Question

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

Explanation:

Step1: Find angle at J

Sum of angles in triangle is $180^\circ$.
$\angle J = 180^\circ - 67^\circ - 74^\circ = 39^\circ$

Step2: Define sides for Law of Sines

Let $JL = k$, $JK = l$, $KL = 2.3$.
Law of Sines: $\frac{\sin(\angle J)}{KL} = \frac{\sin(\angle K)}{JL} = \frac{\sin(\angle L)}{JK}$

Step3: Calculate side JL

Solve for $JL$ using $\angle J, \angle K, KL$.
$\frac{\sin(39^\circ)}{2.3} = \frac{\sin(67^\circ)}{JL}$
$JL = \frac{2.3 \times \sin(67^\circ)}{\sin(39^\circ)} \approx \frac{2.3 \times 0.9205}{0.6293} \approx 3.36$

Step4: Calculate side JK

Solve for $JK$ using $\angle J, \angle L, KL$.
$\frac{\sin(39^\circ)}{2.3} = \frac{\sin(74^\circ)}{JK}$
$JK = \frac{2.3 \times \sin(74^\circ)}{\sin(39^\circ)} \approx \frac{2.3 \times 0.9613}{0.6293} \approx 3.54$

Step5: Compute perimeter

Add all three sides.
Perimeter $= 2.3 + 3.36 + 3.54 = 9.2$

Answer:

9.2 units