Question

The hypotenuse of a right triangle is 10 inches. The lengths of the two legs are given by 2 consecutive even integers. How do you find the lengths of the two legs?

Answer 1

`6,8`

Explanation:

The first thing to tackle here is how to express "two consecutive even integers" algebraically.

`2x` will give an even integer if `x` is also an integer. The next even integer, following `2x`, would be `2x+2`. We can use these as the lengths of our legs, but must remember that this will only hold true if `x` is a (positive) integer.

Apply the Pythagorean theorem:

`(2x)^2+(2x+2)^2=10^2`

`4x^2+4x^2+8x+4=100`

`8x^2+8x-96=0`

`x^2+x-12=0`

`(x+4)(x-3)=0`

`x=-4,3`

Thus, `x=3` since the side lengths of the triangle can't be negative.

The legs are

`2xrArr6`
`2x+2rArr8`
`"hypotenuse"rArr10`

A more intuitive way to do this problem is to recognize that a `6,8,10` triangle is just twice the size of the fundamental `3,4,5` right triangle.