Question

find the lengths of w, x, y, and z shown in the figure below if xy = 65. round your answers to the nearest tenth. note that the figure is not drawn to scale.

Explanation:

Step1: Use trigonometric relation for $w$

In the right - triangle with side 19 and angle $41^{\circ}$, $\sin41^{\circ}=\frac{w}{19}$, so $w = 19\times\sin41^{\circ}$.
$w=19\times0.656059\approx12.5$

Step2: Use trigonometric relation for $x$

$\cos41^{\circ}=\frac{x}{19}$, so $x = 19\times\cos41^{\circ}$.
$x=19\times0.754709\approx14.3$

Step3: Find $y$ using $xy = 65$

Since $xy = 65$ and $x\approx14.3$, then $y=\frac{65}{x}=\frac{65}{14.3}\approx4.5$

Step4: Use Pythagorean theorem for $z$

In the large right - triangle with legs 20 and $x + y$. First, $x + y\approx14.3+4.5 = 18.8$. Then by the Pythagorean theorem $z=\sqrt{20^{2}+(x + y)^{2}}=\sqrt{400+(18.8)^{2}}=\sqrt{400 + 353.44}=\sqrt{753.44}\approx27.4$

Answer:

$w\approx12.5$
$x\approx14.3$
$y\approx4.5$
$z\approx27.4$