Constructions, Loci and Congruence

This lesson covers geometric constructions using a compass and straightedge, loci, congruence and similarity for AQA GCSE Mathematics. Construction questions require precise drawing with the correct equipment, while congruence and similarity questions test your understanding of when shapes are identical or proportional. These topics appear across both Foundation and Higher tiers.

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Equipment for Constructions

For construction questions, you need:

• A sharp pencil (for accuracy)
• A ruler (for straight lines and measuring)
• A pair of compasses (for arcs and circles)
• A protractor (for measuring angles, though true constructions do not use one)

Exam Tip: You MUST show all construction arcs clearly. Do not rub them out. The examiner needs to see them to award the method marks. If you only show the final answer with no arcs, you will score 0 even if the answer is correct.

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Constructing Triangles

Given Three Sides (SSS)

To construct a triangle with sides 6 cm, 5 cm, and 4 cm:

1. Draw the longest side (6 cm) using a ruler. Label the ends A and B.
2. Open compasses to 5 cm. Place the point on A and draw an arc above the line.
3. Open compasses to 4 cm. Place the point on B and draw an arc to cross the first arc.
4. Label the intersection point C. Join AC and BC.

Given Two Sides and the Included Angle (SAS)

To construct a triangle with sides 7 cm, 5 cm, and an included angle of 40 degrees:

1. Draw one side (7 cm) using a ruler. Label ends A and B.
2. At A, use a protractor to measure 40 degrees. Make a faint mark.
3. Draw a line from A through the mark.
4. On this line, measure 5 cm from A. Label this point C.
5. Join BC.

Given Two Angles and a Side (ASA)

To construct a triangle with a base of 8 cm and base angles of 50 degrees and 60 degrees:

1. Draw the base (8 cm). Label ends A and B.
2. At A, use a protractor to draw a line at 50 degrees to the base.
3. At B, use a protractor to draw a line at 60 degrees to the base.
4. Label the intersection point C.

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Key Constructions with Compass and Straightedge

Perpendicular Bisector of a Line Segment

The perpendicular bisector of a line segment is a line that is at right angles to the segment and passes through its midpoint.

1. Open compasses to more than half the length of the line segment AB.
2. Place the compass point on A and draw arcs above and below the line.
3. Without changing the compass width, place the point on B and draw arcs to cross the first arcs.
4. Draw a straight line through the two intersection points.

This line is the perpendicular bisector and every point on it is equidistant from A and B.

Angle Bisector

The angle bisector divides an angle into two equal parts.

1. Place the compass point on the vertex of the angle and draw an arc that crosses both arms.
2. Place the compass point on each intersection and draw two arcs that cross each other.
3. Draw a straight line from the vertex through the intersection of the arcs.

Every point on the angle bisector is equidistant from the two arms of the angle.

Perpendicular from a Point to a Line

1. From the point, draw an arc that crosses the line in two places.
2. From each crossing point, draw arcs below the line (same radius) that intersect.
3. Draw a line from the original point through this intersection.

Perpendicular from a Point on a Line

1. From the point on the line, draw arcs either side (equal distance).
2. From each arc, draw further arcs above (or below) the line that intersect.
3. Draw a line from the point through the intersection.

graph TD
    A[Constructions] --> B[Perpendicular bisector]
    A --> C[Angle bisector]
    A --> D[Perpendicular from point to line]
    A --> E[Perpendicular from point on line]
    B --> B1[Equidistant from two points]
    C --> C1[Equidistant from two lines]

Exam Tip: In the exam, your constructions should be accurate to within 2 mm and 2 degrees. Practise using compasses until your arcs are neat and your intersections are precise. Wobbly arcs suggest poor technique.

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Loci

A locus (plural: loci) is the set of all points that satisfy a given condition.

The Four Standard Loci

Condition	Locus
Fixed distance from a point	A circle centred on the point
Fixed distance from a line	Two parallel lines either side of the line, with semicircular ends (a "stadium" or "racetrack" shape)
Equidistant from two points	The perpendicular bisector of the line joining the two points
Equidistant from two intersecting lines	The angle bisector(s) of the two lines

Worked Example 1

A treasure is buried equidistant from two trees, A and B, and no more than 5 m from a well at point W. Shade the region where the treasure could be.

Solution:

1. Construct the perpendicular bisector of AB (all points equidistant from A and B).
2. Draw a circle of radius 5 m centred on W (all points within 5 m of W).
3. The treasure lies on the perpendicular bisector AND inside (or on) the circle.
4. Shade the segment of the perpendicular bisector that falls within the circle.

Worked Example 2

A dog is tied to a post with a 4 m rope. Describe and sketch the locus of points the dog can reach.

Solution: The locus is a circle with radius 4 m centred on the post. The dog can reach any point within or on the circle.

Exam Tip: Loci questions often ask you to shade a region satisfying multiple conditions. Construct each locus separately first, then identify and shade the overlapping region. Use hatching or shading clearly and label your answer.

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Congruent Shapes

Two shapes are congruent if they are exactly the same shape and size. One can be mapped onto the other by a combination of reflections, rotations, and translations.

Conditions for Congruent Triangles

To prove two triangles are congruent, you need to show one of the following:

Criterion	What You Need
SSS (Side-Side-Side)	All three sides of one triangle equal all three sides of the other
SAS (Side-Angle-Side)	Two sides and the included angle are equal
ASA (Angle-Side-Angle)	Two angles and a corresponding side are equal
RHS (Right angle-Hypotenuse-Side)	Both triangles have a right angle, equal hypotenuses, and one equal side

Important Note

SSA (Side-Side-Angle) where the angle is NOT included is NOT a valid congruence criterion (it can produce two different triangles — the ambiguous case).

Worked Example 3

Triangle ABC has AB = 5 cm, BC = 7 cm, and angle ABC = 60 degrees. Triangle DEF has DE = 5 cm, EF = 7 cm, and angle DEF = 60 degrees.

Are they congruent? State the criterion.

Solution: Yes, the triangles are congruent by SAS (two sides and the included angle are equal: AB = DE, BC = EF, angle ABC = angle DEF).

Worked Example 4

Prove that triangles ABD and CBD are congruent in a kite ABCD where AB = CB, AD = CD, and BD is a shared diagonal.

Solution: In triangles ABD and CBD:

• AB = CB (given — sides of a kite)
• AD = CD (given — sides of a kite)
• BD = BD (common side)

Therefore triangles ABD and CBD are congruent by SSS.

Exam Tip: In congruence proofs, list the three pairs of equal elements clearly and state the congruence criterion at the end. Marks are awarded for each correct pair and for naming the criterion correctly.

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Similar Shapes

Two shapes are similar if they have the same shape but different sizes. Corresponding angles are equal, and corresponding sides are in the same ratio.

Key Properties

• Corresponding angles are equal
• Corresponding sides are in the same ratio (the scale factor)
• The shapes are an enlargement of each other

Scale Factor

Scale factor = (length on larger shape) / (corresponding length on smaller shape)

Worked Example 5

Triangles PQR and XYZ are similar. PQ = 6 cm, QR = 8 cm, and XY = 9 cm. Find YZ.