Given: LN ⊥ KM LN = 16 ft m∠K = 25°, m∠M = 55° Find: Radius R

Question

I drew out triangle LKM in a circle with center O and point N on KM. I tried to use law of sines to get LM and got 27.89514873 but I'm not sure what to do with that or what to do next.

Answer 1

Without seeing the real diagram, there are two interpretation
1. KM is a diameter. But that can't be or else m∠L = 90°.

2. KM is a chord. We know m∠L = 100°
Triangle LNM is tight-angled and tan 55° = 16/NM
NM = 16/ tan 55° = 11.20332... (store that in your calculator's memory)
In the same way, find KN
You now have the chord KM
Let the radius be R
The angle subtended by a chord at the centre is twice the angle at the circle.
So angle KOM = 200°
and by the cosine law
R^2 + R^2 - 2(R)(R)cos 200° = KM^2
solve for R

Answer 2

thanks this was very helpful but i have a question about the method used to find NM and KN for KN, is there a specific name for it for formula? thank you!

Answer 3

Or, using another approach,
16/KL = sin25°
16/LM = sin55°
KM^2 = KL^2 + LM^2 - 2*KL*LM*cos100°
KM = 45.515
which is the same answer you get using Reiny's method

Answer 4

thank you oobleck!

Answer 5

r=19.39057726?

Answer 6

Hmm. That's not what I got. How did you get that?