;лепипедов выделяем подмножестk= 4;о прямоугольl= 5;ых параллелепl= 0;педов и, наконец, в этом подмножестk= 4;е выделяем вс = 77; параллелепl= 0;педы с равными измерениямl= 0; - это и есть подмножестk= 4;о кубов (мы пропустили еще подмножестk= 4;о прямых параллелепl= 0;педов, которое сод = 77;ржит в себе подмножестk= 4;о прямоугольl= 5;ых параллелепl= 0;педов (см. главу 3, определениk= 7; 2)).

На рис. 4.2 вы видите наглядное изображениk= 7; изложенногl= 6; представлеl= 5;ия.

 

 

Глядя на эту картинку, можно сказать, например, чт= 086; все кубы – параллелепl= 0;педы, все параллелепl= 0;педы – призмы, все прямоугольl= 5;ые параллелепl= 0;педы – выпуклые многограннl= 0;ки и так далее.

Ясно, что все свойства прямоугольl= 5;ого параллелепl= 0;педа имеют место = 080; для куба. Если обозна = 95;ить через а длину ребра куба, то формулы для вычисления площади боковой и полной поверхностl= 0; и объёма куб= 072; будут следующие:=

 

Sбок. =3D 4a= 2,  Sполl= 5;. =3D 6a= 2, V =3D a= 3= .

 

Куб является одним из представитk= 7;лей правильных многограннl= 0;ков. О правильны = 93; многограннl= 0;ках мы поговори = 84; позже, однак= 086; сейчас пред = 89;тавляется целесообраk= 9;ным дать опреде = 83;ение правильногl= 6; многограннl= 0;ка.

 

Определ = 77;ние 2.

Много= 075;ранник называется правильным, если все его грани – равные правильные многоугольl= 5;ики и в каждой вершине сходится одно и тоже число рёбер.

 

У правильных многограннl= 0;ков многограннm= 9;е углы тоже правильные, то есть имею= 090; равные плоские и равные двуг = 88;анные углы. У куба каждый трёх = 75;ранный угол имеет прямые плоские и прямые двугранные углы. Правильные многограннl= 0;ки – аналог правильных многоугольl= 5;иков на плоскост = 80;, как, впрочем, и неправильнm= 9;е многограннl= 0;ки. Так, паралле= 083;епипед является аналогом параллелогl= 8;амма; также как параллелогl= 8;амм, параллелепl= 0;пед имеет центр симметрии, в котором пер = 77;секаются и делятся пополам все четыре его диагонали. Прямоугольl= 5;ый параллелепl= 0;пед – аналог прямоугольl= 5;ика, куб – аналог квадрата.=

Отмет= 080;м ещё один факт, которы= 081; делает куб самым важны = 84; из всех многограннl= 0;ков: объем куба с единичным ребром принят за единицу объ = 05;ма.

 

 

 

 

Глава 4.1. Задачи о куб= 077;

 

Задача 1. Ребро куба АВСДА1С1Д1 равно 2 см. Найти расстояние между прямыми АД1 и В1С (рис. 4.3).=

 

        =           

 

 

Решение

 

Прове= 076;ём плоскость через прямы = 77; АД1 и АВ и другую плоскость через прямы = 77; В1С и СД. Эти плоскости пересекутсn= 3; по прямой МК, М и К – центры граней куба АА1Д1Д и ВВ= 1С1С соответствk= 7;нно. Прямая СД лежит в плоскости А1СД и параллельнk= 2; плоскости АВС1Д1, значит, лини= 103; пересечениn= 3; этих плоскостей параллельнk= 2; прямой СД, а так как прямая СД перпендикуl= 3;ярна к граням АА1Д и ВВ1С1С, то и прямая МК будет перпендикуl= 3;ярна к этим граням.

Следо= 074;ательно, прямая МК является общим перпе = 85;дикуляром к скрещивающl= 0;мся прямым АД1 l= 0; В1С и длина этого перпендикуl= 3;яра есть рассто = 03;ние между этими прямыми. Очевидно, чт= 086; МК =3D СД =3D 2 (см).=

 

Задача 2. Площадь сечения куб = 72; плоскостью, проходящей через концы трех рёбер, выходящих и = 79; одной вершины, равна S. Найти длину ребра куба и угол между секущ = 77;й плоскостью = 80; основанием куба (рис. 4. 4).

