Answer:
To find the shorter route, we need to calculate the distances for both routes and compare them.
Route through Towns P and Q:
The distance between Town A and Town P is 3 units to the right and 5 units up, so it is √(3² + 5²) = √34 km.
The distance between Town P and Town Q is 2 units to the right, so it is 2 km.
The distance between Town Q and Town B is 4 units to the right and 4 units down, so it is √(4² + 4²) = 4√2 km.
Therefore, the total distance for this route is √34 + 2 + 4√2 km.
Route through Town R:
The distance between Town A and Town R is 5 units to the right and 3 units up, so it is √(5² + 3²) = √34 km.
The distance between Town R and Town B is 5 units to the right and 7 units down, so it is √(5² + 7²) = √74 km.
Therefore, the total distance for this route is √34 + √74 km.
Comparing the distances, we can see that:
√34 + 2 + 4√2 km ≈ 10.2 km
√34 + √74 km ≈ 12.6 km
Therefore, the shorter route is the one that goes through Towns P and Q, which is approximately 10.2 km long.