Question

in $delta xyz$, $angle x=39^{circ}$ and $angle y=21^{circ}$. $angle xwz=90^{circ}$ and $xy=5$. find the length of $xw$ to the nearest 100th.

Explanation:

Step1: Find ∠XZY

First, calculate the third angle in △XYZ using the angle sum property of triangles.
$\angle XZY = 180^\circ - 39^\circ - 21^\circ = 120^\circ$

Step2: Use Law of Sines to find XZ

Apply the Law of Sines to △XYZ to solve for side XZ.
$\frac{XZ}{\sin\angle Y} = \frac{XY}{\sin\angle XZY}$
$XZ = \frac{XY \cdot \sin\angle Y}{\sin\angle XZY} = \frac{5 \cdot \sin21^\circ}{\sin120^\circ}$
Calculate the values: $\sin21^\circ \approx 0.3584$, $\sin120^\circ \approx 0.8660$
$XZ \approx \frac{5 \cdot 0.3584}{0.8660} \approx 2.064$

Step3: Use cosine to find XW

In right △XWZ, use the cosine of ∠ZXW to find XW.
$\cos\angle X = \frac{XW}{XZ}$
$XW = XZ \cdot \cos39^\circ$
$\cos39^\circ \approx 0.7771$
$XW \approx 2.064 \cdot 0.7771 \approx 1.60$

Answer:

$1.60$