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

Explanation:

Step1: Find side - length $b$ in the first right - triangle

In a 45 - 45 - 90 right - triangle, the two legs are equal. If one leg is 3, using the ratio of side lengths $x:x:x\sqrt{2}$ (where $x$ is the length of each leg and $x\sqrt{2}$ is the hypotenuse), and since the legs of a 45 - 45 - 90 triangle are congruent, $b = 3\sqrt{2}$.

Step2: Find side - length $a$ in the second right - triangle

In a 30 - 60 - 90 right - triangle, if the shorter leg (opposite the 30° angle) is $x$, the hypotenuse is $2x$ and the longer leg (opposite the 60° angle) is $x\sqrt{3}$. Given the shorter leg is 5, and the hypotenuse $a$ is related to the shorter leg by the formula $a = 2\times5$. So $a = 10$.

Answer:

$b = 3\sqrt{2}$, $a = 10$