In a right-angled triangle ABC, angle A = 40, angle B = 90, and in triangle MNK, angles M, N, K

In a right-angled triangle ABC, angle A = 40, angle B = 90, and in triangle MNK, angles M, N, K are related as 5: 9: 4, AB = 3cm, KN = 9cm. What is the ratio of BC to NM?

Let the angle ∠K = 4 * X in the MNK triangle, then the angle ∠M = 5 * X, the angle ∠H = 9 * X.

The sum of the interior angles of a triangle is 180, then:

4 * X + 5 * X + 9 * X = 180.

18 * X = 180.

X – 180/18 = 10.

∠К = 40, ∠М = 50, angle Н = 90.

Then the MNK triangle is right-angled and is similar to the ABC triangle in its acute angle.

The coefficient of similarity is: K = AB / KN = 3/4.

Then BC / NM = 3/4.

Answer: BC / NM = 3/4.