Question

in δxyz, ∠y = 90° and ∠x = 53°. ∠zwy = 76° and xw = 60. find the length of zy to the nearest integer. figure not necessarily drawn to scale.

Explanation:

Step1: Find angle $\angle XZW$

In $\triangle XYZ$, $\angle Y = 90^{\circ}$ and $\angle X=53^{\circ}$, so in $\triangle XWZ$, since $\angle ZWY = 76^{\circ}$ is an exterior - angle, $\angle XZW=\angle ZWY-\angle X$. Then $\angle XZW = 76^{\circ}-53^{\circ}=23^{\circ}$.

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

In $\triangle XWZ$, by the sine - rule $\frac{ZW}{\sin\angle X}=\frac{XW}{\sin\angle XZW}$. Given $XW = 60$, $\angle X = 53^{\circ}$, and $\angle XZW=23^{\circ}$. We know that $\sin53^{\circ}\approx0.7986$ and $\sin23^{\circ}\approx0.3907$. So $ZW=\frac{XW\times\sin\angle X}{\sin\angle XZW}=\frac{60\times\sin53^{\circ}}{\sin23^{\circ}}=\frac{60\times0.7986}{0.3907}\approx122.7$.

Step3: Find the length of $ZY$ in right - triangle $ZWY$

In right - triangle $ZWY$, $\sin\angle ZWY=\frac{ZY}{ZW}$. Given $\angle ZWY = 76^{\circ}$ and $ZW\approx122.7$, and $\sin76^{\circ}\approx0.9703$. Then $ZY = ZW\times\sin\angle ZWY\approx122.7\times0.9703\approx119$.

Answer:

119