The altitude of a right triangle is 16 cm. Let ℎ be the length of the hypotenuse and let 𝑝 be the perimeter of the triangle. Express ℎ as a function of 𝑝.

Answer 1

You are given a right triangle where h is the hypotenuse and the height (altitude) is 16 cm. First, define the other leg of the right triangle as x, and define x and h to also be in cm, so everything is consistent. The perimeter of the triangle p is the sum of the sides, so p= h+16+x. Now, use the Pythagorean theorem on the right triangle, which gives h^2 = (16)^2 + x^2.

Now, solve the formula for the perimeter for x. x= p-16-h. Express this as x = (p-16) - h. (you will see why in a minute)

Plug this expression for x into the Pythagorean theorem formula. To do this, you need to determine x^2.

x^2 = [ (p-16) - h ]^2 = (p-16)^2 - 2h(p-16) + h^2

Put this into the Pythagorean theorem formula

h^2 = 16^2 + (p-16)^2 - 2h(p-16) + h^2

Subtract h^2 from both sides to get

0 = 16^2 + (p-16)^2 - 2h(p-16)

Add 2h(p-16) to both sides to get

2h(p-16) = 16^2 + (p-16)^2

Divide both sides by 2(p-16) to get

h = [ 16^2 + (p-16)^2 ]/[2(p-16)]

h= [ p^2 -32p + 512 ]/[ 2p-32]