for the right triangles below, find the exact values of the side lengths c and a. if necessary, write your…

Explanation:

Step1: Solve for $c$ in the first right - triangle

In a 45 - 45 - 90 right - triangle, the ratio of the sides is $1:1:\sqrt{2}$. If the legs are of length $x$, the hypotenuse $c$ is given by $c = x\sqrt{2}$. Here $x = 8$, so $c=8\sqrt{2}$.

Step2: Solve for $a$ in the second right - triangle

In a 30 - 60 - 90 right - triangle, if the side opposite the 30 - degree angle is $x$, the side opposite the 60 - degree angle is $x\sqrt{3}$ and the hypotenuse is $2x$. Here the side opposite the 30 - degree angle is $a$, and the side opposite the 60 - degree angle is 6. Since $\tan60^{\circ}=\frac{6}{a}=\sqrt{3}$, then $a = \frac{6}{\sqrt{3}}$. Rationalizing the denominator, $a=\frac{6\sqrt{3}}{3}=2\sqrt{3}$.

Answer:

$c = 8\sqrt{2}$ $a = 2\sqrt{3}$