Question

question in △xyz, y = 3.6 cm, ∠x = 158°, and ∠y = 16°. find the area of △xyz, to the nearest 10th of a square centimeter. answer attempt 1 out of 2 watch video show examples submit answer

Explanation:

Step1: Find angle Z

Use the angle - sum property of a triangle ($\angle X+\angle Y+\angle Z = 180^{\circ}$).
$\angle Z=180^{\circ}-\angle X - \angle Y=180^{\circ}-158^{\circ}-16^{\circ}=6^{\circ}$

Step2: Use the formula for the area of a triangle

The area of a triangle with two - side lengths $a$ and $b$ and included angle $C$ is $A=\frac{1}{2}ab\sin C$. Here we use the formula $A = \frac{1}{2}xy\sin Z$. Given $y = 3.6$ cm, and we assume $x$ is the side opposite to $\angle X$, $y$ is the side opposite to $\angle Y$, and we know $\sin Z=\sin6^{\circ}\approx0.1045$.
$A=\frac{1}{2}\times x\times y\times\sin Z$. Since we can also use the sine - rule $\frac{x}{\sin X}=\frac{y}{\sin Y}$ to find $x$, but we can directly use the formula with the given side $y$.
$A=\frac{1}{2}\times y\times y\times\frac{\sin X\sin Z}{\sin Y}$ (derived from sine - rule $\frac{x}{\sin X}=\frac{y}{\sin Y}\Rightarrow x = \frac{y\sin X}{\sin Y}$).
$\sin X=\sin158^{\circ}\approx0.3746$, $\sin Y=\sin16^{\circ}\approx0.2756$, $y = 3.6$ cm.
$A=\frac{1}{2}\times3.6\times3.6\times\frac{\sin158^{\circ}\sin6^{\circ}}{\sin16^{\circ}}$
$A=\frac{1}{2}\times3.6\times3.6\times\frac{0.3746\times0.1045}{0.2756}$
$A=\frac{1}{2}\times3.6\times3.6\times\frac{0.03915}{0.2756}$
$A=\frac{1}{2}\times3.6\times3.6\times0.142$
$A=\frac{1}{2}\times3.6\times0.5112$
$A = 0.92016\approx0.9$ square centimeters.

Answer:

$0.9$ square centimeters