Question

What are the lengths of the two legs of a 30°-60°-90° triangle if the length of the hypotenuse is `12sqrt3`?

Answer 1

The 'short' side is `6sqrt(3)` and the long side is `18`

Explanation:

In a 30-60-90 triangle the sides always work in the following ratio:

short side: `1:1`
hypotenuse: `2:1`
long side: `sqrt(3):1`

This means that the hypotenuse is twice the length of the short side, and the long side is `sqrt(3)` times larger than the short side.

If we know the length of the hypotenuse, we can find the length of the short leg:

`"short"/"hyp"=1/2 rArr "short"="hyp"/2`

`"short"=(12sqrt(3))/2`

`color(blue)("short"=6sqrt(3))`

Now that we know the length of the short side, we can find the length of the long side:

`"long"/"short"=sqrt(3) rArr "long"=sqrt(3)xx"short"`

`"long"=sqrt(3)xx6sqrt(3)=6sqrt(3)^2=6xx3`

`color(green)("long"=18)`