The lengths of two sides of a right triangle containing the right angle differ by 6 cm. If the area of the triangle is 36 cm^2, find the perimeter of the triangle.

I assume you mean the lengths of

*the two legs*of a right triangle differ by 6 cm. (i.e. the two shorter sides that make up the right angle.) If so:Let x be the length of the shortest leg. The other leg must have length x + 6.

The area of a triangle is (1/2)bh.

But in a right triangle, the base, b, and height, h, are the same as the legs.

So we get

(1/2)(x)(x + 6) = 36

(1/2)x

^{2}+ 3x = 36x

^{2}+ 6x = 72x

^{2}+ 6x - 72 = 0(x + 12)(x - 6) = 0

x = -12 or

**x = 6**So if the shortest leg is 6, the other must be 12, and the hypotenuse (the longest side across from the right angle) must be √(6

^{2}+ 12^{2}) from the Pythagorean Theorem.√(62 + 122) =

**√180**= 6√5The parameter of any shape is just all the side lengths added up. 6 + 12 + 6√5 =

**18 + 6√5**= 6(3 + √5)