Direction Sense

Mental model

You are drawing a map from a fixed origin.

Direction problems test your ability to maintain a constant frame of reference. By plotting lines proportionally from a starting origin point and rotating your perspective for turns, complex paths reduce to simple geometry.

Method

1. Draw a standard compass cross on your scratchpad (North up, South down, East right, West left).
2. Start every problem by marking a clear central origin point (the starting point).
3. Trace the path step-by-step. Draw lines that are roughly proportional to the given distances (e.g., a 10m line should look twice as long as a 5m line).
4. For left/right turns, imagine yourself standing on the paper facing the current direction. A 'Right' turn is always 90 degrees clockwise; a 'Left' turn is 90 degrees counter-clockwise.
5. To find the final distance from the start, draw a straight line connecting the end point to the origin. If it's diagonal, drop a perpendicular line to form a right triangle and use Pythagoras' theorem (a^2 + b^2 = c^2).

Worked example

Problem: A person walks 4 km East, then turns Left and walks 3 km. How far and in which direction is they from the starting point?

1. Mark the origin point. Draw a line extending to the right (East) and label it '4'.
2. Facing East, a 'Left' turn means turning counter-clockwise, which is North (up).
3. Draw a line pointing up from the end of the 4km line, and label it '3'.
4. Connect the final point back to the origin. This forms a right-angled triangle with base 4 and height 3.
5. Use Pythagoras: Distance^2 = 4^2 + 3^2 = 16 + 9 = 25. Distance = 5.
6. Determine direction: The end point is above and to the right of the origin, which is North-East.

Answer: 5 km North-East

Common mistakes

• Drawing disproportionate lines: If your 10m line is drawn shorter than your 5m line, you will misjudge the final position relative to the starting point and guess the wrong direction.
• Confusing left/right when facing South: When facing South (down), a 'Right' turn goes West (left on the paper). Not reversing your perspective causes mirrored mistakes.

Glossary

proportional

Drawn so that sizes accurately match the real-world distances.

perpendicular

A line that meets another line at an exact 90-degree corner.

origin

The exact starting point from which a journey or measurement begins.

Recall questions

• What rule of thumb prevents direction errors on left/right turns?
• Why is it important to draw line segments proportionally?
• How do you calculate the shortest distance if the endpoint is diagonal from the start?

Questions & answers

A man walks 5 km South, turns right and walks 3 km, then turns right again and walks 5 km. In which direction is he from the starting point?

West

Approach: He goes South (down) 5. Turning right (clockwise) means going West (left) 3. Turning right again (clockwise) means going North (up) 5. He is now exactly 3 km West of his starting point.

Starting from point X, a girl walks 15m West, turns left and walks 20m, then turns left again and walks 15m. How far is she from X?

20m

Approach: West 15, left (South) 20, left (East) 15. The final East path cancels out the initial West path. She is exactly 20m South of point X.

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