in $\\delta xyz$, $z = 6.8$ inches, $m\\angle z = 26^\\circ$ and $m\\angle x = 132^\\circ$. find the length…
in $\\delta xyz$, $z = 6.8$ inches, $m\\angle z = 26^\\circ$ and $m\\angle x = 132^\\circ$. find the length of $y$, to the nearest 10th of an inch.
Answer
Explanation:
Step1: Find $\angle Y$
The sum of angles in a triangle is $180^\circ$. $$m\angle Y = 180^\circ - 132^\circ - 26^\circ = 22^\circ$$
Step2: Apply Law of Sines
Relate sides and angles: $\frac{y}{\sin Y} = \frac{z}{\sin Z}$ $$y = \frac{z \cdot \sin Y}{\sin Z}$$
Step3: Substitute known values
Plug in $z=6.8$, $\sin22^\circ$, $\sin26^\circ$ $$y = \frac{6.8 \cdot \sin(22^\circ)}{\sin(26^\circ)}$$
Step4: Calculate the value
Use $\sin22^\circ\approx0.3746$, $\sin26^\circ\approx0.4384$ $$y \approx \frac{6.8 \cdot 0.3746}{0.4384} \approx \frac{2.5473}{0.4384} \approx 5.8$$
Answer:
$5.8$ inches