Question

How do you find the remaining side of a `30^circ-60^circ-90^circ` triangle if the side opposite `60^circ` is 4?

Answer 1

`(4sqrt3)/3` and `(8sqrt3)/3`

Explanation:

A `30^@ - 60^@ - 90^@` triangle is a type of special right triangle because the side lengths always occur in the ratio `x - x sqrt3 - 2x`.

The `30^@` angle is opposite to the side length `x`, the `60^@` angle is opposite to `xsqrt3`, and the `90^@` angle is opposite to `2x`, as shown below.

If the side opposite to `60^@` is `4`, then `xsqrt 3 = 4`. To find the remaining sides, first solve for `x`:

`xsqrt 3 = 4`

`x = 4/sqrt3` `->` divide both sides by `sqrt3`

`x = 4/sqrt3 * color(red)(sqrt3/sqrt3)` `->` rationalize the denominator by multiplying by `sqrt3/sqrt3`, which is equal to `1`

`x = (4sqrt3)/3` `->` multiply

Lastly, to find the side opposite to the `90^@` angle, find `2x`:

`2x = 2 * (4sqrt3)/3 = (8sqrt3)/3`

So, the sides are `(4sqrt3)/3, 4,` and `(8sqrt3)/3`.