The height of a right-angled triangle, lowered by the hypotenuse, is 12 cm and divides it into segments

The height of a right-angled triangle, lowered by the hypotenuse, is 12 cm and divides it into segments, the difference between which is 7 cm. Calculate the perimeter of the triangle.

1. Vertices of the triangle – A, B, C. Angle A – straight line. P is the perimeter. AE – height.

2. AE = √BE x CE.

AE² = BE x CE.

3. We take the length of the segment BE as x, the length of the segment CE = x – 7 cm.

x (x – 7) = 144;

x² – 7x – 144 = 0.

x = (7 + √49 + 576) / 2 = (7 + 25) / 2 = 16;

The second value x = (7 – 25) / 2 = (- 9) – is not accepted.

BE = 16 cm, CE = 16 – 7 = 9 cm.

4. AB = √AE² + BE² = √144 + 256 = 20 cm.

5. AC = √AE² + CE² = √144 + 81 = 15 cm.

6. BC = 16 + 9 = 25 cm.

7.R = 20 + 15 + 25 = 60 cm.