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This page sets out the method for loci and constructions in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Loci and constructions topic page.
When you need the line halfway between two points, or points equidistant from A and B.
When you need the line that splits an angle exactly in half, or points equidistant from two lines.
When a question says 'shade the region where…' with two or more conditions.
Points a fixed distance from one point form a circle.
The radius is that distance. The centre is the point.
The locus is the perpendicular bisector of the segment joining them.
Every point on that line is the same distance from both ends.
The locus is the angle bisector.
Construct it with two pairs of arcs so the distances to the lines match.
“Closer to A than to B” is one side of the perpendicular bisector.
Shade the region the question names, and use a solid or dashed boundary if the question includes the boundary.
A construction is marked by the arcs, not by measuring with a ruler.
The arcs are the evidence of the method.
The perpendicular bisector crosses the segment at its midpoint.
The midpoint is halfway, so each half is half the length.
Put the compass point on the fixed point and open it to the given distance. Draw the full circle. Every point on that circle is the right distance from the centre, and every point at that distance lies on the circle. A point 5 cm from O is on the locus “5 cm from O”. A point 7 cm from O is not. If the question says “less than 5 cm from O”, you shade the inside of the circle and the circle itself is the boundary, usually drawn dashed when it is not included.
A distance check can use coordinates. From O, a point that is 3 cm across and 4 cm up is √(3^2 + 4^2) cm away. That is √(9 + 16) = √(25) = 5 cm, so the point sits on the circle of radius 5 cm. The same idea works on a scale drawing: measure from the centre, not from a corner of the page.
Method: 1. Put the compass on the fixed point and open it to the given distance. 2. Draw the circle: every point on it is that distance from the centre. 3. Check a point 3cm across and 4cm up with sqrt3^2+4^2=5cm.
To construct the perpendicular bisector of AB, open the compass to a radius greater than half of AB. Put the point on A and draw an arc above the line and an arc below it. Without changing the radius, put the point on B and draw arcs that cross the first pair. Join the two crossing points with a straight line. That line is the perpendicular bisector. It cuts AB at its midpoint and it is at right angles to AB.
If A and B are 8 cm apart, the midpoint is 8 ÷ 2 = 4 cm from A and 4 cm from B. You do not need to measure that 4 cm with a ruler if the arcs were drawn properly: the construction forces the crossing. Leave the arcs. “Closer to A than to B” is the half of the page that contains A, up to the bisector. Points on the bisector itself are the same distance from A and from B, so they are not closer to either point.
Method: 1. Open the compass past half of AB, draw crossing arcs from A and from B, then join the crossings. 2. The line cuts AB in half and meets it at 90^°. 3. If AB is 8cm, the midpoint is 4cm from A and 4cm from B.
To bisect the angle at C, put the compass point on C and draw an arc that crosses both arms of the angle. From each of those two crossings, draw equal arcs inside the angle so that they meet. Join C to that meeting point. The ray is the angle bisector. If the whole angle is 70^°, the bisector makes two angles of 70^° ÷ 2 = 35^°.
This is the locus of points equidistant from the two lines. It is not the perpendicular bisector of anything unless the question says so. A common mix-up is to join the two crossings on the arms and call that an angle bisector. That chord is a different line. The bisector must pass through the vertex, because the vertex is on both lines and the locus starts there.
Method: 1. From the vertex, mark both arms, then draw equal arcs from those marks so they meet. 2. Join the vertex to that meeting point. 3. An angle of 70^° splits into 70÷ 2=35^° and 35^°.
Read every condition and sketch each boundary before you shade. Suppose AB is 10 cm. The perpendicular bisector sits at the midpoint, 5 cm from A. The region closer to A than to B is on A’s side of that line. A second rule, “inside the circle of radius 6 cm centred at A”, allows points up to 6 cm from A. Along the line AB, the bisector stops you at 5 cm and the circle would have allowed 6 cm. The stricter limit is 5 cm.
The circle reaches 6 - 5 = 1 cm past the bisector, so near the middle of AB the circle is not the rule that bites. A point 4 cm from A along AB is closer to A than to B, and it is inside the circle, so it is in the region. A point 5.5 cm from A is 4.5 cm from B, so it is closer to B and it is outside, even though 5.5 is still less than 6. Shade only the overlap, and label which side you kept.
