Understanding the Policeman's Movement and Distance Calculation

The question asks for the approximate shortest distance between the police station and the restaurant, specifically stating "via the fruit shop". This implies we need to consider the path or the straight-line segments connecting these points, ensuring the fruit shop is included.

Let's trace the policeman's path step by step and map the locations:

• The policeman starts at the Police Station (let's call this point PS).
• He walks 60 m towards the west. Let's call the end of this segment Point A. So, PS → Point A (60 m West).
• From Point A, he takes a left turn. If he was walking west, a left turn is towards the south. He walks 38 m to reach the Fruit Shop (FS). So, Point A → FS (38 m South).
• From the Fruit Shop, he takes a right turn. If he was walking south towards the Fruit Shop, a right turn is towards the west. He walks 60 m. Let's call the end of this segment Point C. So, FS → Point C (60 m West).
• From Point C, he takes a right turn. If he was walking west, a right turn is towards the north. He walks 38 m to reach the Restaurant (R). So, Point C → R (38 m North).

Mapping Locations Using Coordinates

Let's assume the Police Station (PS) is at the origin (0,0) for simplicity. Directions are standard: East (+x), West (-x), North (+y), South (-y).

• Police Station (PS): (0, 0)
• Point A (60m West of PS): (-60, 0)
• Fruit Shop (FS) (38m South of Point A): (-60, -38)
• Point C (60m West of FS): (-60 - 60, -38) = (-120, -38)
• Restaurant (R) (38m North of Point C): (-120, -38 + 38) = (-120, 0)

So, the coordinates are:

Calculating Distance "Via the Fruit Shop"

The phrase "shortest distance between the police station and the restaurant via the fruit shop" typically means the shortest path that includes the fruit shop. This is usually interpreted as the straight-line distance from the Police Station to the Fruit Shop plus the straight-line distance from the Fruit Shop to the Restaurant.

Straight-line Distance from Police Station (0,0) to Fruit Shop (-60,-38)

We can use the distance formula derived from the Pythagorean theorem: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

Distance (PS to FS) = $\sqrt{(-60 - 0)^2 + (-38 - 0)^2}$

Distance (PS to FS) = $\sqrt{(-60)^2 + (-38)^2}$

Distance (PS to FS) = $\sqrt{3600 + 1444}$

Distance (PS to FS) = $\sqrt{5044}$ m

Straight-line Distance from Fruit Shop (-60,-38) to Restaurant (-120,0)

Using the distance formula again:

Distance (FS to R) = $\sqrt{(-120 - (-60))^2 + (0 - (-38))^2}$

Distance (FS to R) = $\sqrt{(-120 + 60)^2 + (0 + 38)^2}$

Distance (FS to R) = $\sqrt{(-60)^2 + (38)^2}$

Distance (FS to R) = $\sqrt{3600 + 1444}$

Distance (FS to R) = $\sqrt{5044}$ m

Total Approximate Shortest Distance via Fruit Shop

Total distance = Distance (PS to FS) + Distance (FS to R)

Total distance = $\sqrt{5044} + \sqrt{5044} = 2 \times \sqrt{5044}$ m

Now, let's approximate the value of $\sqrt{5044}$.

$\sqrt{5044} \approx 71.021$ m

Total distance $\approx 2 \times 71.021 \approx 142.042$ m

Comparing this result to the given options, 142 m is the closest approximate value.

Note: If the question were asking for the total distance covered along the exact path described, it would be 60 m + 38 m + 60 m + 38 m = 196 m, which is not among the options. The interpretation using straight-line distances between the key points (Police Station, Fruit Shop, Restaurant) that form the shortest path via the fruit shop matches one of the options.

Revision Table: Key Concepts in Direction and Distance

Additional Information on Direction Problems

Direction and distance problems often involve visualizing movements on a plane. It's helpful to:

• Draw a diagram to represent the path taken.
• Use a coordinate system to track locations precisely.
• Distinguish between total distance traveled (path length) and displacement (straight-line distance between start and end).
• Remember that a 'left turn' or 'right turn' is relative to the current direction of movement.

In cases asking for the "shortest distance", it usually refers to the straight-line distance (displacement) unless specified otherwise (like "shortest distance along the path"). The phrase "via the fruit shop" guides us to include that specific point in the calculation, typically by summing the straight-line segments connecting the start, the via point, and the end.