📐 Geometrical Constructions: Building Precision in Mathematics

Welcome to the chapter on Geometrical Constructions! This topic is all about building geometric shapes and lines with absolute precision, using only two simple tools: a straight edge (ruler) and a pair of compasses.

Think of yourself as a mathematical architect! You're learning the fundamental rules used by builders, designers, and engineers throughout history. The skills you gain here require care and accuracy, and they teach you how different parts of geometry relate to each other.

Let's put away the protractor (mostly!) and focus on the power of the compass and ruler!

Section 1: Your Essential Toolkit

To master constructions, you must understand the role of the two permitted instruments:

1. The Straight Edge (Ruler)
  • Used only for drawing a straight line connecting two points.
  • Crucially: You should generally not use the measurement marks on the ruler for constructions, except when initially drawing a line of a specific length (e.g., a 7 cm line) or measuring the final result.
2. The Compasses
  • Used for drawing circles or arcs (parts of circles).
  • The compass setting determines the radius of the arc.
  • Important Tip: For a single construction step (like creating a perpendicular bisector), keep the compass opening fixed. Do not accidentally change the radius!

Quick Review: Prerequisite Concepts

  • Perpendicular: Lines that meet at a right angle (\(90^{\circ}\)).
  • Bisect: To cut something exactly in half.

Section 2: Fundamental Constructions

These are the most important techniques you must memorise step-by-step.

1. Constructing a Perpendicular Bisector of a Line Segment (A to B)

The perpendicular bisector is a line that cuts a line segment (like AB) into two equal halves AND crosses it at \(90^{\circ}\).

Analogy: Imagine two friends, A and B, who want to meet exactly halfway between them. The perpendicular bisector shows all possible meeting spots that are equidistant from both A and B.

Step-by-Step Guide:
  1. Place the compass point on A. Open the compass so the radius is more than half the length of AB.
  2. Draw a large arc above and below the line AB.
  3. Keep the compass radius exactly the same. Place the point on B.
  4. Draw another arc above and below the line AB, ensuring these arcs cross the arcs drawn from A.
  5. Use your straight edge to draw a line connecting the two points where the arcs intersect (P and Q).


This new line PQ is the perpendicular bisector of AB.

Key Takeaway: This construction creates a line of equal distance from the endpoints (A and B).

2. Constructing an Angle Bisector

The angle bisector is a line that divides a given angle into two smaller, equal angles.

Analogy: If the angle is a slice of cake, the angle bisector ensures you cut it perfectly in half to share equally!

Step-by-Step Guide:
  1. Place the compass point on the vertex (the corner) of the angle (let's call it O).
  2. Draw an arc that crosses both lines (arms) forming the angle. Let the intersection points be C and D.
  3. Place the compass point on C. Draw an arc inside the angle. (You can adjust the radius now, but keep it fixed for the next step.)
  4. Keep the compass radius the same. Place the point on D. Draw an arc that crosses the arc drawn in step 3. Let the intersection point be E.
  5. Use your straight edge to draw a line from the vertex O through the intersection point E.


The line OE is the angle bisector.

Key Takeaway: This construction creates a line of equal distance from the two arms (sides) of the angle.

Section 3: Constructing Specific Angles

Using only a ruler and compasses, we can accurately construct certain key angles.

1. Constructing a \(60^{\circ}\) Angle

This is based on the fact that all angles in an equilateral triangle are \(60^{\circ}\).

  1. Start with a straight line, mark a point P where the angle will be.
  2. Place the compass point on P and draw an arc that crosses the line (point A).
  3. Keep the compass radius the same. Place the compass point on A and draw a second arc that crosses the first large arc (point Q).
  4. Draw a straight line from P through Q.

The angle formed at P is exactly \(60^{\circ}\).

2. Constructing a \(90^{\circ}\) Angle (Perpendicular at a Point)

This is essentially constructing a perpendicular bisector, but at a specific point on the line, rather than between two end points.