 

 

 

Решение

 

Очеви= 076;но, что сечение является равностороl= 5;ним треугольниl= 2;ом (каждая его сторона ест = 00; диагональ грани куба). Обозначим сторону сечения АВ1С через b, а длину ребра куба – через а. Имеем:

 

b =3D a;

S ΔАВ1С =3D b2 = / 4 =3D а2= = / 2 =3D S,

Откуда находим: а =3D / .

 

 Уго = 83; α между плоскостью сечения и плоскостью основания найдём из формулы, связывающеl= 1; площадь треугольниl= 2;а АВ1С и площадь его проекции на плоскость основания, которой является треугольниl= 2; АВС.

 

S ΔАВС= ; =3D  S ΔАВ1С * cos α         cos = α =3D  S Δ АВС / S Δ <= /span>АВ1С;

 S Δ <= /span>АВС =3D = а2/2 =3D  = S / ;         cosα =3D S /  S= =3D 1/ ;  α =3D  arcos 1/.

 

Задача 3. В кубе АВСДА1В1С1 проведена плоскость через вершину Д и середины ребер АА1 и СС1. Найти площадь полученногl= 6; сечения, острый угол = 074; нём и угол между секущей  плоскостьn= 2; и плоскость = 02; основания куба (рис. 4.5), если ребро куба равно а.

 

            <= /b>

 

Построе = 85;ие сечения

 

Ясно, что прямая К= 052; принадлежиm= 0; секущей пло = 89;кости. Так как АК =3D Мi= 7; и АКMC, то КМAC, а значит, КМ <= /b>А1С1. Итак, прямая КМ лежит в секущей плоскости и параллельнk= 2; плоскости грани А1В1&= #1057;1Д1, следователn= 0;но, линия пересечениn= 3; этих плоско = 89;тей параллельнk= 2; КМ и А1С1. Найдём точк = 91; пересечениn= 3; прямых А1Д1= и КД (точка Н), эта точка принадлежиm= 0; секущей плоскости и плоскости грани А1В1&= #1057;1Д1, значит, она принадлежиm= 0; линии их пересечениn= 3;. Проведём через эту точку пряму = 02; НР, параллельнm= 1;ю прямой А1С1= . Легко доказ = 72;ть, что прямая Н= 056; пересечёт ребро А1В1 в точке В1. Соединив точки К и В1, получим линию пересечениn= 2; секущей плоскости с гранью АВВ1; аналогично точки М и В1 принадлежаm= 0; линии пересечениn= 3; секущей пло = 89;кости с гранью ВВ1С. Сечение построено. Очевидно, чт= 086; оно являетс = 03; ромбом.

 

Решение

 

Найдём сторону сечения КД и его диагонали К = 52; и В1Д:

КД =3D ; КМ =3D АС =3D а; В1Д = =3D √ ВD2 +B1B= 2 =3D a√3

 

S КДМh= 2;1 =3D

 

Так как диагональ К = 52; меньше диагонали В1Д, то угол КДМ будет остры = 81;, обозначим его через α. Из треугольниl= 2;а КОД найдём:

 

Sin    cos .

 

Тогда

Sin α =3D 2sin α/2 cos α/2 =3D 2/5, = α =3D arcsin 2/5,<= span style=3D'mso-spacerun:yes'>  α < 90°.

 

Легко видет= ь, что основание АВСД является проекцией сечения КДМ = 42;1 на плоскост = 00; основания, поэтому

 

SАВСh= 4; =3D Sкдмв1 * cos β, где β – угол между секущей плоскостью = 80; плоскостью основания. Отсюда находим:

Cos β =3D SАВСh= 4; /SКДМh= 2;1 =3D а2 /  β =3D arccos

 

Задача 4. Ребро куба равно a. Найти расстояние от вершины куба до его диагонали, соединяющеl= 1; две другие вершины (рис. 4= .6).

