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

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).

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

Pythagoras' theorem in context

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

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

The distance between the goalkeeper and the striker is 49.24 metres.

The distance between the goalkeeper and the striker is 92.20 metres (to 2 d.p.).

The distance between the goalkeeper and the striker is 98.49 metres (to 2 d.p.).

The ball the goalkeeper kicks will reach the striker.

The ball the goalkeeper kicks will not reach the striker.

On this football pitch, the goalkeeper (at point A) wants to kick the ball to a striker (at point E), however the goalkeeper's longest kick is 91 m. Which of these statements are correct?

A factory produces cylindrical saucepans whose maximum base length is 28 cm. What must the minimum integer height of the saucepan be (in cm) so it can fully submerge a spaghetti noodle.

An archer fires an arrow that travels 20 metres to hit a target that is 9 m above ground. The arrow then drops to the ground. The archer must walk {{ }} m (to 2 d.p.) to pick up the arrow.

The ground the archer walks on to pick up the arrow is flat.

The archer fires the arrow from the ground.

An archer fires an arrow at a target 20 m away. The arrow drops 9 m to the ground. Jun uses Pythagoras' theorem to find how far the archer walks to collect the arrow. What assumptions does Jun make?

The captain (at C) passes the ball to the goalkeeper (at A), who then passes to the midfielder (at D), who then passes to a striker (at F). The total distance the ball travelled is {{}}m (to 2 d.p.).

Two bamboo sticks are placed on the diagonal of this kite to support it during high winds. The length of the vertical bamboo stick is {{}} cm (to 2 d.p.).

Izzy writes down a correct Pythagoras' theorem equation for a triangle. Which of the three sides of this triangle is the hypotenuse?

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

Which of these are correct for this triangle?

What is the value of $$b$$, rounded to 2 d.p.? $$b=$${{}}.

The diagonals of this kite have been drawn on. The value of $$x$$ is {{}}.

The two diagonals of a kite intersect to create two acute angles of equal length

The two diagonals of a kite intersect at right angles

The diagonals of this kite have been drawn on. Which of these statements are correct.

Starter Quiz

Izzy writes down a correct Pythagoras' theorem equation for a triangle. Which of the three sides of this triangle is the hypotenuse?