Question

find the lengths of w, x, y, and z shown in the figure below if wy = 64. round your answers to the nearest tenth. note that the figure is not drawn to scale. (figure shows a right triangle with a height x, base segments w and y, angle 54° at the left end of w, and a side of length 9 opposite to some part, and a hypotenuse z for the larger triangle. there are input boxes for w, x, y, z.)

Explanation:

Step1: Find \( w \) using cosine

In the left - right triangle with hypotenuse 9 and angle \( 54^{\circ} \), \( \cos(54^{\circ})=\frac{w}{9} \). So \( w = 9\times\cos(54^{\circ}) \). Calculate \( \cos(54^{\circ})\approx0.5878 \), then \( w\approx9\times0.5878 = 5.2902\approx5.3 \)

Step2: Find \( x \) using sine

In the left - right triangle with hypotenuse 9 and angle \( 54^{\circ} \), \( \sin(54^{\circ})=\frac{x}{9} \). So \( x = 9\times\sin(54^{\circ}) \). Calculate \( \sin(54^{\circ})\approx0.8090 \), then \( x\approx9\times0.8090 = 7.281\approx7.3 \)

Step3: Find \( y \) using \( wy = 64 \)

We know that \( w\approx5.3 \) and \( wy = 64 \), so \( y=\frac{64}{w}\approx\frac{64}{5.3}\approx12.1 \)

Step4: Find \( z \) using Pythagorean theorem or trigonometry

We can use the Pythagorean theorem in the right - triangle with legs \( x\approx7.3 \) and \( y\approx12.1 \). \( z=\sqrt{x^{2}+y^{2}}=\sqrt{7.3^{2}+12.1^{2}}=\sqrt{53.29 + 146.41}=\sqrt{199.7}\approx14.1 \) (or we can use trigonometry, since the large triangle has angle \( 54^{\circ} \) and adjacent side \( w + y=5.3 + 12.1 = 17.4 \), \( \cos(54^{\circ})=\frac{w + y}{hypotenuse} \), but the first method is more straightforward here)

Answer:

\( w\approx5.3 \), \( x\approx7.3 \), \( y\approx12.1 \), \( z\approx14.1 \)