Question

in △xyz, ∠x = 58° and ∠y = 63°. ∠xwz = 90° and xy = 77. find the length of xw to the nearest integer.

Explanation:

Step1: Find angle Z in triangle XYZ

In a triangle, the sum of interior angles is 180°. So in $\triangle XYZ$, $\angle Z=180^{\circ}-\angle X - \angle Y=180^{\circ}-58^{\circ}-63^{\circ}=59^{\circ}$.

Step2: Use the sine - rule in $\triangle XYZ$

Let's consider the relationship between the sides and angles of $\triangle XYZ$. We know that $\frac{XZ}{\sin Y}=\frac{XY}{\sin Z}$. First, we find $XZ$. Given $XY = 77$, $\sin Y=\sin63^{\circ}\approx0.891$, $\sin Z=\sin59^{\circ}\approx0.857$. Then $XZ=\frac{XY\times\sin Y}{\sin Z}=\frac{77\times0.891}{0.857}\approx80.17$.

Step3: Use right - triangle trigonometry in $\triangle XWZ$

In right - triangle $\triangle XWZ$ with $\angle XWZ = 90^{\circ}$, $\cos X=\frac{XW}{XZ}$. Since $\angle X = 58^{\circ}$ and $\cos58^{\circ}\approx0.5299$ and $XZ\approx80.17$. Then $XW=XZ\times\cos X\approx80.17\times0.5299\approx42$.

Answer:

42