Question

На сторонах MK и KL треугольника KLM отмечены точки N и P соответственно так, что отрезок NP делится биссектрисой угла К пополам; MN:NK=4:3, NM:PL=2:1, NK=3см.Найдите длину отрезка KL. Ребят пожалуйста пожалуйста Даю 100 баллов

Answer 1

В треугольнике NKP биссектриса является медианой (делит основание пополам) => треугольник равнобедренный.

KP=NK=3x, MN=4x, PL=MN/2=2x

NK=3x=3 см => x=1 см

KL=KP+PL=5x =5 см

Answer 2

Problem Analysis

We are given a triangle KLM with points N and P marked on the sides MK and KL, respectively. The segment NP divides the angle K in half. We are also given the following ratios: MN:NK = 4:3, NM:PL = 2:1, and NK = 3 cm. We need to find the length of segment KL.

Solution

To find the length of segment KL, we can use the fact that NP divides angle K in half. This means that the ratio of the lengths of MK:KN and LP:PL is the same. Let's denote the length of KL as x.

Using the given ratios, we can set up the following equations:

1. MN:NK = 4:3 - MN = 4/7 * NK - MN = 4/7 * 3 cm - MN = 12/7 cm

2. NM:PL = 2:1 - NM = 2/3 * PL - NM = 2/3 * (x - PL) - NM = 2/3 * x - 2/3 * PL

Since NP divides angle K in half, we can set up the following equation:

3. MK:KN = LP:PL - MK = x - PL - KN = PL - MK:KN = (x - PL):PL

Now, we can substitute the values of MN and MK:KN into equation 3:

(x - PL):PL = 12/7:3

Cross-multiplying, we get:

7(x - PL) = 12PL

Simplifying, we have:

7x - 7PL = 12PL

Combining like terms, we get:

7x = 19PL

Now, let's substitute the value of NM from equation 2 into the equation above:

7x = 19(2/3 * x - 2/3 * PL)

Simplifying further, we have:

7x = 38/3 * x - 38/3 * PL

Multiplying both sides by 3 to eliminate the fraction, we get:

21x = 38x - 38PL

Combining like terms, we have:

17x = 38PL

Finally, let's substitute the value of NK (3 cm) into the equation above:

17x = 38(3)

Simplifying, we have:

17x = 114

Dividing both sides by 17, we get:

x = 6

Therefore, the length of segment KL is 6 cm.

Conclusion

The length of segment KL in triangle KLM, given the conditions mentioned, is 6 cm.