Question

В правильной четырехугольной пирамиде MABCD с вершиной M стороны основания равны 4, а боковые ребра 8. Найти площадь сечения пирамиды плоскостью проходящей через точку B и середину ребра МD параллельно прямой AC. (Если не сложно, то с рисунком, хотя и за решение буду очень рад) !

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 the vertex M and the base sides equal to 4, and the lateral edges equal to 8. We need to find the area of the section of the pyramid with 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, we need to determine the shape of the section. Let's analyze the given information and solve the problem step by step.

Step 1: Determine the shape of the section The section of the pyramid with a plane passing through point B and the midpoint of edge MD, parallel to the line AC, will be a trapezoid. This is because the plane intersects the pyramid in a way that the resulting shape has two parallel sides (the bases of the trapezoid) and two non-parallel sides (the lateral sides of the trapezoid).

Step 2: Find the length of the parallel sides of the trapezoid To find the length of the parallel sides of the trapezoid, we need to determine the lengths of the segments that form the bases of the trapezoid. Let's denote the midpoint of edge MD as E.

From the given information, we know that the base sides of the pyramid are equal to 4. Therefore, the length of segment AB is 4.

To find the length of segment DE, we can use the fact that E is the midpoint of segment MD. Since the lateral edges of the pyramid are equal to 8, we can conclude that the length of segment MD is 8. Therefore, the length of segment DE is half of 8, which is 4.

So, the lengths of the parallel sides of the trapezoid are AB = 4 and DE = 4.

Step 3: Find the height of the trapezoid To find the height of the trapezoid, we need to determine the distance between the parallel sides. Let's denote the intersection point of the plane and the lateral edge MC as F.

From the given information, we know that the lateral edges of the pyramid are equal to 8. Therefore, the length of segment MC is 8.

To find the height of the trapezoid, we need to find the distance between the parallel sides AB and DE. This distance is equal to the length of segment BF.

To find the length of segment BF, we can use the fact that the plane is parallel to the line AC. Therefore, the length of segment BF is equal to the length of segment CF.

From the given information, we don't have the exact length of segment CF. However, we can use the fact that the plane passes through point B and the midpoint of segment MD. Therefore, the line segment BF is the perpendicular bisector of segment MD.

Since E is the midpoint of segment MD, we can conclude that the length of segment BE is equal to the length of segment ME. Therefore, the length of segment BF is equal to the length of segment CF.

So, the height of the trapezoid is equal to the length of segment CF.

Step 4: Find the length of segment CF To find the length of segment CF, we need to use the given information and apply geometric reasoning.

From the given information, we know that the base sides of the pyramid are equal to 4. Therefore, the length of segment BC is 4.

To find the length of segment CF, we can use the fact that the line segment BF is the perpendicular bisector of segment MD.

Since E is the midpoint of segment MD, we can conclude that the length of segment BE is equal to the length of segment ME. Therefore, the length of segment BF is equal to the length of segment CF.

Since the line segment BF is the perpendicular bisector of segment MD, we can conclude that the length of segment BC is equal to the length of segment CF.

So, the length of segment CF is 4.

Step 5: Calculate the area of the trapezoid Now that we have the lengths of the parallel sides (AB = 4, DE = 4) and the height (CF = 4) of the trapezoid, we can calculate its area using the formula for the area of a trapezoid:

Area = (sum of the lengths of the parallel sides) * (height) / 2

Substituting the values, we get:

Area = (AB + DE) * CF / 2 = (4 + 4) * 4 / 2 = 8 * 4 / 2 = 32 / 2 = 16

Therefore, the area of the section of the pyramid with a plane passing through point B and the midpoint of edge MD, parallel to the line AC, is 16 square units.

Conclusion

The area of the section of the pyramid with a plane passing through point B and the midpoint of edge MD, parallel to the line AC, is 16 square units.