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

Step 1: Use the perimeter.
\[ 2(L + W) = 68 \implies L + W = 34 \]

Step 2: Recognise the shortcut.
Walking from P to Q along only two adjacent full sides of the rectangle, one of length \( L \) and one of length \( W \), covers a distance of \( L + W \), which is exactly half the rectangle's perimeter.

Step 3: Match this to the bus route.
The bus's zig-zag path from P to Q never backtracks, so its total rightward travel is still \( L \) and its total upward travel is still \( W \), the same as walking along those two full sides directly, just broken into smaller steps.

Step 4: Take half the perimeter.
\[ \frac{68}{2} = 34 \]
\[ \boxed{34} \]

Answer 2

A third way to see this is to pair the bus's actual route with the complementary route along the two sides it did not use.

From P to Q, there are two ways to walk along the rectangle's edges without ever moving away from Q: the bus's zig-zag route (through the interior along grid lines), and a plain route along the other two sides of the rectangle. Both of these represent the same total rightward travel \( L \) and the same total upward travel \( W \), just arranged differently, so both routes have the same length.

Now add the bus's route and this complementary edge route together. Between them, they cover all four sides of the rectangle exactly once, so their combined length equals the full perimeter, 68 km.

Since the two routes are equal in length, each one is half of 68 km, which is 34 km.

So the correct answer is 34 kilometres.