Question

what is the approximate length of kl? use the law of sines to find the answer. law of sines: $\frac{sin(a)}{a}=\frac{sin(b)}{b}=\frac{sin(c)}{c}$ 1.8 units 2.0 units 3.2 units 3.7 units

Explanation:

Step1: Find angle J and angle K

Since the sum of angles in a triangle is 180°, and angle L = 105°, assume angle J and angle K are equal (isosceles - like situation for simplicity as no other angle info given). So angle J=angle K=(180 - 105)/2 = 37.5°.

Step2: Apply the law of sines

We know that \(\frac{\sin(L)}{JK}=\frac{\sin(K)}{JL}\). Given \(JK = 4.7\), \(JL=2.7\), \(\sin(L)=\sin(105^{\circ})\approx0.966\), \(\sin(K)=\sin(37.5^{\circ})\approx0.609\). Substituting into \(\frac{\sin(L)}{JK}=\frac{\sin(K)}{JL}\), we can also use \(\frac{\sin(L)}{JK}=\frac{\sin(J)}{KL}\). Let's use \(\frac{\sin(L)}{JK}=\frac{\sin(J)}{KL}\), then \(KL=\frac{JK\times\sin(J)}{\sin(L)}\). Substituting the values: \(KL=\frac{4.7\times\sin(37.5^{\circ})}{\sin(105^{\circ})}=\frac{4.7\times0.609}{0.966}\approx2.97\approx3.0\) (closest value among options is 3.2).

Answer:

C. 3.2 units