Question

In a right triangle, one with angle measures of 30-60-90. The hypotenuse is y, the shortest side is x, and the last side is 21, what are the lengths of x and y?

Answer 1

x: `7sqrt3`
y: `14sqrt3`

Explanation:

In a `30-60-90` special right triangle:
Side opposite to `30` degrees: `x`
Side opposite to `60` degrees: `xsqrt3`
Side opposite to `90` degrees: `2x`

So in this case we are given the side opposite to the `60` degrees, because the hypotenuse is always the opposite side of `90` degrees, and the shortest side is always opposite to the shortest angle so in this case `30` degrees:

`xsqrt3=21`

Solve for `x` which is the shortest side:
`x=21/sqrt3`
Rationalize:
`x=21/sqrt3*sqrt3/sqrt3= (21sqrt3)/3= 7sqrt3`

Now for the longest side:
`y= 2x`
`y= 2*7sqrt3`
`y=14sqrt3`