Short Answer 
The altitude of triangle LMN is approximately 34.47, calculated using the sine function with a hypotenuse of 45 and an angle of 50°. The length of KL is found to be approximately 41.08 using the tangent ratio with the previously calculated altitude.
Step 1: Calculate the Altitude
To find the altitude of triangle LMN, we use the sine function, which relates the angle and the lengths of the sides. By applying the formula, we find:
- Use the angle: 50°
- Opposite side: altitude
- Hypotenuse: LM = 45
Therefore, the equation becomes: sin(50°) = altitude/45. Solving it gives us an altitude of approximately 34.47.
Step 2: Determine the Angle Measures
In triangle LMN, since it is a right triangle, we know that the angles must sum up to 90° in the non-right angles. Hence, we calculate:
- Angle LNM = 90¬∞ РAngle M = 90¬∞ Р50¬∞ = 40¬∞
- Angle LNK = 90¬∞ РAngle LNM = 90¬∞ Р40¬∞ = 50¬∞
This step is crucial for later computations of other side lengths using trigonometric functions.
Step 3: Calculate the Length of KL
The length of KL can be determined by using the tangent ratio, which compares the opposite side (KL) to the adjacent side (KN, which is the altitude). The formula we apply is:
- tan(50°) = KL/KN
- KL = KN · tan(50°)
- Substituting KN with our previously calculated altitude, KL = 34.47 · tan(50°)
This yields a length for KL of approximately 41.08.