The perimeter of a right triangle is 90 cm and its area is 270 cm ^ 2. Find the lengths of the sides

The perimeter of a right triangle is 90 cm and its area is 270 cm ^ 2. Find the lengths of the sides of the triangle (using the system).

It is known that the perimeter of a right-angled triangle is P = 90 cm, and the area is S = 270 cm2.

It is necessary to calculate all the sides of the triangle.

Let us write down the formulas for the perimeter and area for a right-angled triangle, which has legs A and B, and C is the hypotenuse.

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

P = A + B + C, which implies (A + B) = 90 – C.

S = ½ * A * B, whence we obtain AB = 540 cm2.

Let’s transform the first equation.

C ^ 2 = A2 + 2AB + B ^ 2 – 2AB.

C ^ 2 = (A + B) ^ 2 – 2AB.

Let’s substitute the values.

C ^ 2 = (90 – C) ^ 2 – 2 * 540.

C ^ 2 = 8100 – 180C + C ^ 2 – 1080.

180C = 7020.

C = 39.

Then it turns out that the hypotenuse of the triangle is 39 cm.

A + B = 90 – C = 90 – 39 = 51.

1) A + B = 51.

2) AB = 540.

Let us square the first equation and subtract the quadrupled value of the second equation from it.

(A + B) ^ 2 – 4AB = 51 ^ 2 – 4 * 540.

A2 + 2AB + B ^ 2 – 4AB = 2601 – 2160.

(A – B) 2 = 212.

A – B = 21.

A + B = 51.

Let’s add both equations.

2A = 72.

A = 36.

B = 51 – A.

B = 51 – 36.

B = 15.

Thus, the lengths of the legs of the triangle were 15 cm and 36 cm.

Answer: the sides of the triangle are 15 cm, 36 cm, 39 cm.