Question

The hypotenuse of a right triangle is 39 inches, and the length of one leg is 6 inches longer than twice the other leg. How do you find the length of each leg?

Answer 1

The legs are of length `15` and `36`

Explanation:

Method 1 - Familiar triangles

The first few right angled triangles with an odd length side are:

`3, 4, 5`

`5, 12, 13`

`7, 24, 25`

Notice that `39 = 3 * 13`, so will a triangle with the following sides work:

`15, 36, 39`

i.e. `3` times larger than a `5, 12, 13` triangle ?

Twice `15` is `30`, plus `6` is `36` - Yes.

`color(white)()`
Method 2 - Pythagoras formula and a little algebra

If the smaller leg is of length `x`, then the larger leg is of length `2x+6` and the hypotenuse is:

`39 = sqrt(x^2 + (2x+6)^2)`

`color(white)(39) = sqrt(5x^2+24x+36)`

Square both ends to get:

`1521 = 5x^2+24x+36`

Subtract `1521` from both sides to get:

`0 = 5x^2+24x-1485`

Multiply both sides by `5` to get:

`0 = 25x^2+120x-7425`

`color(white)(0) = (5x+12)^2-144-7425`

`color(white)(0) = (5x+12)^2-7569`

`color(white)(0) = (5x+12)^2-87^2`

`color(white)(0) = ((5x+12)-87)((5x+12)+87)`

`color(white)(0) = (5x-75)(5x+99)`

`color(white)(0) = 5(x-15)(5x+99)`

Hence `x = 15` or `x = -99/5`

Discard the negative solution since we are seeking the length of the side of a triangle.

Hence the smallest leg is of length `15` and the other is `2*15+6 = 36`