Method: 1. Sketch each boundary, then shade only the overlap. 2. On a 10cm line AB, stop at the bisector, which is 5cm from each end. 3. A circle of radius 6cm reaches further, so along AB the stricter limit is 5cm.
A locus question starts with a sentence such as "the same distance from A and B". The first job is to turn that sentence into a shape. "A fixed distance from a point" is a circle. "The same distance from two points" is the perpendicular bisector of the line joining them. "The same distance from two lines that meet" is the angle bisector. "A fixed distance from a line" is a pair of parallel lines with a semicircle at each end.
Say the shape in words before you draw or calculate. In an exam you may be asked to describe the locus, to construct it, or to use it in a calculation. Many wrong answers come from using the wrong shape, for example drawing a circle when the question wanted a bisector. Write a short label next to your work, such as "bisector of AB" or "circle, centre O, radius 4 cm".
Method: 1. Underline the rule in the question, such as "same distance from" or "less than". 2. Match it to a circle, a perpendicular bisector, an angle bisector or a parallel line shape. 3. Write the name of the shape and the key numbers next to it. 4. Only then draw, construct or calculate.
Tip: A locus is always a line or a curve. A region is an area, and you need shading for it.
A region question adds a direction to the locus. "Less than 4 cm from A" is the inside of the circle of radius 4 cm. "At least 4 cm from A" is the circle and everything outside it. "Closer to A than to B" is the side of the perpendicular bisector that contains A. When there are two rules, the answer is where both are true, so you shade only the overlap.
Draw each boundary lightly first, and decide which side of each boundary is wanted. A dashed line can show a boundary that is not included, such as "less than", and a solid line can show one that is, such as "at most". Then shade the overlap and label it, for example "R". Examiners give a mark for each correct boundary and a mark for the correct shading.
Method: 1. Draw each boundary lightly with a ruler or compasses. 2. Decide which side of each boundary the words ask for. 3. Shade only where all the rules are true at once. 4. Label the region if the question asks you to.
Constructions must use compasses and a straight edge, and you must leave your construction arcs visible. To construct the perpendicular bisector of AB, set the compasses to more than half of AB, draw an arc from A above and below the line, then draw an arc with the same radius from B so that the arcs cross. Join the two crossing points with a ruler. If the compasses are set to less than half of AB, the arcs will not cross.
To bisect an angle, draw an arc from the vertex that cuts both arms, then draw arcs of equal radius from the two cut points so that they cross, and join the crossing point to the vertex. To make a 60^° angle, construct an equilateral triangle. Keep the compass width fixed within each step and never rub out the arcs, since the marks are awarded for the arcs as well as the final line.
Method: 1. Set the compasses to a suitable radius, more than half of the line for a bisector. 2. Draw the arcs from each end, keeping the same radius. 3. Join the points where the arcs cross with a ruler. 4. Leave all construction arcs visible and label your final line or region.
Tip: If your arcs do not cross, increase the compass width. Never rub out your arcs.
Many exam questions ask for a length or area linked to a locus. A point on the perpendicular bisector of AB is the same distance from A and B, so the bisector, half of AB and the distance to the point form a right angled triangle. Use Pythagoras to find unknown lengths. A rope or a sprinkler gives a circle, so area is π r^2 and the circumference is 2π r, often left in terms of π.
A tethered animal at the corner of a building can only reach part of a circle. At an outside corner of a shed that is a 270^° sector, which is 3/4 of a full circle. If the rope is long enough to go around a corner, the part left over gives another sector with a smaller radius. Draw a quick sketch and write the radius of each sector before you calculate its area.
Method: 1. Draw a sketch and mark every distance and radius. 2. Find any missing length with Pythagoras. 3. Work out each sector or circle area using the fraction of the full circle. 4. Add the areas and leave the answer in terms of π if it asks for that.
AQA-style and Edexcel-style questions. Yumio is not affiliated with, endorsed by or connected to AQA, Pearson Edexcel or any other exam board.