Question

1. use the pattern you saw in problems 10 - 15 to generate the blank spaces of this triangle with side length x.

Explanation:

Step1: Recall 30 - 60 - 90 triangle ratios

In a 30 - 60 - 90 triangle, if the side opposite the 30 - degree angle is $a$, the side opposite the 60 - degree angle is $a\sqrt{3}$, and the hypotenuse is $2a$. Here the side opposite the 60 - degree angle is $x$.

Step2: Find the side opposite 30 - degree angle

Let the side opposite the 30 - degree angle be $y$. Since $\frac{y}{x}=\frac{1}{\sqrt{3}}$, then $y = \frac{x}{\sqrt{3}}=\frac{\sqrt{3}x}{3}$.

Step3: Find the hypotenuse

Let the hypotenuse be $z$. Since $\frac{x}{z}=\frac{\sqrt{3}}{2}$, then $z=\frac{2x}{\sqrt{3}}=\frac{2\sqrt{3}x}{3}$.

Answer:

The side opposite the 30 - degree angle is $\frac{\sqrt{3}x}{3}$ and the hypotenuse is $\frac{2\sqrt{3}x}{3}$