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)

Answer 2

The length of KL and KM from the given diagram are 10√2 and 27.32 respectively

The sum of an interior angle of a triangle is 180 degrees

From the triangle:

<K + <L + <M = 180

<K + 105 + 30 = 180

<K + 135 = 180

<K = 180 - 135

<K = 45 degrees

$\frac{LM}{sin45} =\frac{KL}{sin30}\\ \frac{20}{sin45}=\frac{KL}{sin30}\\KL =\frac{20\times sin30}{sin45}\\KL = \frac{10}{\frac{1}{\sqrt{2} } } \\KL = 10\sqrt{2}$

Get the length of KM;

$\frac{KL}{sin30}= \frac{KM}{sin105} \\\frac{10\sqrt{2} }{sin30}= \frac{KM}{sin105} \\20\sqrt{2} = \frac{KM}{sin105} \\KM = 20\sqrt{2}sin105\\KM= 20\sqrt{2}\times 0.9659\\KM=27.32$

Hence the length of KL and KM from the given diagram are 10√2 and 27.32 respectively

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