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

Explanation:

Step1: Find side - length (d) using 45 - 45 - 90 triangle properties

In a 45 - 45 - 90 triangle, the ratio of the sides is (1:1:\sqrt{2}). If the legs have length (x), the hypotenuse (c) is (c = x\sqrt{2}). Here, (x = 5), so (d=5\sqrt{2}).

Step2: Find side - length (a) using 30 - 60 - 90 triangle properties

In a 30 - 60 - 90 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}), and (\tan60^{\circ}=\sqrt{3}), we have (a = \frac{6}{\sqrt{3}}). Rationalizing the denominator, (a=\frac{6\sqrt{3}}{3}=2\sqrt{3}).

Answer:

(d = 5\sqrt{2}) (a = 2\sqrt{3})