Question

В прямом параллелепипеде с высотой корень из 15 , стороны основания АВСД равны 2м и 4 м. Диагональ АС равна 5 м. Найдите S диагонального сечения параллелепипеда , проходящего через вершины ВД (Желательно с рисунком)

Answer 1

BB₁D₁D - диагональное сечение, проходящее через вершины B и D.
Сечение имеет форму прямоугольника, так как BB₁║ D₁D и они перпендикулярны BD.

В основании лежит параллелограмм. Сумма квадратов диагоналей параллелограмма равна сумме квадратов его сторон:
AC² + BD² = 2·(AB² + BC²)
25 + BD² = 2(16 + 4)
BD² = 40 - 25
BD² = 15
BD = √15 м

Sсеч = BD · DD₁ = √15 ·√15 = 15 м²

Answer 2

Calculation of the Diagonal Section Area of the Parallelepiped

To find the area of the diagonal section of the parallelepiped, we need to determine the shape of the section. Since the diagonal section passes through the vertices B and D, it will be a rectangle.

Let's denote the length of the rectangle as a and the width as b.

To find the values of a and b, we can use the information given in the problem:

- The height of the parallelepiped is the square root of 15. - The sides of the base ABCD are 2m and 4m. - The diagonal AC is 5m.

Using these values, we can apply the Pythagorean theorem to find the lengths of a and b.

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.

In our case, the diagonal AC is the hypotenuse, and the sides of the base AB and BC are the other two sides of the right triangle.

Let's calculate the lengths of AB and BC:

- AB = 2m - BC = 4m

Using the Pythagorean theorem, we can find the length of AC:

AC^2 = AB^2 + BC^2

AC^2 = 2^2 + 4^2

AC^2 = 4 + 16

AC^2 = 20

AC = √20

AC = 2√5

Since the diagonal AC is 5m, we have:

2√5 = 5

Now, we can set up the equation for the rectangle's dimensions:

a = AB = 2m

b = BC = 4m

Therefore, the area of the diagonal section of the parallelepiped is:

S = a * b = 2m * 4m = 8 square meters.

Here is a diagram to help visualize the parallelepiped and the diagonal section:

       A _________ B
       /|       /|
      / |      / |
     /  |     /  |
    /___|____/   |
   D    |   C    |
    |   |____|___|
    |  /      |  /
    | /       | /
    |/________|/

Please note that the diagram is not to scale and is only for illustrative purposes.

I hope this helps! Let me know if you have any further questions.