Constructing perpendicular and angle bisectors, perpendiculars from a point, and finding loci of points satisfying a given condition.

What this dot point is asking

AQA wants you to perform the standard ruler-and-compasses constructions (perpendicular bisector, angle bisector, perpendicular from or to a point) leaving your construction arcs visible, and to find loci, the set of all points satisfying a distance or position condition. Construction questions reward accuracy and visible arcs; loci questions combine several constructions to shade a region. These appear at both tiers and require precise compass work.

The standard constructions

All constructions must keep the compass arcs visible, because the arcs are how the examiner awards method marks. Use a sharp pencil and do not erase the arcs.

• Perpendicular bisector of a line $AB$: open the compasses to more than half of $AB$. From $A$ draw arcs above and below the line, then with the same radius from $B$ draw arcs that cross them. Join the two crossing points; this line is perpendicular to $AB$ and passes through its midpoint.
• Angle bisector at a vertex: place the compass at the vertex and draw an arc crossing both arms. From each of those two crossing points, draw equal arcs that meet inside the angle. Draw a line from the vertex through that meeting point.
• Perpendicular from a point to a line: with the compass on the point, draw an arc crossing the line twice, then bisect the segment between those two crossings.

Loci

A locus is the set of all points obeying a condition. The standard loci are worth memorising.

Combining loci to find a region

Many exam questions ask for the region satisfying two or three conditions at once, such as "within $5\,\text{cm}$ of $A$ and nearer to line $PQ$ than line $PR$". Draw each locus separately (a circle, a bisector, a parallel pair), then shade only where all the conditions overlap. State whether boundaries are included by using a solid line for "at most" and being careful with "less than".

Constructing accurate triangles and angles

Beyond the bisectors, you should be able to construct a triangle from given information using ruler and compasses. Given three sides (SSS), draw the base, then use the compass set to each of the other two side lengths to swing arcs from the two ends; their intersection is the third vertex. The same compass-and-arc approach constructs a $60^\circ$ angle (the angle of an equilateral triangle) without a protractor, and bisecting that gives $30^\circ$. These exact constructions are a regular exam requirement, and accuracy to within about a millimetre or a degree is expected.

Why the bisector loci work

The two key loci are worth understanding, not just performing. The set of points equidistant from two fixed points $A$ and $B$ is exactly the perpendicular bisector of $AB$, because any point on that line is the same distance from each end by symmetry. The set of points equidistant from two intersecting lines is the angle bisector between them, since reflecting across the bisector swaps the two lines. Knowing the meaning lets you translate a worded condition ("equally far from the two roads") straight into the right construction without guessing.