The perimeter of a right-angled triangle is 84 cm, and its hypotenuse is 37 cm. Find the legs of the triangle.

1. The perimeter of a right-angled triangle is 84 cm.

The hypotenuse is 37 cm.

Let’s find the sum of the legs.

84 – 37 = 47 cm.

2. Let X cm be the length of one leg.

Then (47 – X) cm is the length of the second.

According to the Pythagorean theorem, the sum of the squares of the legs is equal to the square of the hypotenuse.

X * X + (47 – X) * (47 – X) = 37 * 37.

X * X + 47 * 47 – 2 * 47 * X + X * X = 37 * 37.

2 * X * X – 2 * 47 * X + 2209 – 1369 = 0.

X * X – 49 * X = 420 = 0.

Find the discriminant of the quadratic equation.

D = 49 * 49 – 4 * 420 = 2209 – 1680 = 529.

X1 = (49 + 23) / 2 = 36 cm – first leg.

47 – 36 = 13 – the second leg.

X2 = (49 – 23) / 2 = 13 cm – first leg.

47 – 13 = 36 – second.

We got 2 identical answers.

Answer: the legs of the triangle are 36 cm and 13 cm.