# In a right-angled triangle, the height is 24 cm, we draw from the right angle to the hypotenuse

**In a right-angled triangle, the height is 24 cm, we draw from the right angle to the hypotenuse, divides the hypotenuse into 2 segments, the difference of which is 14 cm.Find the area of the triangle**

The smallest of the hypotenuse segments will be denoted by x cm, then the larger one will be (x + 14) cm. In a right-angled triangle, the square of the height is equal to the product of the projections of the legs to the hypotenuse, that is, our two segments.

24² = x * (x + 14)

x² + 14x – 576 = 0

D = b² – 4 * a * c = 196 + 2304 = 2500, D> 0, two roots of the equation.

x1 = (-14 + 50) / 2 = 18;

x2 = (-14 – 50) / 2 = -32, negative root is not suitable.

18 + 14 = 32 (cm).

18 + 32 = 50 (cm) – hypotenuse.

Find the area of a right-angled triangle through the hypotenuse and height:

S = 1/2 * c * h = 1/2 * 50 * 24 = 600 (cm²).

Answer: the area is 600 cm².