  1. Start with a line segment and mark the point P where you want the \(90^{\circ}\) angle.
  2. Place the compass point on P. Draw arcs of equal radius on both sides of P, crossing the line (points A and B).
  3. Now, treat A and B as the ends of a line segment. Place the compass point on A, open it to a radius greater than AP, and draw an arc above P.
  4. Keep the compass radius the same. Place the point on B and draw an arc crossing the arc from step 3 (point Q).
  5. Draw a line from P through Q.

The angle is \(90^{\circ}\).

🛑 Common Mistake Alert!
Do not confuse the Perpendicular Bisector (which cuts a line in half) with constructing a Perpendicular at a point (which creates \(90^{\circ}\) at a specific spot). The techniques are similar but serve different goals!
3. Constructing \(45^{\circ}\) and \(30^{\circ}\) Angles

We achieve these by simply bisecting the angles we already know how to make:

  • To construct a \(45^{\circ}\) angle: Construct a \(90^{\circ}\) angle, then use the angle bisector technique on it.
  • To construct a \(30^{\circ}\) angle: Construct a \(60^{\circ}\) angle, then use the angle bisector technique on it.

Did you know? Using only a compass and straight edge, it is impossible to perfectly trisect (cut into three equal parts) a general angle! This famous problem baffled mathematicians for centuries.

Section 4: Loci (Singular: Locus)

Construction skills are essential when dealing with Loci. A locus is simply the path traced by a moving point, or the set of all points, that satisfy a specific condition.

When a question asks you to find a locus, it is often followed by a request to shade a region (a concept called "Region Constraints").

Locus Type 1: Equidistant from Two Points (A and B)

The locus of points equidistant from two fixed points A and B is the perpendicular bisector of the line segment AB.


If you are asked to shade a region that is 'Nearer to A than to B', you must construct the perpendicular bisector and shade the side that contains point A.

Locus Type 2: Equidistant from Two Intersecting Lines

The locus of points equidistant from two intersecting lines is the angle bisector of the angle between them.


If you are asked to shade a region 'Nearer to line X than to line Y', you construct the angle bisector and shade the side containing line X.

Locus Type 3: Fixed Distance from a Point (P)

The locus of points at a fixed distance, \(d\), from a fixed point P is a circle with centre P and radius \(d\).


If you are asked to shade the region 'Less than 5 cm from P', you open your compass to 5 cm, draw the circle centred at P, and shade the area *inside* the circle.

Locus Type 4: Fixed Distance from a Straight Line (L)

The locus of points at a fixed distance, \(d\), from a straight line L consists of two lines parallel to L, separated by the distance \(d\), connected at the ends by semicircles of radius \(d\).

Example: A dog tied to a fence (the line L) by a rope of length \(d\). The path it can walk is bounded by two parallel lines and two semicircles at the ends of the fence segment.

Putting it all Together: Shading Regions

Often, examination questions combine several locus conditions (e.g., nearer to A than B, and less than 3 cm from C).

When shading regions, follow these steps precisely:

  1. Identify the boundary: Determine which construction defines the edge of the required region (perpendicular bisector, angle bisector, or circle).
  2. Construct the boundary: Draw this boundary line/curve accurately using ruler and compasses.
  3. Determine the 'wanted' side: Decide which side of the boundary satisfies the condition (e.g., nearer to A, or outside the circle).
  4. Shade the unwanted region: It is often easier and safer to shade the part of the diagram that is NOT required. The clear, unshaded region is your final answer.
🎯 Memory Aid: The Bisector Rules
  • Perpendicular Bisector: Equidistant from Points. (Used for A and B).
  • Angle Bisector: Equidistant from Lines. (Used for line X and line Y).

Key Takeaway Summary

Geometrical constructions in IGCSE Mathematics demand high accuracy. You must be able to perform the perpendicular bisector and angle bisector constructions perfectly, as they are the foundations for constructing specific angles and solving complex locus problems. Practice makes perfect—ensure your compasses are tight and your pencil is sharp!