The height drawn from the vertex of the right angle of the triangle CDO

The height drawn from the vertex of the right angle of the triangle CDO is 6 cm and divides the hypotenuse into segments, one of which is 9 cm larger than the other. Find DN and ON

1. СN – height. ∠С = 90 °.

2. The height СN, drawn from the top of the right angle, is calculated by the formula:

CN = √DN x ON. СN² = DN х ОN.

3. We take the length of the segment DN as x (cm). The length of the segment ON is (x + 9) cm.

4. Let’s make the equation: x (x + 9) = 36;

x² + 9x = 36;

x² + 9x – 36 = 0;

The first value x = (- 9 + √81 + 4 x 36) / 2 = (- 9 + √225) / 2 = (-9 + 15) / 2 = 3 cm.

The second value x = (-9 – 15) / 2 = – 12 cm. Does not satisfy the condition of the problem.

The length of the segment ON = 3 + 9 = 12 cm.

Answer: DN = 3 cm, ON = 12 cm.