PYTHAGORAS

Aim: To be able to know Pythagoras’ Theorem

All: Will be able to recall theorem.Most: Will be able to use to find the length of hypotenuse.Some: Will be able to use it to find the length of the shorter side.

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Pythagoras’ TheoremTrigonometry

Polygon anglesRatio & Proportion

Pythagoras’ Theorem

Only works in right angled triangles Nothing to do with angles Two main types

of questions Type 1 Type 2

Hypotenuse

Hypotenuse

The hypotenuse is the longest side in a right angled triangle.

It is always the side opposite the right angle.

hypotenuse

Spotting the Hypotenuse

hypotenuse

Pythagoras’ Theorem ‘In a right-angled triangle, the area of the square on

the side opposite to the right angle is equal to the sum of the squares on the sides forming the right angle.’

Pythagoras’ Theorem

Pythagoras’ Theorem states that: ‘The sum of the squares of the lengths of the sides

containing the right angle is equal to the square of the hypotenuse.’

In other words:

a2 + b2 = c2

a

b

c

A

BC

a2 + b2 = c2 ‘c’ must be the hypotenuse

You must square the numbers first, and then add

Remember that ‘square’ means to multiply the number by itself (32 = 3x3 = 9)

Type 1 (Finding The Hypotenuse)

a² + b ² Square, square Add Square root 10

18

?

Find the missing side.

102 = 100, 182 = 324

100 + 324 = 424

424 = 20.6 (3 sf)

Type 1Find the missing sides.

Give your answers to 3 sf.8c

m

10cm

11m

7m

24km

5km

102 = 100, 82 = 64100 + 64 = 164164 = 12.8cm

112 = 121, 72 = 49121 + 49 = 170170 = 13.0m

242 = 576, 52 = 25576 + 25 = 601601 = 24.5km

Type 2 (Finding A Leg)

Square, square Take away Square root

Find the missing side.

3.12 = 9.61, 22 = 4

9.61 – 4 = 5.61

5.61 = 2.37 miles (3 sf)

2 miles

3.1 miles

?

Type 2Find the missing sides.

Give your answers to 3 sf.9c

m

15cm

13m6m

24km

9km

152 = 225, 92 = 81225 – 81 = 144

144 = 12.0cm

132 = 169, 62 = 36169 – 36 = 133

133 = 11.5m

242 = 576, 92 = 81576 – 81 = 495

495 = 22.2km

Navigation (1)

Navigation problems are often solved using Pythagoras’ Theorem.

Make sure you know which way North, South, East and West point!

N

S

EW

Navigation (2)A plane leave an airport and travels 32km west then it turns and travels 41km north. It develops a problem and has to return to the airport. How far is it?

Step 1. Draw a diagram

32km Airport

?

Step 2. Use Pythagoras

This is Type 1. We have to find the hypotenuse.

41km 322 = 1024, 412 =

16811024 + 1681 = 27052705 =

52.0km

Word Problems (1)

Sometimes it is not obvious that you need to use Pythagoras’ Theorem.

If you draw a diagram you might spot a right angled triangle you can use…

Word Problems (2)Farmer Giles wants to cross to the diagonally opposite corner of his rectangular marrow field. The field measures 400m by 500m. How much distance does he save by going across the field rather than going around it?

Step 1. Draw a diagram

500m

400m?

Word Problems (3)Step 2. Use Pythagoras

400

500

5002 = 250 000, 4002 = 160 000

250 000 + 160 000 = 410 000

410000 = 640.3m

640.3

Step 3. The final answer

Distance round outside = 500 + 400 = 900m.

So Farmer Giles saves 900 – 640.3 = 260m (3 sf)

Word Problems (4)A ladder rests against a wall. For safety reasons the base of the ladder must be at least 2m from the wall. The ladder is 6.2m long. How high up the wall can the ladder reach?

Step 1. Draw a diagram

?

2m

6.2m

Word Problems (5)Step 2. Use Pythagoras

2m

5.87

m This is a Type 2 problem.

6.22 = 38.44, 22 = 4

34.44 = 5.87m (3 sf)

38.44 – 4 = 34.44

6.2m

And finally ….

Pythagoras’ Theorem only works for a particular type of triangle, which type?

If you are finding the hypotenuse, do you add or subtract the shorter sides squared?

What is meant by “hypotenuse”?

I wish I had worked

harder at school!