 

        =   

 

 

Решение

 

Пусть надо найти расстояние от вершины А до диагонал = 80; ВД1 . Проведем через прямы = 77; АВ и ВД1 плоскость, АВД1. В этом треугольниl= 2;е проведём высоту АК, длина отрезка АК и есть искомо = 77; расстояние. Имеем:

 

.

С другой стороны,

,

откуда находим:

.

 

Задача 5. Три латунных куба с ребрами 3 см, 4 см и 5 см переплавлеl= 5;ы в один куб. Какую длину имеет ребро полученногl= 6; куба?

 

Решение

 

Понятно, что объем получ = 77;нного куба равен сумме объёмов трё = 93; первоначалn= 0;ных кубов, то есть

 

V=3D 33 + 43 + 53 =3D 216

 

Отсюда находим длину ребра этого куба:

 

.

 

Задача 6. Если каждое ребр = 86; куба увеличить н = 72; 2 см, то его объём увеличится на 98 куб. см. Чему равна длина ребра первоначалn= 0;ного куба?

 

Решение

 

Обозначим искомую длину ребра через х, запи&#= 1096;ем условие задачи:

(х + 2)3 =3D х3 + 98.

Решив это ур= ;авнение, найдём: х =3D 3 (см).=

 

Задача 7. Если каждое ребр = 86; куба увеличить н = 72; 1 м, т= ;о его объём увеличится = 74; 125 раз. Найти первоначалn= 0;ную длину ребра.

 

Решение

 

Обозначим первоначалn= 0;ную длину ребра куба через х, запишем условие задачи:

 

(х + 1)3 = =3D 125х3.

 

Поделим обе части этого уравнения н = 72; х3:

 

.

Далее имеем= :

 

 

х + 1=3D 5х, х =3D 0,25 (м).<= /b>

 

Задача 8. В кубе найти расстояние между диагональю = 86;снования и непересекаn= 2;щейся с ней диагон= 072;лью куба, если ребро куба равно а (рис. 4.7).

 

 

 

Решение

 

Докаж= 077;м, что прямая В= 044; перпендикуl= 3;ярна плоскости А = 40;1С. Плоскость А = 40;1С перпендикуl= 3;ярна плоскости основания, т= 072;к как проходи = 90; через пряму = 02; АА1, перпендикуl= 3;ярную плоскости основания. В &#= 1090;аком случае, если из точки О провести перпендикуl= 3;яр к плоскости АА1С, то он будет лежат = 00; в плоскости основания, а так как этот перпендикуl= 3;яр должен быть = 087;ерпендикул&= #1103;рен прямой АС, то это будет прямая ОД, что и требовалосn= 0; доказать. Проведём в треугольниl= 2;е АА1С ОК перпендикуl= 3;ярно А1С; вследствие доказанногl= 6; выше длина отрезка ОК будет искомым расстояниеl= 4;. Из треуголь = 85;ика А1АС находим:

 

; .

 

Из треугольниl= 2;а ОКС находим:

 

.

 

Задача 9. Найти угол между диагоналямl= 0; куба (рис. 4.8).

 

 

   Реше= ние

 

Диагонали куба равны и в точке пересечениn= 3; делятся пополам, поэтому АО=3DВО=3D (&#= 1072; – ребро). Из треугольниl= 2;а АОВ по теореме кос = 80;нусов имеем:

 

АВ2 =3D АО2 + В = 54;2 – 2ּАОּ= ВОּcosα

 

или

 

а2 =3D

 

откуда находим

 

, .

Как видите, угол между диагоналямl= 0; куба не зави= 089;ит от длины его ребра.

 

= Примечание: в рукописи книги задач = 80; о кубе есть ещё, их всего 4= 0.

 

 

Главk= 2; 4.2. Приложения=

 

А) Д= ополнительl= 5;ые сведения о кубе

 

Решая задачи, мы узнали ряд свойств куб = 72;, которые сей = 95;ас перечислим. Во всех приведённыm= 3; ниже формулах а обозначает длину ребра куба.

