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

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

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).

Length of a shorter side

Maths

Checking understanding of similarity

Checking understanding of congruence

Similarity in shapes

Congruence in shapes

Congruent, similar or neither

Rotational symmetry

Congruent triangles (SSS)

Congruent triangles (SAS)

Congruent triangles (ASA and AAS)

Congruent triangles (RHS)

Applying the criteria for congruence

Demonstrating Pythagoras' theorem

Further demonstrating of Pythagoras' theorem

Length of the hypotenuse

Determining which side

Pythagoras' theorem in context

Problem solving with similarity and Pythagoras' theorem

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

Find the length of the shorter side of this right-angled triangle, labelled $$b$$.

The length of the shorter side of this right-angled triangle, labelled $$c$$ is {{}} cm.

The length of the shorter side of this right-angled triangle, labelled $$d$$, is {{ }} cm, rounded to 2 decimal places.

The length of the side $$x$$ cm is {{ }} cm, rounded to 2 decimal places.

A right-angled triangle has a hypotenuse of 130 cm. The length of one of its shorter sides is 66 cm. Using Pythagoras' theorem, the perimeter of this triangle is {{}} cm.

Calculate the area of this right-angled triangle.

The triangle formed from these three squares is right-angled. Find the area of the square labelled $$h$$ units².

The triangle formed from these three squares is right-angled. The area of the square labelled $$p$$ units²  is {{}} units².

The area, $$w$$, of the largest square in this diagram, is {{ }} cm².

Find the length of the hypotenuse of this triangle.

The length of the hypotenuse for this triangle is {{}} units.

The length of the hypotenuse of this triangle is {{ }} cm.

Starter Quiz

The triangle formed from these three squares is right-angled. Find the area of the square labelled $$h$$ units².