A perpendicular bisector is drawn by striking equal-radius arcs from both ends of a segment and joining where the arcs cross: every point on that line is equidistant from the two endpoints. An angle bisector is drawn by striking an arc across both arms of the angle, then equal-radius arcs from those two crossing points: every point on that line is equidistant from the two arms. Keep every arc on the page for full construction marks.
Constructions questions are not really about drawing a neat line. They are about proving, through the visible arcs, that you used compasses rather than a protractor or a ruler to estimate the halfway point. Examiners mark the method, not just the final line, so a construction with no arcs showing usually loses marks even if the line itself lands in the right place.
The second common failure is changing the compass radius partway through. If the two arcs from the ends of a segment are drawn with different radii, the intersections will not sit on the true perpendicular bisector, and the resulting line will be tilted. The construction only works because both arcs share exactly one setting.
Perpendicular bisector, step 1
Open compasses to more than half of AB
Perpendicular bisector, step 2
Arc from A, then same radius arc from B
Angle bisector, step 1
Arc centred at vertex, crossing both arms
Angle bisector, step 2
Equal arcs from those 2 points, meeting inside
Both constructions rely on the same idea: two arcs drawn with an identical radius from two different centres meet only at points that are the same distance from both centres. Join those meeting points and you have drawn a line made entirely of equidistant points, which is exactly what a bisector is.
Construct the perpendicular bisector of a line segment AB that is 7 cm long, and state what is true of every point on your construction line.
Solution
AB = 7 cm using a ruler.AB, so more than 3.5 cm. A radius of 5 cm works well.A and draw an arc above the line and an arc below the line, without changing the radius.B, keep the same 5 cm radius, and draw two more arcs that cross the first pair, one above and one below AB.AB: it crosses AB at its midpoint, at 90°.The property to state: every point on the perpendicular bisector of AB is equidistant from A and from B. This one sentence is often worth its own mark, separate from the marks for the construction lines themselves.
Angle PQR = 60°, formed by two straight lines QP and QR meeting at Q. Bisect angle PQR, then use your bisector to mark a point X, 4 cm from Q, that is equidistant from lines QP and QR.
Solution
Q, draw a single arc of any convenient radius, say 4 cm, that crosses both arm QP and arm QR. Label the crossing points M on QP and N on QR.M and draw an arc inside the angle.N, and draw a second arc inside the angle so that it crosses the arc from step 2.Q to the point where the two arcs from step 2 and step 3 cross. This ray bisects angle PQR, splitting it into two 30° angles.4 cm along the bisector from Q and mark this as point X. Because X lies on the angle bisector, it is equidistant from lines QP and QR.The step students get wrong
Two mistakes recur here. The first is rubbing out the arcs once the final line looks right: examiners award marks specifically for visible construction lines, so an unmarked, arc-free line earns little even when it is drawn in the correct place. The second is changing the compass radius partway through a construction, for instance opening the compasses wider for the second arc than the first. Any change in radius between the two arcs from A and B, or between the arcs from M and N, moves the intersection off the true bisector, and everything drawn from that point onward is wrong even though the method looks correct on paper.
Both bisectors reappear immediately in loci questions, where you are asked to shade a region satisfying two or more conditions at once, such as "equidistant from two lines and less than 4 cm from a point." See our loci problems guide for how to combine these conditions. For now, the two equidistance properties above (from a perpendicular bisector, and from an angle bisector) are the two constructions almost every loci question is built from. Constructions also rarely appear alone in an exam paper: the same diagram usually asks you to calculate an angle as well, so keep the standard angle facts fresh, both alternate, corresponding and co-interior angles on parallel lines and the interior and exterior angles of a polygon. These two bisector constructions also show up as a step inside larger triangle and quadrilateral constructions, once the main shape is drawn and a follow-on part asks for a locus.
Why does the perpendicular bisector construction actually work?
Any point that is the same distance from both A and B must lie on the perpendicular bisector of AB, and any point on the perpendicular bisector is the same distance from both A and B. The arcs drawn with equal radius from A and from B only cross at points that satisfy this equal-distance condition, which is why joining those crossing points always produces the correct line.
Does the radius I choose for the angle bisector arcs matter?
No, as long as the first arc from the vertex is large enough to cross both arms clearly, and the second pair of arcs (from the two crossing points) use one shared radius. The exact size of that shared radius does not change where the bisector lands, since the construction depends only on the two radii being equal to each other, not on any specific length.
Can I use a protractor instead of compasses for these questions?
No. Constructions questions specifically test the compasses-and-ruler method, and marks are awarded for the arcs themselves. A line at the correct angle but drawn with a protractor, with no construction arcs, typically scores zero on a "construct" question even if the angle is accurate.
How is this different from finding a midpoint using coordinates?
Finding a midpoint by calculation is a coordinate geometry skill using an (x, y) formula, not this drawing skill. This construction is the physical, compasses-based way to locate the same midpoint, and additionally produces the whole perpendicular line through it, not just the single midpoint.
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