Given: KLMN is a trapezoid, KL = MN,

Question

m∠LKM=m∠MKN, LM:KN = 8:9 ,
Perimeter KLMN = 132
LM and KN are parallel
Find: The length of the legs.

I reposted this question because the last time was not vary clear, and I got 49 but it is wrong

Answer 1

Since LM and KN are parallel, alternate interior angles ∠LKM and ∠LMK are congruent. So, ∆LKM is isosceles. That means that KL=LM.

So, since LM:KN=8:9, we can say that LM=8x and KN=9x. That means that KL=MN=8x, and thus

8x+9x+8x+8x = 132
33x = 132
x = 4

The legs KL and MN are 32.

Just to check, we need to make sure that this means that KM bisects ∠LKN.

It is easy to get the height of the trapezoid: √(32^2-2^2) = √1020

So,
tan∠LKN = √1020/2
tan∠MKN = √1020/34
And you can verify that ∠MKN is 1/2 ∠LKN

Answer 2

The length of the midsegment is 34