A man faces towards north. Turning to his right, he walks 25 metres. He then…

CONCEPT: In a direction-sense (turn-and-walk) problem, track the walker's facing using the compass rule -- a right turn rotates the current facing 90 degrees clockwise (North to East to South to West to North) and a left turn rotates it 90 degrees counter-clockwise -- and record every straight leg as a displacement along the East-West axis (East positive) and the North-South axis (North positive). The final direction from the start is read off the SIGN of the two net displacement totals: East-positive with North-positive falls in the North-East quadrant, East-positive with North-negative falls in the South-East quadrant, and so on.

APPLICATION: Starting at the origin (0, 0) facing North, apply each turn and leg in order:

1. Turn right (North to East) and walk 25 m: now at (25, 0), facing East.

2. Turn left (East to North) and walk 30 m: now at (25, 30), facing North.

3. Turn right (North to East) and walk 25 m: now at (50, 30), facing East.

4. Turn right (East to South) and walk 55 m: now at (50, 30 - 55) = (50, -25), facing South.

5. Turn right (South to West) and walk 40 m: now at (50 - 40, -25) = (10, -25), facing West.

CROSS-CHECK: Net East-West displacement = 25 + 25 - 40 = 10 m East of the start; net North-South displacement = 30 - 55 = 25 m South of the start. Both totals match the running coordinates above -- the man ends up 10 m East and 25 m South of his starting point, which is the South-East quadrant. As an independent check, the straight-line distance from the start works out to √(102 + 252) = √725 ≈ 26.9 m, along that South-East bearing.

So the man is to the South-East of his starting point.