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.

Problem solving with similarity and Pythagoras' theorem

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

Length of a shorter side

Determining which side

Pythagoras' theorem in context

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

These two triangles are similar to each other and are both in the same orientation. The length of the hypotenuse marked $$h$$ is {{ }} cm.

You need to use Pythagoras' theorem to find the size of angle $$g$$.

You do not need to use Pythagoras' theorem to find the size of angle $$g$$.

Which of these statements are correct for this triangle?

The perimeter of this triangle is 16 + 25.4 + 19.7 = 61.1 cm (to 1 d.p.).

The perimeter of this triangle is 16 + 25.4 + 25.4 = 92.4 cm (to 1 d.p.).

The area of this triangle is 158 cm² (to the nearest cm²).

The area of this triangle is 203.2 cm² (to 1 d.p.).

$$f°$$ = 180° − 90° − 51° = 39°.

Each square is 1 unit in length. The shortest distance from point A to point B is {{ }} units (rounded to 2 d.p.).

Point C is at the coordinate (12, 3). The perimeter of a triangle whose vertices are at points A, B, and C is {{}} units (to the nearest unit).

Alex rearranges a correct Pythagoras' theorem equation for a triangle. Which of these statements are correct?

The perimeter of this right-angled triangle is {{}} cm.

Adhesive tape is placed along the two diagonals of this rectangular picture frame. How much adhesive tape is needed, rounded to the nearest cm? {{ }} cm of tape.

Andeep and Sofia need to split a 3.6 metre roll of adhesive tape in the ratio 7 : 5. The length of the adhesive tape that Andeep receives is {{ }} centimetres.

These two triangles are similar to each other. The side marked $$u$$ cm is {{ }} cm long.

An isosceles triangle has an angle of 22°. Which of these are possible sizes of one of the other angles?

Problem solving with similarity and Pythagoras' theorem

Lesson outcome