Question

В правильной четырехугольной пирамиде SABCD высота SO равна 13, диагональ основания BD равна 8. Точки К и М - середины ребер CD и ВС соответственно. Найдите тангенс угла между плоскостью SMK и плоскостью основания АВС.

Answer 1

Проведем в пирамиде  сечение МSK.
МК - средняя линия треугольника CDB, она параллельна DB и равна ее половине.
Диагональ АС квадрата АВСD равна диагонали DB
ОР - четверть этой диагонали и равна 8:4=2-это мы видим из треугольника CDB, в котором   высота делится отрезком МК пополам.
SР- высота, биссектриса и медиана треугольного сечения МSK.
 tg ∠ SPo=SP:OP=13:2=6,5

Answer 2

Given Information

We are given the following information about a right quadrilateral pyramid SABCD: - Height SO = 13 - Diagonal of the base BD = 8 - Points K and M are the midpoints of the edges CD and BC, respectively.

We need to find the tangent of the angle between the plane SMK and the plane of the base ABC.

Solution

To find the tangent of the angle between the plane SMK and the plane of the base ABC, we need to find the angle between the two planes first.

Let's break down the solution step by step:

1. Find the length of the diagonal AC of the base ABC. 2. Find the length of the side AB of the base ABC. 3. Use the lengths of AC and AB to find the angle between the plane SMK and the plane of the base ABC. 4. Finally, find the tangent of the angle.

Step 1: Find the length of diagonal AC of the base ABC

To find the length of diagonal AC, we can use the Pythagorean theorem in triangle ABC.

Let's denote the midpoint of AB as N. Since K is the midpoint of CD, we can consider the quadrilateral ABCD as a trapezoid with bases AB and CD. Therefore, the diagonal AC is equal to the sum of the diagonals of the trapezoid.

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

AC = √(AB^2 + BC^2)

Step 2: Find the length of side AB of the base ABC

To find the length of side AB, we can use the fact that the diagonal BD is equal to the sum of the diagonals of the trapezoid ABCD.

Using the Pythagorean theorem, we can find the length of AB as follows:

AB = √(BD^2 - AC^2)

Step 3: Find the angle between the plane SMK and the plane of the base ABC

To find the angle between the plane SMK and the plane of the base ABC, we can use the fact that the angle between two planes is equal to the angle between their normal vectors.

The normal vector of the plane SMK is perpendicular to the line MK, which lies in the plane of the base ABC. Therefore, the normal vector of the plane SMK is parallel to the normal vector of the plane ABC.

Since the normal vector of the plane ABC is perpendicular to the base AB, we can find the angle between the two planes by finding the angle between the normal vector of the plane ABC and the line AB.

Step 4: Find the tangent of the angle

Once we have the angle between the two planes, we can find the tangent of the angle using trigonometric functions.

Let's calculate the values step by step.

Step 1: Find the length of diagonal AC of the base ABC

Using the Pythagorean theorem in triangle ABC, we have:

AC = √(AB^2 + BC^2)

Step 2: Find the length of side AB of the base ABC

Using the fact that the diagonal BD is equal to the sum of the diagonals of the trapezoid ABCD, we have:

AB = √(BD^2 - AC^2)

Step 3: Find the angle between the plane SMK and the plane of the base ABC

Since the normal vector of the plane SMK is parallel to the normal vector of the plane ABC, we can find the angle between the two planes by finding the angle between the normal vector of the plane ABC and the line AB.

Let's denote the angle between the normal vector of the plane ABC and the line AB as θ.

Step 4: Find the tangent of the angle

Once we have the angle θ, we can find the tangent of the angle using the formula:

tangent(θ) = sin(θ) / cos(θ)

Now, let's calculate the values.

Calculation

Using the given information, we can calculate the values as follows:

1. Length of diagonal AC: - AC = √(AB^2 + BC^2) 2. Length of side AB: - AB = √(BD^2 - AC^2) 3. Angle between the plane SMK and the plane of the base ABC: - Let's denote the angle between the normal vector of the plane ABC and the line AB as θ.

4. Tangent of the angle: - tangent(θ) = sin(θ) / cos(θ)

Let's calculate the values using the given information.

Calculation

Using the given information, we can calculate the values as follows:

1. Length of diagonal AC: - AC = √(AB^2 + BC^2) - AC = √(6^2 + (8/2)^2) - AC = √(36 + 16) - AC = √52 - AC ≈ 7.211

2. Length of side AB: - AB = √(BD^2 - AC^2) - AB = √(8^2 - 52) - AB = √(64 - 52) - AB = √12 - AB ≈ 3.464

3. Angle between the plane SMK and the plane of the base ABC: - Let's denote the angle between the normal vector of the plane ABC and the line AB as θ. - Since the normal vector of the plane ABC is perpendicular to the base AB, the angle between the normal vector and the line AB is 90 degrees.

4. Tangent of the angle: - tangent(θ) = sin(θ) / cos(θ) - Since the angle θ is 90 degrees, the tangent of the angle is undefined.

Therefore, the tangent of the angle between the plane SMK and the plane of the base ABC is undefined.

Please let me know if I can help you with anything else.