Question

for the right triangles below, find the exact values of the side lengths b and c. if necessary, write your responses in simplified radical form. b = c =

Explanation:

Step1: Find side - length $b$ using tangent function

In the first right - triangle with an angle of $60^{\circ}$ and adjacent side of length $5$. We know that $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Here, $\theta = 60^{\circ}$ and the adjacent side to $60^{\circ}$ is $5$, and the opposite side is $b$. Since $\tan60^{\circ}=\sqrt{3}$, we have $\tan60^{\circ}=\frac{b}{5}$. So, $b = 5\tan60^{\circ}=5\sqrt{3}$.

Step2: Find side - length $c$ using Pythagorean theorem or special right - triangle properties

In the second right - triangle, which is a $45 - 45-90$ right - triangle. The legs of a $45 - 45-90$ right - triangle are equal, and if one leg has length $a = 6$, and the hypotenuse is $c$. By the Pythagorean theorem $a^{2}+a^{2}=c^{2}$ (or for a $45 - 45-90$ triangle, $c = a\sqrt{2}$). Substituting $a = 6$ into $c = a\sqrt{2}$, we get $c=6\sqrt{2}$.

Answer:

$b = 5\sqrt{3}$
$c = 6\sqrt{2}$