Question

what is the approximate perimeter of the triangle? use the law of sines to find the answer. o 4.6 units o 5.7 units o 6.9 units o 9.2 units law of sines: $\frac{sin(a)}{a}=\frac{sin(b)}{b}=\frac{sin(c)}{c}$

Explanation:

Step1: Find the third - angle

The sum of angles in a triangle is 180°. Let the third - angle be $\angle J$. So, $\angle J=180-(67 + 74)=39^{\circ}$.

Step2: Use the law of sines to find side $JK$

By the law of sines, $\frac{JK}{\sin L}=\frac{KL}{\sin J}$. We know that $KL = 2.3$, $\angle L=74^{\circ}$, and $\angle J = 39^{\circ}$. Then $JK=\frac{2.3\sin74^{\circ}}{\sin39^{\circ}}$. Since $\sin74^{\circ}\approx0.9613$ and $\sin39^{\circ}\approx0.6293$, $JK=\frac{2.3\times0.9613}{0.6293}=\frac{2.21099}{0.6293}\approx3.51$.

Step3: Use the law of sines to find side $JL$

By the law of sines, $\frac{JL}{\sin K}=\frac{KL}{\sin J}$. We know that $\angle K = 67^{\circ}$, $\angle J=39^{\circ}$, and $KL = 2.3$. Then $JL=\frac{2.3\sin67^{\circ}}{\sin39^{\circ}}$. Since $\sin67^{\circ}\approx0.9205$ and $\sin39^{\circ}\approx0.6293$, $JL=\frac{2.3\times0.9205}{0.6293}=\frac{2.11715}{0.6293}\approx3.36$.

Step4: Calculate the perimeter

The perimeter $P$ of $\triangle JKL$ is $P=JK + KL+JL$. Substituting the values: $P\approx3.51+2.3 + 3.36=9.17\approx9.2$ units.

Answer:

9.2 units