Question

Incline mats, or triangle mats, are offered with different levels of incline to help gymnasts learn basic moves. As the name may suggest, two sides of the mat are right triangles. If the height of the mat is 14 inches shorter than the length of the mat and the hypotenuse is 4 inches longer than the length of the mat, what is the length of the mat?

Answer 1

The length of the mat is 30 inches.

Explanation:

Let H be the hypotenuse of the incline mat, L its length and h its height.

It is given that H = L + 4 and h = L - 14. Now, using Pythagoras' theorem for right-angled triangles, H² = L² + h²

(L + 4)² = L² + (L - 14)²

Expanding, we have

L² + 8L + 16 = L² + L² -28L + 196

collecting all the terms to the right side, we have

L² + L² -28L + 196 - L² - 8L - 16 = 0

L² - 36L + 180 = 0

Using the quadratic formula, L =

L
`\frac{-(-36) +/- \sqrt{(-36)^(2) -4X1X180} )}{2X1} = (36 +/- √(1296 - 720) )/(2) \\= (36 +/- √(576) )/(2) \\= (36 +/- 24 ))/(2)\\=(36 + 24)/(2) or (36 - 24)/(2) \\=(60)/(2) or (12)/(2) \\= 30 or 6`

We use the larger answer L = 30 inches since L = 6 would give h = L -14 = 6 - 14 = -8 .

So, the length of the mat is 30 inches.