Loci

Constructing a Perpendicular Bisector

• Step 1: Set the compass to a width greater than half the length of the line segment $AB$.
• Step 2: Place the compass point on $A$ and draw arcs above and below the line.
• Step 3: Keeping the same width, place the point on $B$ and draw arcs that intersect the first two.
• Step 4: Draw a straight line through the two intersection points of the arcs.

Constructing an Angle Bisector

• Step 1: Place the compass point at the vertex of the angle and draw an arc that crosses both lines forming the angle.
• Step 2: Place the compass point at each intersection on the lines and draw two arcs that cross each other in the middle of the angle.
• Step 3: Draw a straight line from the vertex through the intersection of these two arcs.

• Perpendicular Bisector is used when comparing distances to two distinct locations (e.g., 'closer to Town A than Town B').

• Angle Bisector is used when comparing distances to two boundaries or paths (e.g., 'closer to the fence than the road').

• Keep Construction Lines: Never rub out your arcs. Examiners need to see these to award marks for the correct method, even if the final line is slightly off.

• Check the Scale: Always convert real-world distances (e.g., 5 km) into drawing distances (e.g., 5 cm) using the provided scale before setting your compass.

• Boundary Styles: Use a solid line if the condition is 'less than or equal to' ($\le$) and a dashed line if the condition is strictly 'less than' ($<$).

• Shading Regions: If multiple conditions apply, identify the valid region for each one separately (perhaps with small ticks) and only shade the final area where all conditions overlap.

• The 'Sausage' Error: When drawing a locus at a fixed distance from a line segment, students often forget the semi-circular ends. A point at the very end of the line segment still has a 'fixed distance' radius extending around it.

• Compass Slip: Ensure the compass hinge is tight. If the width changes between drawing the first and second set of arcs, the bisector will be inaccurate.

• Wrong Bisector: Confusing the perpendicular bisector (for points) with the angle bisector (for lines) is a frequent error in word problems.