Question

ЕЩЁ СРОЧНЯК!!! В правильной четырехугольной пирамиде MABCD с вершиной M стороны основания равны 4, а боковые ребра равны 8. Найдите площадь сечения пирамиды плоскостью, проходящей через точку B и середину ребра MD параллельно прямой AC.

Answer 1

искомое сечение -  симметричный четырехугольник  BPKL

диагонали  PL , BK  пересекаются под углом 90 град

по условию

стороны основания  AB=BC=CD=AD =4

боковые ребра  MA=MB=MC=MD =8

точка К - середина ребра MD ;  KD = MD /2 = 8/2=4

ABCD -квадрат

диагональ  AC = BD =  4√2

пересечение диагоналей  точка  F  :  BF =FD = BD/2 =4√2 /2 =2√2

BK - медиана треугольника  MBD

длина медианы  BK = 1/2 √(2 BM^2 +2 BD^2  - MD^2 ) =1/2 √(2*8^2 +2*(4√2)^2  - 8^2 ) =4√2

по теореме косинусов

cos KBD = ( KD^2 - (BK^2+BD^2) )/ (-2*BK*BD)= ( 4^2 - ((4√2)^2+(4√2)^2) )/ (-2*4√2*4√2)= 3/4

MF - высота

треугольник  EBF - прямоугольный

BE = BF / cos KBD = 2√2 / 3/4 = 8√2/3

KE = BK - BE =4√2 -8√2/3 =4√2/3

по теореме Пифагора EF =√(BE^2 - BF^2) =√( (8√2/3)^2 - (2√2)^2) =2√14/2

MF - высота

треугольник  MFB - прямоугольный

по теореме Пифагора MF =√( MB^2 -BF^2) =√( 8^2- (2√2)^2 ) =2√14

ME =MF -EF =2√14 -2√14/2 = 2√14/2

треугольники  MPL  ~ MCA    подобные

PL / AC = ME /MF ; PL = AC * ME /MF = 4√2 * 2√14/2 /2√14 =2√2

площадь   сечения(четырехугольника  BPKL)     

Sс = PL*BK *sin<BEP /2 = 2√2*4√2*sin90 /2 = 8

Ответ  8

Answer 2

Problem Analysis

We are given a right quadrilateral pyramid MABCD with vertex M, where the sides of the base are equal to 4 and the lateral edges are equal to 8. We need to find the area of the section of the pyramid formed by a plane passing through point B and the midpoint of edge MD, parallel to the line AC.

Solution

To find the area of the section of the pyramid, we need to determine the shape of the section. Since the section is formed by a plane passing through point B and the midpoint of edge MD, parallel to the line AC, it will be a trapezoid.

To find the area of the trapezoid, we need to know the lengths of its bases and its height.

Let's start by finding the lengths of the bases of the trapezoid.

The base of the trapezoid parallel to the line AC is formed by the intersection of the plane with the base of the pyramid. Since the sides of the base are equal to 4, the length of this base will also be 4.

The other base of the trapezoid is formed by the intersection of the plane with the lateral edge MB. To find the length of this base, we need to find the distance between point B and the midpoint of edge MD.

To find the midpoint of edge MD, we need to find the coordinates of points M and D. Unfortunately, the coordinates of these points are not provided in the given information. Therefore, we cannot determine the exact length of the other base of the trapezoid.

Without the exact length of the other base, we cannot calculate the area of the trapezoid. Therefore, we cannot find the area of the section of the pyramid with the given information.

Please provide the coordinates of points M and D, or any additional information that can help us determine the length of the other base of the trapezoid.