Question

Given: △KLM, LM=20 3

Answer 1

Split the triangle into a 30-60-90 triangle. Since you have LM from that triangle, you can figure out the other sides. Then, using the leg from that triangle, the one next to it is a 45-45-90 triangle, meaning it's isosceles.

KM = 30 + 10√3

KL = 10√6

(sorry i'm not too sure if my "reasons" for statement/reason is correct)

Answer 2

Consider the attached diagram. Segment KO is an altitude of the triangle, so KO ⊥ LM. ∠L has measure 180° -105° -30° = 45°, so ΔKOL is an isosceles right triangle.

If we let segment LO have measure 1, then KO also has measure 1 and KL has measure √(1²+1²) = √2 by the Pythagorean theorem.

ΔKMO is half of an equilateral triangle, so KM has measure 2, and MO has measure √(2²-1²) = √3 by the Pythagorean theorem.

Then the ratio of KM to LM is 2:(1+√3) and the ratio of KL to LM is √2:(1+√3). That is, ...

... KL = LM×(√2)/(1+√3) = (20√3)(√2)/(1 +√3)

... KL = (20√6)/(√3 +1) = (20√6)(√3 -1)/(3 -1) . . . . . with denominator rationalized

... KL = 30√2 -10√6 ≈ 17.9315

and

... KM = LM×2/(1+√3) = KL×√2

... KM = (30√2 -10√6)√2

... KM = 60 -20√3 ≈ 25.3590

== == == == == ==

The Law of Sines tells you ...

... KL/sin(M) = KM/sin(L) = LM/sin(K)

Then ...

... KL = sin(M)·LM/sin(K) = sin(30°)·20√3/sin(105°) ≈ 17.9315

... KM = sin(L)·LM/sin(K) = sin(45°)·20√3/sin(105°) ≈ 25.3590