One leg of a right-angled triangle is 5 cm smaller than the other. Find the length of each leg if the area

One leg of a right-angled triangle is 5 cm smaller than the other. Find the length of each leg if the area of this triangle is 45 cm ^ 2.

To solve the problem, we apply the area formula for right-angled triangles using legs and sine: S = (a * b * sin a) * (1/2), but sin a in a right-angled triangle is 1, then S = (a * b) * (1/2).
Let the first leg = a, and the second equal to b. Then, by hypothesis, a = b – 5.
Substitute all values under the area formula and calculate: S = (a * b) * (1/2); 45 = ((b – 5) * b) * (1/2); 45 = ((b ^ 2 – 5b) / 2; 90 = b ^ 2 – 5b. It follows that b = 12 cm, a = 7 cm (approximate values).
Answer: 12; 7.