 

1. Сечение куба плоскостью, проходящей через концы трех рёбер, выходящих и = 79; одной вершины, является пр = 72;вильным треугольниl= 2;ом. Его площадь равна /2, = а угол, образуемый секущей плоскостью = 89; плоскостью основания куба, равен ;

 

2. Сечение куба плоскостью, проходящей через центр куба перпендикуl= 3;ярно его диагона = 83;и, является правильным треугольниl= 2;ом. Его площадь равна , &#= 1072; угол, образуемый секущей плоскостью = 89; плоскостью основания куба, равен ;

 

3. Угол между диагоналямl= 0; куба равен ;

 

4. Концы двух непараллелn= 0;ных диагоналей противолежk= 2;щих граней куба являются вершинами правильногl= 6; тетраэдра (четырехгра = 85;ника), полная поверхностn= 0; которого равна , &#= 1072; объём равен = 072;3/3;

 

5. Центры гра = 85;ей куба являются вершинами октаэдра (пр= 072;вильного восьмигранl= 5;ика), полная поверхностn= 0; которого равна а2ּ= , &#= 1072; объём равен = 072;3/6;

 

6. Наименьшеk= 7; число тетраэдров, на которое м= 086;жно разбить куб, равно пяти;

 

7. Куб можно разбить на три равные четырехугоl= 3;ьные пирамиды.=

 

В дополнение = 82; свойствам 1 и 2 заметим, что ещё только квадрат (правильный четырехугоl= 3;ьник) может быть сечением куба плоскостью. = 053;икакой другой правильный многоугольl= 5;ик не может быт= 100; сечением куба плоскостью.

 

Б) Понятие вk= 7;ктора

 

Векто= 088;ной величиной, или векторо = 84; (в широком смысле), называется всякая величина, обладающая направлениk= 7;м. Так, например, сила, действующаn= 3; на материал = 00;ную точку, есть вектор, так как она обла= 076;ает направлениk= 7;м. Скорость материальнl= 6;й точки – тоже вектор. Векторную величину мо = 78;но полностью охарактериk= 9;овать направленнm= 9;м отрезком, предваритеl= 3;ьно задав линей = 85;ый масштаб.

В геометрии вектором (в узком смысл = 77;) называется всякий направленнm= 9;й отрезок. Вектор, началом которого служит точк = 72; А, а концом – точка В, обозначаетl= 9;я. Под длиной вектора = 87;онимается длина отрезка АВ. Длина векто = 88;а называется также его модулем, обозначаетl= 9;я модуль вектора так:

или просто – АВ.

Прямо= 091;гольными координатаl= 4;и вектора наз = 99;ваются алгебраичеl= 9;кие проекции этого векто = 88;а на оси координат (рис. 4.1.1.). Координаты вектора записывают рядом с обоз= 085;ачением вектора в фигурных скобках. Оче= 074;идно, что

, .

 

 

 

Аналогичны&= #1077; формулы имеют место для вектора = 074; пространстk= 4;е (рис. 4.1.2):

=  

, .

 

Так как расстояние от точки А до точки В равн= 086; длине отрезка АВ, то последня = 03; формула служит для определениn= 3; расстояния между двумя точками пространстk= 4;а, если известны координаты этих точек.

Угол α между векторами  можно найти по следующей формуле:

 

(1) .

 

Если вы посмотрите на эту формулу применителn= 0;но к двум векторам на = 087;лоскости, то легко узнаете в не= 081; хорошо изве = 89;тную теорему косинусов.=

А теперь на примерах покажем практическl= 6;е применение = 87;риведённых формул.

 

Пример 1. В кубе АВСДА1В1С1 найти расстояние = 72;) между центрами граней АА1Д= ;1Д и А1В1С1Д1; б) между серединами = 89;крещивающи&#= 1093;ся ребер; в) от вершины С1 до центра грани АА1Д1= Д; г) между точками M и N, если А= 1M= =3DBN= =3D1/3a, где а – длина ребра куба; д) между точками А и Е, если В1Е:ЕС= 1=3Dn= :m=

 

 

Как вы решали бы эт= 080; задачи, не зная формул = 99; расстояния между двумя точками? Кон= 077;чно, построили один или два прямоугольl= 5;ых треугольниl= 2;а и по теореме Пифагора нашли нужно = 77; расстояние. Теперь нет необходимоl= 9;ти строить треугольниl= 2;и и применять теорему Пифагора, которая, безусловно, является самой заслуженноl= 1; в геометрии. Отметим, что расстояние между точка = 84;и А и В обозначаетl= 9;я символом ρ(А,= 042;) и приступим = 082; решению задачи.

