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

Explanation:

Step1: Solve for (b) in the first triangle

In a 45 - 45 - 90 right - triangle, the legs are equal. Using the property that if the hypotenuse (c) and legs (a = b) in a 45 - 45 - 90 triangle, (c=\sqrt{2}a). Here (c = 3), and since (a = b), we have (3=\sqrt{2}b), so (b=\frac{3}{\sqrt{2}}=\frac{3\sqrt{2}}{2}).

Step2: Solve for (d) in the second triangle

In a 30 - 60 - 90 right - triangle, if the shorter leg (opposite 30°) is (x), the hypotenuse is (2x) and the longer leg (opposite 60°) is (x\sqrt{3}). Here the shorter leg is 5, and (d) is the hypotenuse. So (d = 2\times5=10).

Answer:

(b=\frac{3\sqrt{2}}{2}), (d = 10)