The hypotenuse of a right-angled triangle is 26cm, and the area is 120cm2. Find the smaller leg.

Let the hypotenuse of the triangle be c, and the legs a and b.

Then c = 26 and the area S = 1/2 * a * b = 120, a * b = 240.

By the Pythagorean theorem we have:

c ^ 2 = a ^ 2 + b ^ 2,

26 ^ 2 = a ^ 2 + (240 / a) ^ 2,

676 * a ^ 2 = a ^ 4 + 57600,

a ^ 4 – 676 * a ^ 2 + 57600 = 0.

Let x = a ^ 2, then

x ^ 2 – 676 * x + 57600 = 0.

By Vieta’s theorem, we note that the roots of this equation are:

x1 = 576, x2 = 100.

Then b ^ 2 = 576 and b = 24, and if b ^ 2 = 100, then b = 10.

Thus, the legs of the triangle are 10 and 24.

Answer: the smaller leg is 10 cm.