а) Пусть точки О1 и О2 – центры граней АА1Д= ;1Д и А1В1С1Д1 соответствk= 7;нно. Запишем координаты этих точек:

 

, ;

.

 

б) Пусть точки К и L – середины скрещивающl= 0;хся ребер АД и СС1 соответствk= 7;нно. Запишем их координаты = 80; найдём расстояние между ними:

 

, ;

.

 

 

в) , ;

.

 

г) , ;

.

 

д) A (a; 0; 0),   E (0, an/(n + m); a= )

 

ρ(A= ,E) =3D a/(n+m)√2(n+m)2 + n2

 

Если применить последнюю формулу к задаче 13 главы 4, где В1<= /sub>Е:ЕС1=3D2:1, то получим:

 

ρ(A,E) =3D= a√22/3 .

 

Пример г) тоже взят из этой задачи. Итак, мы науч&#= 1080;лись находить расстояние между двумя точками пространстk= 4;а по их координатаl= 4;.

 

Пример 2. Используя формулу (1), решить задачу 27 главы 4.

 

Смотрите на рис. 4.2.8 и 4.1.3. Чтоб= 099; найти угол между прямыми, над= 086; найти угол между векторами, лежащими на этих прямых. Так, в случае а) найдём угол между векторами А = 40;1 и. Имеем:

 

;

; ;

, .

Сразу заметим, что если изменить направлениk= 7; одного из векторов, то косинус угл = 72; у вас получи= 090;ся отрицательl= 5;ым, а абсолютно = 77; значение ег = 86; будет таким же, это значит, что вы нашли не острый угол между прямы = 84;и, а его дополнение до 180 градусов= .

В случае б) искомый уго = 83; найдём как угол между векторами  и .

; ;

; ;

 cos α =3D 1/2; α =3D 60º.

 

И, наконец, в случае в) найдём искомый уго = 83; как угол между векторами   и .

, ; ; ;

, &#= 945; =3D 90º.

 

Пример 3. Решить с применениеl= 4; формулы (1) задачу 9 глав&#= 1099; 4 (рис. 4.1.4).

 

 

Рис. 4.1= .4.

 

Задача 9 решена с применениеl= 4; теоремы кос = 80;нусов. Возьмем теперь друг = 91;ю пару диагоналей куба и решим задачу по фо= 088;муле (1).

=  

= ; ; ;

= ;

= ;

= ;

= ; .

= Сделаем последнее полезное замечание: е= 089;ли вектор , то вектор .

 

 

 

Фрагмен = 90; последний раз редактировk= 2;лся 22 мая 2009 г.

 

На одном форум = 77; вела переговоры = 89; участником = 86; том, чтобы сделать хорошие иллюстрациl= 0; к этой книге. &= #1054;н вроде сначала согласился = 80; даже присла = 83; несколько образцов иллюстрациl= 1;, но потом замолчал. По= 082;ажу его иллюстрациl= 0;.

 

 

 

Очен = 00; хотелось бы сделать хот = 03; бы электрон = 85;ую версию этой книги, ведь книга полез = 85;ая. На форуме, например, не раз видела просьбы пом = 86;чь в решении задач по стереометрl= 0;и, в том числе и о многограннl= 0;ках. Жалко, что рукопись пропадёт. Даже одна гл= 072;ва “Куб” из рук= 086;писи представлеl= 5;а частично. В главе сорок задач, а представлеl= 5;о только 9.

Я не умею набирать математичеl= 9;кие формулы. Дан= 085;ый фрагмент мн = 77; набирали в литературнl= 6;м агентстве. П= 088;и наборе было сделано очень много ошибок. Я уже несколько раз редакти = 88;овала текст, но так и не исправила все ошибки, где-то прост= 086; не могу это сделать, потому что ошибка в фор= 084;уле, а я не могу её редактировk= 2;ть.

Если найдутся желающие набрать текст книги, сделать красивые иллюстрациl= 0;, буду очень признательl= 5;а.

 

Наталия Макарова<= /i>

 

 

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