Question

В правильной четырёхугольной пирамиде sabcd все рёбра равны между собой.Точки k m лежат на рёбрах sa и sb, при этом sk/ka = sm/mb = 6/7. Найдите угол между прямыми km и sс. ответ дайте в градусах.

Answer 1

Прикрепляю решение фотографией

Answer 2

Given Information:

We are given a right quadrilateral pyramid with the vertices SABCD, where all the edges are equal. Points K and M lie on the edges SA and SB, respectively, such that SK/KA = SM/MB = 6/7. We need to find the angle between the lines KM and SC.

Solution:

To find the angle between the lines KM and SC, we can use the concept of similar triangles. Let's consider triangle SKM and triangle SAB.

Since SK/KA = SM/MB = 6/7, we can conclude that triangles SKM and SAB are similar by the Side-Side-Side (SSS) similarity criterion.

Now, let's consider the angle between KM and SC. We can find this angle by finding the angle between the corresponding sides of the similar triangles SKM and SAB.

Let's denote the angle between KM and SC as θ.

According to the properties of similar triangles, the corresponding angles of similar triangles are equal. Therefore, the angle between KM and SC is equal to the angle between the corresponding sides of triangles SKM and SAB.

Let's denote the angle between the corresponding sides of triangles SKM and SAB as α.

Now, we can set up the following proportion:

SK/SA = KM/AB

Since SK/SA = 6/7, we can rewrite the proportion as:

6/7 = KM/AB

We can solve this proportion for KM:

KM = (6/7) * AB

Since all the edges of the pyramid are equal, AB = SA = SB.

Therefore, KM = (6/7) * SA = (6/7) * SB

Now, we can use the fact that the angle between the corresponding sides of similar triangles SKM and SAB is equal to the angle between KM and SC.

Let's denote the length of SC as x.

Since the angle between KM and SC is equal to the angle between the corresponding sides of triangles SKM and SAB, we can set up the following proportion:

KM/SC = AB/SA

Substituting the values we found earlier:

(6/7) * SB / x = SB / SA

Simplifying the equation:

(6/7) * SB * SA = SB * x

Dividing both sides by SB:

(6/7) * SA = x

Therefore, the length of SC is (6/7) times the length of SA.

Now, we can find the angle between KM and SC by using the properties of right triangles.

In triangle SCA, we have a right angle at C. Let's denote the angle between KM and SC as θ.

Using trigonometric ratios, we can write:

tan(θ) = KM/SC

Substituting the values we found earlier:

tan(θ) = (6/7) * SA / ((6/7) * SA)

Simplifying the equation:

tan(θ) = 1

To find the angle θ, we can take the inverse tangent (arctan) of both sides:

θ = arctan(1)

Using a calculator, we find that θ is approximately 45 degrees.

Therefore, the angle between the lines KM and SC is approximately 45 degrees.

Note: The above solution assumes that the given pyramid is a right quadrilateral pyramid, which means it has a rectangular base. If the pyramid is not right or not quadrilateral, the solution may be different.