Constructions: bisectors, angles and loci

What you'll learn

This topic covers accurate geometric constructions using only a pencil, ruler and pair of compasses. Constructions questions appear regularly in Edexcel GCSE Mathematics Paper 2 and Paper 3, typically worth 2-4 marks. You must demonstrate precise construction techniques including perpendicular bisectors, angle bisectors, standard angles (60°, 90°), and loci representing sets of points satisfying specific conditions.

Key terms and definitions

Construction — a precise geometric drawing created using only a straight edge (ruler) and compasses, with construction arcs left visible as evidence of method.

Bisector — a line that divides an angle or line segment into two equal parts.

Perpendicular bisector — a line at right angles (90°) to a line segment, passing through its midpoint, equidistant from both endpoints.

Angle bisector — a line that divides an angle exactly in half, equidistant from both arms of the angle.

Locus (plural: loci) — the complete set of points satisfying a particular rule or condition, forming a path or region.

Equidistant — at equal distance from two or more points or lines.

Construction arcs — the circular marks made by compasses that must remain visible on your diagram to show your construction method.

Perpendicular — at right angles (90°) to a given line or surface.

Core concepts

Equipment and examination requirements

For constructions in Edexcel GCSE Mathematics, you must use:

• A sharp pencil (HB or H recommended)
• A ruler with clear mm markings
• A pair of compasses with a pencil attachment
• An eraser (but do NOT erase construction arcs)

Critical examination rule: Construction arcs must remain visible. Examiners award marks based on seeing correct construction method, not just the final answer. Erasing your arcs will lose method marks even if your answer is correct.

Constructing a perpendicular bisector

The perpendicular bisector of a line segment AB creates a line perpendicular to AB passing through its midpoint. Every point on this perpendicular bisector is equidistant from A and B.

Method:

1. Place compass point on A and set radius to more than half the distance AB
2. Draw arcs above and below the line segment AB
3. Keep the same compass radius, place compass point on B
4. Draw arcs above and below AB to intersect the first pair of arcs
5. Use a ruler to join the two intersection points with a straight line
6. This line is the perpendicular bisector of AB

Key applications: Finding the centre of a circle passing through two points, creating equal distances from two locations, or dividing a line segment exactly in half.

Constructing an angle bisector

The angle bisector divides any angle into two equal parts. Every point on an angle bisector is equidistant from both arms of the angle.

Method for bisecting angle ABC (where B is the vertex):

1. Place compass point on vertex B
2. Draw an arc crossing both arms of the angle (creating points D and E)
3. Place compass point on D and draw an arc inside the angle
4. Keep the same radius, place compass point on E
5. Draw another arc to intersect the previous arc (at point F)
6. Draw a straight line from B through F
7. BF is the angle bisector

Common exam context: Used to find points equidistant from two boundaries, walls or edges.

Constructing a 60° angle

Constructing a 60° angle uses the property that an equilateral triangle has all angles equal to 60°.

Method from point A on line AB:

1. Place compass point on A
2. Draw an arc crossing the line at point B (any convenient radius)
3. Keep the same radius, place compass point on B
4. Draw an arc to intersect the first arc at point C
5. Draw a straight line from A through C
6. Angle BAC = 60°

Extension: To construct 30°, bisect the 60° angle. To construct 120°, extend the construction or use supplementary angle properties.

Constructing a perpendicular from a point to a line

Two variations exist depending on whether the point lies on the line or away from it.

Method for perpendicular from point P not on line L:

1. Place compass point on P
2. Draw an arc crossing line L at two points (A and B)
3. Increase compass radius beyond half of AB
4. Place compass point on A, draw an arc below the line
5. Keep same radius, place compass point on B, draw arc to intersect previous arc at point C
6. Draw straight line from P through C
7. PC is perpendicular to line L

Method for perpendicular from point P on line L:

1. Place compass point on P with any convenient radius
2. Draw arcs on both sides along line L (creating points D and E)
3. Increase compass radius
4. Place compass point on D, draw arc above or below the line
5. Keep same radius, place compass point on E, draw arc intersecting previous arc
6. Draw line from P through intersection point

Constructing a 90° angle (perpendicular)

A 90° angle can be constructed by creating a perpendicular to a line. Use either perpendicular construction method above, or recognise that bisecting a 180° straight line creates 90°.

