Question

in $delta xyz$, $angle y=90^circ$ and $angle x=68^circ$. $angle zwy=80^circ$ and $xw=5.3$. find the length of $zy$ to the nearest 100th. figure not necessarily drawn to scale.

Explanation:

Step1: Find ∠XZW in ΔXZW

In $\Delta XZW$, sum of angles is $180^\circ$.
$\angle XZW = 180^\circ - 68^\circ - 80^\circ = 32^\circ$

Step2: Use Law of Sines on ΔXZW

Relate sides and angles to find $ZW$.
$\frac{ZW}{\sin(68^\circ)} = \frac{XW}{\sin(32^\circ)}$
$ZW = \frac{5.3 \times \sin(68^\circ)}{\sin(32^\circ)}$
$ZW \approx \frac{5.3 \times 0.9272}{0.5299} \approx 9.37$

Step3: Find ∠ZYW in ΔZYW

$\angle ZYW = 90^\circ$, $\angle ZWY = 80^\circ$, so $\angle YZW = 10^\circ$.

Step4: Use Law of Sines on ΔZYW

Solve for $ZY$ using $ZW$.
$\frac{ZY}{\sin(80^\circ)} = \frac{ZW}{\sin(90^\circ)}$
$ZY = 9.37 \times \sin(80^\circ)$
$ZY \approx 9.37 \times 0.9848 \approx 9.23$

Answer:

9.23