# The perimeter of a right-angled triangle is 24 cm, and the area is 24 cm ^ 2. Find all the sides of the triangle.

It is known that the perimeter of a right-angled triangle is P = 24 cm, and the area is S = 24 cm2.

It is necessary to determine the lengths of the sides of a right-angled triangle, where the legs are A, B, and the hypotenuse is C.

We write down the formula for finding the perimeter and area, and also express C through A and B.

1) P = A + B + C.

Whence we get that A + B = 24 – C.

2) S = ½ * A * B.

Whence we get that AB = 24: ½ = 48.

3) C ^ 2 = A ^ 2 + B ^ 2.

We transform the third equality by adding 2 * A * B to both sides of it.

C ^ 2 + 2AB = A ^ 2 + B ^ 2 + 2AB.

C ^ 2 + 2AB = (A + B) 2.

Substitute the values from expressions 1 and 2.

C ^ 2 + 2 * 48 = (24 – C) 2.

Let’s open the brackets and solve the equation.

C ^ 2 + 96 = 576 – 48C + C2.

48C = 576 – 96.

48C = 480.

C = 10.

It turns out that the length of the hypotenuse is 10 cm.

Now we will find the sides by solving the system of equations.

3) A ^ 2 + B ^ 2 = 10 ^ 2.

4) AB = 48.

Multiply the 4th equation by -2 and add to the 3rd.

A2 – 2AB + B ^ 2 = 10 ^ 2 – 48 * 2.

(A – B) ^ 2 = 22.

A – B = 2.

Recall that A + B = 10.

Let’s add the last 2 equations.

2A = 12.

A = 6.

Now we define B.

B = 10 – A.

B = 10 – 6.

B = 4.

Answer: the lengths of the sides of a right-angled triangle are 4, 6, 10.