In a 30-60-90 triangle, the hypotenuse is twice the side opposite the $30^\circ$ angle. How can you use the given hypotenuse length of $10$ with this fact?

Let $x$ be the length of the side opposite the $30^\circ$ angle. Use the relationship between that side and the hypotenuse in a 30-60-90 triangle to write an equation involving $x$ and $10$.

Make sure Desmos is in Degree mode (gear icon → Angle Measure → Degrees), then in a new expression line type 10*sin(30). The value that Desmos outputs is the length of the side opposite the $30^\circ$ angle in a right triangle with hypotenuse $10$.

A right triangle with a $30^\circ$ angle is a 30-60-90 triangle, which has fixed side ratios:

• side opposite $30^\circ$: $1$
• side opposite $60^\circ$: $\sqrt{3}$
• hypotenuse: $2$

So the side opposite $30^\circ$ is always half the hypotenuse.

In this triangle, the hypotenuse is $10$.

In the 30-60-90 ratio, hypotenuse $2$ corresponds to actual length $10$. So the scale factor from the ratio to this triangle is:

That means each ratio unit of length $1$ becomes $5$ units in the actual triangle.

The side opposite $30^\circ$ corresponds to $1$ in the 30-60-90 ratio.

• Ratio length: $1$
• Actual length: $1 \times 5 = 5$

So the side opposite the $30^\circ$ angle has length $5$, which matches choice A) $5$.