Question

find the indicated lengths: given KL parallel NM and LM=45, the measure of angle M=50 degrees, and KN perpindicular NM. and NL perpindicular LM find KN and KL.

Answer 1

Interpret the figure as triangle LMN with LN ⟂ LM (so ∠L = 90°) and ∠M = 50°. Thus LM = 45 is the leg adjacent to ∠M.

Compute first:
- MN (the hypotenuse) = LM / cos 50° = 45 / cos50°
- LN (the other leg) = LM · tan 50° = 45·tan50°

Now place L at the origin, LM on the x-axis and N on the y-axis. The line KL is drawn through L parallel to MN, and KN is drawn through N perpendicular to MN; their intersection is K. Solving (or using similar/right-triangle projections) gives

- KN = LM · sin 50° = 45·sin50° ≈ 34.47
- KL = LM · sin 50° · tan 50° = 45·sin50°·tan50° ≈ 41.07

So KN = 45 sin50° ≈ 34.47 and KL = 45 sin50° tan50° ≈ 41.07.