Given triangle imj with altitude jl, jl = 32, and il =24, find ij, jm, lm, and im.

Okay, so I really hope that you can read all my work. I just spent the last 45 mins doing this problem as neatly as possible with as much detail as possible. So, I really hope my work doesn't confuse you. 

My Final Answers were:
JI= 40 units (found using the Pythagorean theorem)
IM= 66.66 units = 66 and 2/3rds (found by dividing the length of JL by the cosine of ∠JIM )
LM=42.66 units = 42 and 2/3rds (found by IM= 24+LM; solve for LM since we know IM=66.66)
JM= 53.33 units= 53 and (1/3rd) (found by using the Pythagorean theorem; this time using JI as "a" and IM as c)

Hope this helped and all made sense!

(Pythagorean theorem is a²+b²=c²)

Part A:

From the given figure, IJ represents the hypothenuse of the right triangle IJL.

By the Pythagoras theorem,



Part B:

From the given figure, angle J is obtained as follows:



Line JM can be obtained as follows:



Part C:

From the given triangle, LM is one of the legs of the right triangle JLM with the other leg, JL = 32 and the hypothenuse, JM = 160/3.

By the Pythagoras theorem,



Part D:

From the figure, IM = IL + LM