Question

for the right triangles below, find the exact values of the side lengths a and d. the figures are not drawn to scale.

Explanation:

Step1: Use sine - cosine for 45 - 45 - 90 triangle

In a 45 - 45 - 90 triangle, if the hypotenuse $c = 1$, and we want to find the leg $a$. We know that $\sin45^{\circ}=\frac{a}{c}$. Since $\sin45^{\circ}=\frac{\sqrt{2}}{2}$ and $c = 1$, then $a = c\sin45^{\circ}$.
$a=1\times\frac{\sqrt{2}}{2}=\frac{\sqrt{2}}{2}$

Step2: Use sine for 30 - 60 - 90 triangle

In a 30 - 60 - 90 triangle, if the hypotenuse $c = 1$ and we want to find the side opposite the 30 - degree angle $d$. We know that $\sin30^{\circ}=\frac{d}{c}$. Since $\sin30^{\circ}=\frac{1}{2}$ and $c = 1$, then $d = c\sin30^{\circ}$.
$d = 1\times\frac{1}{2}=\frac{1}{2}$

Answer:

(a) $a=\frac{\sqrt{2}}{2}$
(b) $d=\frac{1}{2}$