A loci question is solved by turning each condition into a shape you can draw, one condition at a time, then shading only the region where every drawn shape overlaps. Most O-Level loci questions combine two of four standard loci: a circle (fixed distance from a point), a strip (fixed distance from a line), a perpendicular bisector (equidistant from two points), or an angle bisector (equidistant from two lines). Draw each accurately with a compass and ruler, then find the overlap.
A loci question usually reads as one long sentence with two or three conditions joined by "and", for example "the point is less than 5 cm from A, and nearer to B than to C." Students who try to picture the whole shaded region in their head before drawing anything tend to freeze, because the final answer is a small overlap of shapes that only makes sense once each shape exists on paper.
The fix is to stop trying to see the answer immediately. Draw one condition at a time, exactly as if it were the only condition in the question, then move to the next. The final region only needs to be identified after every individual locus is on the diagram, not before.
Step 1
Identify which of the 4 standard loci each condition describes
Step 2
Draw each locus accurately with compass and ruler
Step 3
Decide boundary line, region, or both, per condition
Step 4
Shade only where every condition overlaps
r from a point Pr, centre Pd from a straight lined away (a rounded strip if the line is a segment, not an infinite line)A and BABPoint P must be less than 4 cm from a fixed point A, and nearer to point B than to point A, where A and B are 6 cm apart. Describe and shade the region containing all possible positions of P.
Solution
A. Draw this circle with a compass set to 4 cm.AB. Points on the B side of this line are nearer to B.AB: draw two arcs of equal radius (greater than 3 cm, half of AB) from A and from B, one above the line and one below, then join the two arc intersections with a straight line.B side of the perpendicular bisector. Both conditions here are strict: "less than 4 cm" and "nearer to B" each exclude their own boundary, so draw the circle and the bisector as dashed lines. Had the question said "at most 4 cm" or "at least as near to B", those boundaries would be solid.Quick check: pick a point inside your shaded region and check it against both conditions in words. If it fails either one, your shading has drifted onto the wrong side of a boundary. Here the shaded piece is a thin sliver sitting between 3 cm and 4 cm from A, so B itself is not inside it: AB is 6 cm, which is more than 4 cm. Shading all the way across to B is the giveaway that the circle condition was dropped.
A goat is tied so that it stays within 3 m of a straight fence XY, and closer to fence XY than to a second fence XZ, where the two fences meet at X at an angle. Describe the region the goat can graze in.
Solution
XY, one 3 m away on each side, plus a rounded end at any free end of the segment (a point near the end can be within 3 m of the endpoint itself, not just of the line). If the diagram fences the goat into the region between XY and XZ, only the half of that strip on the XZ side is available, but draw the full strip first and rule out the rest afterwards.YXZ. Points on the XY side of this bisector are closer to XY.YXZ: from X, draw one arc crossing both XY and XZ; from each of those two crossing points draw a second arc of equal radius so the arcs intersect; join X to that intersection.XY and on the XY side of the angle bisector.The step students get wrong
Shading the wrong side of a boundary line is the single most common error, especially for "equidistant from two points" and "equidistant from two lines" conditions, because both a perpendicular bisector and an angle bisector look like an ordinary straight line once drawn, with no visual hint about which side is which. Test one clearly labelled point on each side of every boundary against the original wording before shading, rather than shading by instinct.
Whether a boundary is included in the region depends on the wording of the condition, not on the shape of the locus itself.
Exam tip: if a question asks only for the locus itself, for example "the locus of points equidistant from A and B," the answer is a single line or curve, not a shaded region. Shading is only needed once the question uses words like "region", "less than", or "nearer to".
How many conditions can a loci question combine?
Most O-Level questions combine two conditions, but some combine three. The method does not change: draw each locus separately in any order, then shade only where every single drawn shape overlaps at the same time.
Do I need to construct every locus with compass and ruler, or can I estimate?
Construct it properly. Examiners check for correct construction lines and arcs, and a freehand or estimated boundary in roughly the right place earns few or no marks even if the final shading looks correct.
What if a condition describes distance from a line segment, not an infinite line?
The strip locus gets rounded ends at each endpoint of the segment, because a point can be within the given distance of an endpoint even after it has moved past the end of the line itself. Picture the strip as a rectangle with a semicircle capping each end.
How is this different from the constructions I already know?
Loci questions use exactly the same two constructions as perpendicular and angle bisector constructions, plus circles and parallel strips. The new skill in loci is combining two or more of these constructions on one diagram and correctly identifying the overlap, not learning new constructions. The diagram itself is often a shape you have already built using the methods in constructing triangles and quadrilaterals, with the loci conditions added on top.
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Mr. Gan works with students who want a reliable way to combine loci conditions, not guesswork under exam pressure.
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