In a right-angled triangle ABC, the height CH, drawn to the hypotenuse AB, is 12 cm

In a right-angled triangle ABC, the height CH, drawn to the hypotenuse AB, is 12 cm. Find the leg of the triangle if the angle A is 30 degrees.

The height of CH, lowered to the hypotenuse AB, forms a right-angled triangle AНС. Therefore, sin A = sin 30 ° = CH / AC = 12 / AC = 1/2, whence AC = 24.

Because in the triangle BC lies opposite the angle A = 30 °, then BC = 0.5 * AB.

Let BC – x, then by the Pythagorean theorem we get:

(2 * x) ² = x² + AC²,

4 * x² – x² = 24² = 576,

3 * x² = 576,

x = 8 * √3, => BC = 8 * √3.

Then AB = 2 * BC = 16 * √3.

Answer: AC = 24 cm, BC = 8 * √3 cm.