Pythagoras theorem 1.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Pythagoras' theorem 1

Angles and trigonometry

Lesson 4 of 7

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Before We Start

Angles and trigonometry

Lesson 4 of 7

Answer each one, then check.

1

Work out 13².

2

Work out √144.

3

Round 7.2111 to 1 decimal place.

4

Solve x² = 81 (positive answer).

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Before We Start - Answers

Angles and trigonometry

Lesson 4 of 7

1

Work out 13².

169

2

Work out √144.

12

3

Round 7.2111 to 1 decimal place.

7.2

4

Solve x² = 81 (positive answer).

x = 9

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Learning Objectives

Angles and trigonometry

Lesson 4 of 7

1

Identify the hypotenuse of a right-angled triangle.

2

Use Pythagoras' theorem to find the hypotenuse.

3

Use Pythagoras' theorem to find a shorter side.

4

Decide whether a triangle is right-angled.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

The Squares on the Sides

Angles and trigonometry

Lesson 4 of 7

3² + 4² = 5²: the two smaller squares fill the biggest one.

The hypotenuse is the longest side, opposite the right angle. Pythagoras' theorem says the area of the square on the hypotenuse equals the areas of the other two squares added together: a² + b² = c².

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

The Theorem

Angles and trigonometry

Lesson 4 of 7

a² + b² = c², where c is the hypotenuse. It works ONLY in right-angled triangles.

Find the hypotenuse

Square the two shorter sides, ADD, then square root.

Find a shorter side

Square the hypotenuse and the known side, SUBTRACT, then square root.

Sense check

The hypotenuse is always the longest side. If your "shorter side" is longer than the hypotenuse, you added when you should have subtracted.

Exact answers

On the non-calculator paper, leave an answer like √89 as a surd unless told to round.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Finding the Hypotenuse

Angles and trigonometry

Lesson 4 of 7

A right-angled triangle has shorter sides 8 cm and 5 cm. Work out the length of the hypotenuse. Give your answer to 1 decimal place.

1

Write the theorem

c² = 8² + 5²

2

Square and add

c² = 64 + 25 = 89

3

Square root

c = √89 = 9.433…

ANSWER

9.4 cm

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Finding a Shorter Side

Angles and trigonometry

Lesson 4 of 7

A right-angled triangle has hypotenuse 13 cm and one side 7 cm. Work out the length of the other side, to 1 decimal place.

1

Write the theorem with the hypotenuse on its own

a² + 7² = 13²

2

Subtract

a² = 169 − 49 = 120

3

Square root

a = √120 = 10.954…

4

Check: shorter than 13

Yes

ANSWER

11.0 cm

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Is It Right-Angled?

Angles and trigonometry

Lesson 4 of 7

The theorem also works backwards.

The test

Square the three sides. If the two smaller squares add up to the largest square, the triangle is right-angled.

Example

5, 12, 13: 25 + 144 = 169 = 13², so it is right-angled.

Counter-example

9, 12, 16: 81 + 144 = 225, but 16² = 256, so it is NOT right-angled.

Pythagorean triples

Whole-number sides that fit: 3, 4, 5; 5, 12, 13; 8, 15, 17; 7, 24, 25 - and any multiple, such as 6, 8, 10.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Angles and trigonometry

Lesson 4 of 7

CASE STUDY

Plimpton 322

Pythagoras lived around 500 BC, but the idea is much older. A Babylonian clay tablet called Plimpton 322, written about 1800 BC and now in a library in New York, lists fifteen rows of numbers written in base 60. Read correctly, each row gives the sides of a right-angled triangle with whole-number (or simple) sides - including triples as large as 12 709, 13 500 and 18 541. Nobody knows exactly how the Babylonians found them, but they clearly understood the rule more than a thousand years before the Greeks proved it.

About 1800 BC

When Plimpton 322 was written

15 rows

Of Pythagorean triples, in base 60

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Key Terms

Angles and trigonometry

Lesson 4 of 7

Hypotenuse

The longest side of a right-angled triangle, opposite the right angle.

Pythagoras' theorem

In a right-angled triangle, a² + b² = c², where c is the hypotenuse.

Pythagorean triple

Three whole numbers that fit the theorem, such as 3, 4, 5.

Surd

A square root that cannot be written exactly as a fraction, such as √2.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Angles and trigonometry

Lesson 4 of 7

YOUR TASK

Triple Hunt

10 minutes

(a) Show that 8, 15, 17 is a Pythagorean triple. (b) Multiply 3, 4, 5 by 2, 3 and 10 and check that each result is a triple. (c) Find the missing number in the triple 20, 21, ?. (d) Explain why no triple can have all three numbers odd.

1

Square each number.

2

Add the two smaller squares.

3

Compare with the largest.

WHAT A GOOD ANSWER SHOWS

(a) 64 + 225 = 289 = 17². (b) 6, 8, 10; 9, 12, 15; 30, 40, 50 all work. (c) 400 + 441 = 841, so 29. (d) Odd squared is odd, and odd + odd = even, so the third square would be even - and then the third number would be even.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Can I...?

Angles and trigonometry

Lesson 4 of 7

Identify the hypotenuse.

Find the hypotenuse.

Find a shorter side.

Give an answer to a sensible accuracy.

Leave an answer as a surd.

Decide whether a triangle is right-angled.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Summary & Exam Focus

a² + b² = c², with c the hypotenuse, opposite the right angle.

Longest side: square, add, square root.

Shorter side: square, subtract, square root.

Converse: if a² + b² = c², the triangle is right-angled.

EXAM FOCUS

A ladder 5 m long leans against a vertical wall. The foot of the ladder is 1.8 m from the wall on horizontal ground. How far up the wall does the ladder reach? Give your answer to 2 decimal places. (3 marks)

Draw and label a sketch, and mark the hypotenuse first. Then decide: finding the hypotenuse means add, finding a shorter side means subtract.

openrevise.com