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

Explanation:

Step1: Find side - length (a) in the first right - triangle

In a (45 - 45-90) right - triangle, the ratio of the sides is (1:1:\sqrt{2}). If the hypotenuse (c = 6), and the legs are of length (x) (in this case (a)), then (c=\sqrt{2}x). So, (a=\frac{6}{\sqrt{2}}). Rationalizing the denominator, we multiply the numerator and denominator by (\sqrt{2}), getting (a = \frac{6\sqrt{2}}{2}=3\sqrt{2}).

Step2: Find side - length (d) in the second right - triangle

In a (30 - 60 - 90) right - triangle, the ratio of the sides is (1:\sqrt{3}:2). The side opposite the (30^{\circ}) angle is (2), and the side opposite the (60^{\circ}) angle is (d). The relationship between the side opposite the (30^{\circ}) angle ((x = 2)) and the side opposite the (60^{\circ}) angle is (d=\sqrt{3}x). So, (d = 2\sqrt{3}).

Answer:

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