Question

The figure shows an urban road map of a city. The boundary of the city is a perfect rectangle as indicated by the black dotted line. The grey lines indicate the major roads that run parallel to the edges of the city. The red line shows the route taken by a bus from point P to point Q. If the perimeter of the boundary is 68 km, what is the distance travelled by the bus in kilometres?

Hint:

For any path on a grid that moves from one corner of a rectangle to the opposite corner without backtracking (i.e., moving only right/left and up/down in a consistent direction), the total path length is always the same: Length + Width of the rectangle. This is half the perimeter.

Answer 1

Correct Answer: 34

Step 1: Set up coordinates.
Place \(P\) at the origin \((0, 0)\) and \(Q\) at \((L, W)\), where \(L\) and \(W\) are the rectangle's length and width.

Step 2: Use the taxicab distance formula.
Since the bus can only move along the grid roads (horizontally or vertically, never diagonally, and always toward \(Q\)), its total distance matches the taxicab distance between \(P\) and \(Q\):
\[ \text{Distance} = |x_Q - x_P| + |y_Q - y_P| = L + W \]

Step 3: Find \(L + W\) from the perimeter.
\[ 2(L + W) = 68 \implies L + W = 34 \]

Step 4: Final Answer.
\[ \text{Bus distance} = \boxed{34 \text{ km}} \]

Answer 2

A more formal way to justify the answer is to prove first that every valid bus route between \(P\) and \(Q\) has the same length, no matter which particular sequence of turns it takes, and only then work out that shared length from the perimeter.

Any route the bus can take is made up of a series of moves, each one either a step to the right (along a horizontal road) or a step up (along a vertical road), since the bus is not allowed to move away from \(Q\) once it has committed to reaching it.
Swapping the order of any two consecutive moves, one right-step and one up-step, does not change how many right-steps or up-steps are used overall, it only changes the shape of the path.
Since any two valid routes can be turned into each other by repeatedly swapping the order of adjacent moves like this, every valid route uses exactly the same number of right-steps and the same number of up-steps.
The total right-step distance in any route must add up to the rectangle's full length \(L\) (since the bus starts at one side and ends at the other), and the total up-step distance must add up to the full width \(W\).
So every valid route, including the one shown in red, has the same total length: \[ \text{Distance} = L + W \]
From the given perimeter, \(2(L+W) = 68\), so \(L + W = 34\). \[ \text{Bus distance} = 34 \text{ km} \]

So the correct answer is 34 km.