In a right-angled triangle ABC, angle A is straight, AB = 40 cm, height AD is 24 cm. Find AC and cosC.

In a right-angled triangle ADB we find the leg BD (by the Pythagorean theorem)
BD = √ (AB² – AD²) = √ (1600 – 576) = √1024 = 32 (cm).
Consider two right-angled triangles ABC and ADB, they are similar (acute angle B is common).
Similar triangles have angles equal:
∠ ACB = ∠ BAD;
Cos ACB = cos BAD = AD / AB = 24/40 = 3/5 = 0.6.
From the similarity of triangles, we write down the aspect ratio:
BD / AB = AB / BC;
BC = AB * AB / BD = 40 * 40/32 = 50 (cm).
By the Pythagorean theorem, we find AC:
AC = √ (BC² – AB²) = √ (2500 – 1600) = √900 = 30 (cm).
Answer: the AC leg is 30 cm, cos C = 0.6.