Tom runs 16.5 m South-West, 2 m East, 9 m North, 2 m East, and finally 7 m…

Concept

The direction of one point from another is fixed entirely by the signs of the net displacement between them: resolve every leg of a journey into a north–south part and an east–west part, add those parts with their signs, and the resulting pair of signs names the quadrant. A leg along a corner bearing such as South-West lies at 45° to both axes, so it contributes an equal share to each — a length L splits into L/√2 south and L/√2 west. One further rule closes problems of this kind: the direction of A from B is always the exact reverse of the direction of B from A.

Application

Place the dining table (the start) at the origin, take north as positive on the vertical axis and east as positive on the horizontal axis, and tally the five legs in order.

1. 16.5 m South-West lies at 45°, so it splits into 16.5/√2 ≈ 11.67 m west and 16.5/√2 ≈ 11.67 m south.

2. 2 m East adds 2 m to the eastward column.

3. 9 m North adds 9 m to the northward column.

4. 2 m East brings the eastward column to 2 + 2 = 4 m.

5. 7 m North brings the northward column to 9 + 7 = 16 m.

6. East–west balance: 4 m of eastward travel against 11.67 m of westward travel leaves 11.67 − 4 = 7.67 m west.

7. North–south balance: 16 m of northward travel against 11.67 m of southward travel leaves 16 − 11.67 = 4.33 m north.

8. The door therefore finishes 7.67 m west and 4.33 m north of the dining table, so the door lies in the North-West quadrant as seen from the table.

9. The stem asks for the reverse relation, so flip both components: the dining table lies 7.67 m east and 4.33 m south of the door, which is the South-East quadrant.

Cross-check

Walk the claim backwards. If the dining table sits South-East of the door, then travelling from the table to the door has to be a net move north and west — and the tally does give 4.33 m north and 7.67 m west, so the two readings agree. The result is also not delicate about the exact corner bearing: the journey stays net-north for any south-west leg whose southward part is under 16 m, and net-west for any whose westward part exceeds 4 m, and the 11.67 m split clears both margins comfortably. The usual slip is to compute where the door ended up relative to the table and stop there; the stem asks the opposite question, and answering it flips both components.

Result: the dining table is South-East of the door.