Question

JL is a common tangent to circles M and K at point J. If angle MLK measures 61ᵒ, what is the length of radius MJ? Round to the nearest hundredth.

Answer 1

3.122

The process to getting this comes in many steps. Firstly, you need to find the angles for JLK and MLJ. To find JLK use the arcsin function using the opposite side and the hypotenuse. 

Arcsin(Opp/Hype) = JLK
Arcsin(.5) = JLK
30 degrees = JLK

This means MLJ = 31 degrees since they add up to 61 degrees. 

Now we need to find the length of LJ, which we can do using the Pythagorean Theorem. 

3^2 + JL^2 = 6^2
9 + JL^2 = 36
JL^2 = 27
JL = $\sqrt{27}$

Now that we have the angle of MLJ and the length of JL, we can use the tangent function to find MJ.

Tan(angle) = opp/adj
Tan(31) = MJ/$\sqrt{27}$
$\sqrt{27}$Tan(31) = MJ
3.122 = MJ

Answer 2

ΔJKL, sin(JLK)=JK/KL=3/6=0.5
<JKL=30deg.

cos(JLK)=JL/KL
JL=KL*cos(JLK)=6cos(30deg.)

<JLK + <JLM = <MLK
30 + <JLM = 61
<JLM=31deg

in ΔMJL
tan(JLM)=JM/JL
JM=JL*tan(JLM)
=6cos(30deg.)*tan(31deg)
=3.12