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

Explanation:

Step1: Solve for $a$ using 45 - 45 - 90 triangle ratio

In a 45 - 45 - 90 triangle, the legs are equal and if the hypotenuse is $c$ and the legs are $x$, then $c=\sqrt{2}x$. Given $c = 6$, we have $6=\sqrt{2}a$, so $a=\frac{6}{\sqrt{2}}=\frac{6\sqrt{2}}{2}=3\sqrt{2}$.

Step2: Solve for $b$ using 30 - 60 - 90 triangle ratio

In a 30 - 60 - 90 triangle, if the hypotenuse is $c = 5$, and the side opposite the 30 - degree angle is $x$, and the side opposite the 60 - degree angle is $y$, then $c = 2x$ and $y=\sqrt{3}x$. Here $x$ is the side adjacent to the 60 - degree angle. Since $c = 5$, the side adjacent to the 60 - degree angle (the shorter leg) is $\frac{5}{2}$, and $b=\frac{5\sqrt{3}}{2}$.

Answer:

$a = 3\sqrt{2}$ $b=\frac{5\sqrt{3}}{2}$