In a right-angled triangle ABC, the height BD is 56 cm and cuts off a segment DC

In a right-angled triangle ABC, the height BD is 56 cm and cuts off a segment DC equal to 33 cm from the gyrotenuse AC. Find AB and cos of angle A.

In the triangle BDC, we find the hypotenuse BC (by the Pythagorean theorem):
ВС = √ (BD² + DC²) = √ (3136 + 1089) = √4225 = 65 (cm).
Consider right-angled triangles ABC and BDC. They are similar (acute angle C is common).
The similarity of triangles allows us to write down the aspect ratio:
AB / BC = BD / DC → AB = BC * BD / DC = 65 * 56/33 = 3640/33 = 110 10/33 (cm).
Similar triangles have equal angles, so we have:
Coc A = cos CBD = BD / BC = 56/65 = 0.861538 ≈ 0.86.
Answer: leg AB is 110 10/33 cm, cos A is 0.86.