Finding length of side in triangle

Question

Problem

In the following triangle, show that the length $\text{CD} = \frac{\sqrt3}{2}a$.

My solution

Using the Pythagorean Theorem on a 30-60-90 triangle, we get

$$CD = \sqrt{a^2 - \frac{a^2}{4}} = \frac{\sqrt3}{2}a$$

However, it was pointed out to me that this property of 30-60-90 triangles is outside the curriculum.

Question

Can someone outline or show me another solution to this problem? Apparently, I used the "wrong" one.

Answer 1

By definition of cosine function, $$ CD = a\cos30^\circ = {\sqrt3\over2}a $$ from the right triangle BCD. (In this triangle, $a$ is hypotenuse, and $CD$ is the leg adjacent to the $30^\circ$ angle.)

Answer 2

As a student you have a moral obligation to obliterate the confines of your curriculum:

This equilateral triangle is a nice 'cheat sheet' in case you forget the values of $\sin 60^\circ,\sin 30^\circ$ and the cosine of those angles. By drawing an equilateral triangle with side lengths $2$, and bisecting it, all the remaining information follows.