The object is the starting figure before a transformation has been applied.

The image is the resulting figure after a transformation has been applied.

Two shapes are similar if the only difference between them is their size. Their side lengths are in the same proportions.

Scale factor is the multiplier between similar shapes that describes how large one shape is compared to the other.

A property of a shape is invariant if that property has not changed after the shape is transformed.

Checking understanding of similarity

Maths

In this unit, pupils continued to build on their understanding of congruence and similarity with right-angled triangles. In the future unit, similarity is used to extend a right-angled triangle in the unit circle to any other right-angled triangle.

In the prior unit, pupils explored the effects of different transformations and whether the transformed shape was congruent or similar to the original shape. This is built on in this unit where pupils formalise the criteria for determining congruence in triangles and study rotational symmetry.

This unit explores the concepts of congruence and similarity, and ways to prove or disprove similarity between shapes. We then explore further properties of triangles with Pythagoras' Theorem.

Geometrical properties: similarity and Pythagoras' theorem

If one shape can fit exactly on top of another using rotation, reflection or translation, then the shapes are congruent.

Checking understanding of congruence

Similarity in shapes

Co-interior angles are on the same side of the transversal line and in between the two other lines.

Congruence in shapes

Congruent, similar or neither

Rotational symmetry describes when a shape appears the same after some rotation. The order of rotational symmetry is the number of times the image appears the same as the object in one full turn.

Rotational symmetry

Congruent triangles (SSS)

Congruent triangles (SAS)

Congruent triangles (ASA and AAS)

The hypotenuse is the side of a right-angled triangle which is opposite the right angle.

Congruent triangles (RHS)

Applying the criteria for congruence

Pythagoras’ theorem shows that the sum of the squares of the two shorter sides of a right-angled triangle is equal to the square of its longest side (the hypotenuse).

Demonstrating Pythagoras' theorem

A hypotenuse is the side of the right-angle triangle which is opposite the right-angle.

A right-angled triangle has exactly one 90° interior angle.

Length of the hypotenuse

Length of a shorter side

Determining which side

The perpendicular height is the perpendicular distance from the base to the opposite vertex.

Pythagoras' theorem in context

Pythagoras’ theorem states that the sum of the squares of the two shorter sides of a right-angled triangle is equal to the square of the hypotenuse.

Geometrical properties: similarity and Pythagoras' theorem

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These lessons will be removed by end of Summer Term 2025.