Find the area of a right-angled triangle, the hypotenuse is 16 cm, and the sum of the legs is 24 cm.

Let us denote the lengths of the legs of this right-angled triangle through a and b.

According to the condition of the problem, the sum of the legs of this right-angled triangle is 24 cm, therefore, the following relation holds:

a + b = 24.

Also in the condition of the problem it is said that the hypotenuse of this right-angled triangle is 16 cm? therefore, the following relation holds:

a ^ 2 + b ^ 2 = 16 ^ 2.

We square both sides of the first ratio and transform as follows:

(a + b) ^ 2 = 24 ^ 2;

a ^ 2 + 2ab + b ^ 2 = 576;

2ab = 576 – (a ^ 2 + b ^ 2);

2ab / 4 = (576 – (a ^ 2 + b ^ 2)) / 4;

ab / 2 = 144 – (a ^ 2 + b ^ 2) / 4.

Since a ^ 2 + b ^ 2 = 16 ^ 2 = 256, and the area S of this triangle is ab / 2, we get:

S = 144 – 256/4 = 144 – 64 = 80 cm ^ 2.

Answer: 80 cm ^ 2.