Questions: Find the exact value of each part labeled with a variable in the figure. (Simplify your answers, including any radicals. Use integers or fractions for any numbers in the expressions.)

Solution Steps

In the left-hand triangle, the angle opposite side with length 24 measures 45°. The angle opposite _r_ measures 30°. Using the sine rule:

$24/\sin(45^{\circ}) = r/\sin(30^{\circ})$ $r = 24\sin(30^{\circ})/\sin(45^{\circ})$ $r = 24(1/2)/(\sqrt{2}/2)$ $r = 12/(\sqrt{2}/2)$ $r = 24/\sqrt{2}$ $r = 12\sqrt{2}$

In the left-hand triangle, the angle opposite side with length 24 measures 45°. The angle opposite _p_ measures 90°. Using the sine rule:

$24/\sin(45^{\circ}) = p/\sin(90^{\circ})$ $p = 24\sin(90^{\circ})/\sin(45^{\circ})$ $p = 24(1)/(\sqrt{2}/2)$ $p = 48/\sqrt{2}$ $p = 24\sqrt{2}$

In the right-hand triangle, the angle opposite _q_ measures 45°. The angle opposite _p_ measures 90°. Using the sine rule:

$q/\sin(45^{\circ}) = p/\sin(90^{\circ})$ $q = p\sin(45^{\circ})/\sin(90^{\circ})$ $q = p(\sqrt{2}/2)/(1)$ $q = (24\sqrt{2})(\sqrt{2}/2)$ $q = 24$

Final Answer: