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

Explanation:

Step1: Find side - length (c) using trigonometric ratios

In a right - triangle with angles (30^{\circ}), (60^{\circ}), and (90^{\circ}), if the side opposite the (30^{\circ}) angle is (a), the side opposite the (60^{\circ}) angle is (b), and the hypotenuse is (c), the ratio of the sides is (a:b:c = 1:\sqrt{3}:2). Given that the side opposite the (60^{\circ}) angle (b = 6). Let the side opposite the (30^{\circ}) angle be (x). Then (\tan60^{\circ}=\frac{6}{x}), and (\sin60^{\circ}=\frac{6}{c}). Since (\sin60^{\circ}=\frac{\sqrt{3}}{2}), we have (\frac{6}{c}=\frac{\sqrt{3}}{2}). Cross - multiply: (c\times\sqrt{3}=12), so (c = 4\sqrt{3}).

Step2: Find side - length (d) using trigonometric ratios

In a right - triangle with angles (45^{\circ}), (45^{\circ}), and (90^{\circ}), the ratio of the sides is (a:a:c = 1:1:\sqrt{2}), where the legs are of equal length and the hypotenuse is (\sqrt{2}) times the length of a leg. Given the hypotenuse (c = 2). Let the length of each leg be (d). Then, using the Pythagorean theorem (d^{2}+d^{2}=2^{2}), or (2d^{2}=4), (d^{2}=2), so (d=\sqrt{2}).

Answer:

(c = 4\sqrt{3}), (d=\sqrt{2})