Question

В треугольнике ABC на стороне AB взяли точку K, а на стороне BC взяли точку N так, что угол BAN = углу BCK. AK = 5 см; KB = 3 см; BN = 2 см. Найдите NC. Даю 50 баллов. За спам бан. Подробное решение

Answer 1

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Answer 2

Problem Analysis

We are given a triangle ABC, where point K is taken on side AB and point N is taken on side BC such that angle BAN is equal to angle BCK. We are also given that AK = 5 cm, KB = 3 cm, and BN = 2 cm. We need to find the length of NC.

Solution

To solve this problem, we can use the angle bisector theorem. According to the angle bisector theorem, in a triangle, if a line divides one side of the triangle into two segments, then the ratio of the lengths of those segments is equal to the ratio of the lengths of the other two sides of the triangle.

Let's apply the angle bisector theorem to triangle ABC. We have AK = 5 cm, KB = 3 cm, and BN = 2 cm. Let NC = x cm.

According to the angle bisector theorem, we have:

AK/KB = AN/BN

Substituting the given values, we get:

5/3 = AN/2

Cross-multiplying, we get:

2 * 5 = 3 * AN

10 = 3 * AN

Dividing both sides by 3, we get:

AN = 10/3 cm

Now, we can find the length of NC by using the fact that the sum of the lengths of the corresponding sides of two similar triangles is proportional. Triangle ABC is similar to triangle ANC, so we can set up the following proportion:

AB/AN = BC/NC

Substituting the given values, we get:

8/(10/3) = 3/x

Cross-multiplying, we get:

8 * (3/x) = 10

24/x = 10

Dividing both sides by 24, we get:

x = 24/10

Simplifying, we get:

x = 12/5 cm

Therefore, NC = 12/5 cm.

Answer

The length of NC is 12/5 cm.

Note: The solution provided above is based on the angle bisector theorem and the given information.