Question

Given: △KLM, LM=20 3 m∠K=105°, m∠M=30° Find: KL and KM

Answer 1

Okay, so the angle of K is 45° (180 - 135).

Draw a line that goes straight down from angle L and let's say that point is point Z. You can see that this triangle can be separated into two special triangles: the 45 45 90 triangle on the left (ΔLKZ) and the 30 60 90 triangle (ΔLMZ) on the right.

I don't know if you've learned this yet (just google them if you're confused):                                                                       - for a 45 45 90 triangle the sides are x, x, and x√2.                                                 - for a 30 60 90 triangle the sides are x, 2x, and x√3.

Since we know that LM is 20√3, using the 30 60 90 trick, you can see that LZ is 10√3 and MZ is 30.

Next, since LZ is 10 sqrt(3), using the 45 45 90 trick, KL is 10√6 and KZ is 10 sqrt(3). Therefore, KM = KZ + MZ = 10√3 + 30

The answer: KL is 10√6, KM is 10√3 + 30

(I'm sorry it's such a long explanation, pictures would have been better)