In triangle ABC, the height CD lowered from the vertex of the right angle C divides the hypotenuse AB

In triangle ABC, the height CD lowered from the vertex of the right angle C divides the hypotenuse AB into segments AD = 9 and DB = 16. Cathet BC = 20. Find the leg AC and the height CD of this triangle.

CD – is the height, so the resulting triangles ACD and CBD are rectangular.

To find the side of CD, we apply the Pythagorean theorem, find the square root of the difference between the square of the hypotenuse CB and the square of the leg BD:

√ (20 * 20 – 16 * 16) = √144 = 12.

Method 1: Now, also by the Pythagorean theorem, we find AC, in a right-angled triangle ACD it is a hypotenuse:

√ (12 * 12 + 9 * 9) = √225 = 15.

Or 2 way:

In the triangle ABC, AC – leg, AB = AD + DB, AB = 25:

√ (25 * 25 – 20 * 20) = √225 = 15.

Answer: The speaker side is 15 cm, the CD height is 12 cm.