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)x2 + 3x = 36
x2 + 6x = 72
x2 + 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 √(62 + 122) from the Pythagorean Theorem.
√(62 + 122) = √180 = 6√5
The parameter of any shape is just all the side lengths added up. 6 + 12 + 6√5 = 18 + 6√5 = 6(3 + √5)