Understanding loci

A locus describes all points satisfying specific conditions. Four standard loci appear in Edexcel GCSE Mathematics:

1. Locus equidistant from a fixed point

• Forms a circle with the fixed point as centre
• Radius equals the specified distance
• Example: "The locus of points 3 cm from point A" is a circle, centre A, radius 3 cm

2. Locus equidistant from two fixed points

• Forms the perpendicular bisector of the line segment joining the two points
• Constructed using perpendicular bisector method
• Example: Mobile phone mast positioning equidistant from two towns

3. Locus equidistant from a fixed line

• Forms two parallel lines on either side of the given line
• Each parallel line is the specified distance away
• Include semicircular ends if the line segment is finite
• Example: "Points 2 cm from line segment AB" creates a 'racetrack' shape

4. Locus equidistant from two intersecting lines

• Forms the angle bisector (actually two bisectors for the four angles created)
• Constructed using angle bisector method
• Example: Finding the best position for a lamppost equidistant from two walls

Combining loci to solve problems

Real examination questions combine multiple loci to identify a specific region or set of points.

Typical problem structure:

"A garden contains a tree T and a flowerbed F. A sprinkler must be:

• More than 3 m from the tree
• Less than 5 m from point P
• Closer to fence A than fence B

Shade the region where the sprinkler can be placed."

Solution approach:

1. Identify each condition and its corresponding locus
2. Construct each locus accurately with visible arcs
3. Use inequality symbols to determine correct side of each locus
4. Shade only the region satisfying ALL conditions simultaneously
5. Label the shaded region clearly (often marked as 'R' in exam questions)

Inequality interpretation:

• "More than 3 m from T" → outside the circle radius 3 m
• "Less than 5 m from P" → inside the circle radius 5 m
• "Closer to A than B" → on A's side of the perpendicular bisector of AB

Worked examples

Example 1: Perpendicular bisector (3 marks)

Question: Using ruler and compasses only, construct the perpendicular bisector of line segment AB. You must show all construction arcs.

[Line segment AB shown, length 8 cm]

Solution:

Step 1: Place compass point on A, set radius greater than 4 cm (more than half of AB). Draw arcs above and below AB. [1 mark for arcs from A]

Step 2: Without changing compass radius, place compass point on B. Draw arcs above and below AB intersecting the first pair of arcs. [1 mark for arcs from B]

Step 3: Use ruler to draw straight line through both intersection points. This line is the perpendicular bisector. [1 mark for ruled line through intersections]

Mark scheme notes: All construction arcs must be visible. Faint arcs that cannot be seen score 0 marks for method. The perpendicular bisector must extend beyond the line segment AB.

Example 2: Combined loci problem (5 marks)

Question: The scale diagram shows a rectangular garden ABCD. A garden ornament is to be placed so that it is:

• Closer to side AB than to side AD
• More than 2 m from corner A

Using a scale of 1 cm to 1 m, shade the region where the ornament can be placed. Label your region R.

[Rectangle ABCD shown: AB = 6 cm, AD = 4 cm]

Solution:

Step 1: "Closer to AB than to AD" means construct the angle bisector of angle DAB. Place compass on A, draw arc crossing both AB and AD. From those intersection points, draw arcs inside the rectangle intersecting each other. Draw line from A through intersection. [1 mark]

Step 2: The ornament must be on the AB side of this bisector (closer to AB). [1 mark for correct side identified]

Step 3: "More than 2 m from A" means construct circle centre A, radius 2 cm (scale: 2 m = 2 cm). [1 mark]

Step 4: The ornament must be outside this circle (more than 2 m away). [1 mark for correct region]

Step 5: Shade the region that is both on the AB side of the angle bisector AND outside the circle, within the rectangle boundaries. Label as R. [1 mark]

Common errors: Students often shade "less than 2 m" (inside the circle) instead of "more than 2 m" (outside). Read inequality words carefully.

