Question

in $delta rst$, $t = 59$ cm, $angle s = 102^{circ}$ and $angle t = 27^{circ}$. find the area of $delta rst$, to the nearest square centimeter.

Explanation:

Step1: Find angle R

The sum of angles in a triangle is 180°. So, $\angle R=180^{\circ}-\angle S - \angle T=180^{\circ}-102^{\circ}-27^{\circ}=51^{\circ}$.

Step2: Use the sine - rule to find side s

By the sine - rule $\frac{s}{\sin S}=\frac{t}{\sin T}$. Substituting the given values: $\frac{s}{\sin102^{\circ}}=\frac{59}{\sin27^{\circ}}$. Then $s = \frac{59\times\sin102^{\circ}}{\sin27^{\circ}}$.
Since $\sin102^{\circ}\approx0.9781$ and $\sin27^{\circ}\approx0.4540$, $s=\frac{59\times0.9781}{0.4540}=\frac{57.7079}{0.4540}\approx127.11$.

Step3: Calculate the area of the triangle

The area of a triangle is given by $A=\frac{1}{2}st\sin R$.
Substitute $s\approx127.11$, $t = 59$, and $\sin R=\sin51^{\circ}\approx0.7771$ into the formula.
$A=\frac{1}{2}\times127.11\times59\times0.7771$.
$A=\frac{1}{2}\times127.11\times46.8589$.
$A=\frac{1}{2}\times5958.43$.
$A\approx2979$.

Answer:

2979