Let's go step-by-step through what is shown in the image:

Part (iii) - Calculating the Length of LM

The task is to calculate the length of LM to the nearest kilometer.

From the image, the calculations are:

1. $80^2 = 6400$
2. $120^2 = 14400$
3. $2 \times 80 \times 120 = 19200$

It seems like a cosine or distance formula is being used here, possibly related to vectors or trigonometry.

Continuing:

4. $19200 \times (\cos L) = 14717.76$
5. To calculate $LM^2$: $LM^2 = 6400 + 14400 - 14717.76$
6. This yields: $LM = \sqrt{35517.76} \approx 188.49 \, \text{km}$

Rounding to the nearest kilometer, the answer is:

$LM \approx 188 \, \text{km}$

Part (iv) - Measure of $\angle LKM$

The angle $\angle LKM$ is calculated to be $140^\circ$.

The answer to this part is simply:

$\angle LKM = 140^\circ$

Part (v) - Bearing of M from K

The task is to calculate the bearing of point M from point K.

Bearings are usually measured in degrees clockwise from North. Since part (iv) gave $\angle LKM = 140^\circ$, I can assume there might be additional information, such as a direction indicator or coordinates, to calculate the bearing.

However, it's not fully clear from the given work, so if you have additional context, please share it, and I'll be happy to help.

Do You Want More Details?

Do you need more detailed explanations for any of these calculations, or help with a specific method used here?

Relative Questions to Expand Your Understanding:

1. What trigonometric relationships can be used to solve for lengths and angles in triangles?
2. How do you apply the cosine rule to calculate the side length of a triangle?
3. What is the standard approach for rounding distances to the nearest kilometer?
4. How can bearings be interpreted and calculated from given angles?
5. What are the common methods used to solve for an angle when side lengths are known?

Tip: Always keep track of rounding instructions—whether the result needs to be rounded up or down can make a significant difference in some practical contexts, like navigation.