Example 3: 60° angle construction (2 marks)

Question: Using ruler and compasses only, construct an angle of 60° at point P on the line below. You must show all construction arcs.

[Horizontal line with point P marked on it]

Solution:

Step 1: Place compass point on P. Draw arc crossing the line (creating point Q). Keep compass radius the same. [1 mark for first arc]

Step 2: Place compass point on Q with same radius. Draw arc intersecting the first arc at point R. Draw straight line from P through R. Angle QPR = 60°. [1 mark for second arc and ruled line]

Mark scheme notes: The compass radius must remain unchanged between both arcs. Changing radius loses the method mark.

Common mistakes and how to avoid them

• Erasing construction arcs — Students erase their working to make diagrams "neater", but construction arcs are essential evidence of method. Even with the correct final answer, erased arcs lose method marks. Correction: Leave all arcs visible, ensuring they are clear enough to be seen by the examiner.

• Insufficient arc length — Drawing tiny arcs that barely intersect makes constructions inaccurate and demonstrates poor technique. Correction: Draw generous arcs that clearly extend beyond intersection points, typically at least 2-3 cm in length.

• Changing compass radius mid-construction — When constructing bisectors, students accidentally alter their compass width between steps. Correction: Set your compass radius carefully and keep your hand steady. Check the radius hasn't changed by testing against the original distance.

• Confusing "closer to" with "further from" — In loci problems, "closer to A than B" means on A's side of the perpendicular bisector, but students shade the wrong side. Correction: Mark a test point clearly on each side and measure distances to verify which side is closer before shading.

• Shading outside the boundaries — Combined loci questions usually have a defined region (garden, field, room) but students shade beyond these limits. Correction: The final shaded region must satisfy the loci conditions AND remain within the stated boundaries. Add a key to clarify your shading.

• Using a protractor instead of constructions — Some students measure angles with protractors despite instructions stating "using ruler and compasses only". Correction: When a question specifies equipment, you must use only those tools. Protractor measurements in a constructions question typically score zero marks.

Exam technique for "Constructions: bisectors, angles and loci"

• Command words matter: "Construct" means use ruler and compasses only with visible arcs. "Draw" allows any method including protractors. "Shade the region" requires clear, ruled boundaries and often benefits from labeling (usually 'R'). Questions worth 3+ marks always require construction arcs to be shown.

• Accuracy tolerances: Edexcel mark schemes typically allow ±2 mm for position and ±2° for angles when marking final answers. However, correct construction method can earn full marks even if final accuracy is slightly outside tolerance, so always show your working.

• Mark allocation patterns: Perpendicular bisectors usually earn 2-3 marks (1 mark per pair of arcs, 1 for ruled line). Angle bisectors typically 2 marks. Combined loci problems range from 4-6 marks (1-2 marks per locus construction, 1-2 marks for correctly shaded region). Budget approximately 3-4 minutes per mark on these questions.

• Ruler discipline: Always use a ruler for straight lines. Freehand "straight" lines in constructions lose marks. Keep your pencil sharp throughout to maintain accuracy, particularly for marking intersection points of arcs.

Quick revision summary

Constructions require ruler and compasses only, with all arcs visible. Master four key constructions: perpendicular bisector (equidistant from two points), angle bisector (equidistant from two lines), 60° angle (equilateral triangle property), and perpendiculars to lines. Four standard loci: circle (equidistant from fixed point), perpendicular bisector (equidistant from two points), parallel lines (equidistant from a line), angle bisector (equidistant from two lines). Combined loci problems require identifying multiple conditions, constructing each accurately, interpreting inequalities correctly, then shading the region satisfying all conditions within stated boundaries. Construction arcs are method marks—never erase them.