In a right-angled triangle ABC, angle C = 90, height CD = 12cm, AB = 25cm. Find AD.

Given: triangle ABC;
angle c = 90 degrees;
CD-height;
CD = 12 cm;
AB = 25 cm.
Find: АD.
Decision:
AB is the hypotenuse of the right-angled triangle ABC. (by construction)
CD height. (by condition)
CD = root of (AD * DB). (formula for finding the length of the height of a right triangle)
DB = AB-AD, hence CD = √ (AD * (AB-AD)). (since the height is drawn to the AB side)
CD = root of (AD * AB-AD ^ 2);
CD ^ 2 = AD * AB-AD ^ 2;
12 ^ 2 = 25 * AD-AD ^ 2;
AD ^ 2-25AD + 144 = 0; (we solve the quadratic equation)
Discriminant = (- 25) ^ 2-4 * (1) * 144 = 625-576 = 49;
AD = (25 + 7) / 2 = 16 cm or
AD = (25-7) / 2 = 9 cm.
Answer: 16 centimeters or 9 centimeters.