Pythagoras TheoremPythagoras Theorem

Reminder of square numbers:

12 = 1 x 1 = 122 = 2 x 2 = 432 = 3 x 3 = 942 = 4 x 4 = 16

32

Base number

Index number

The index number tells us howmany times the base numberis multiplied by itself.e.g. 34 means 3 x 3 x 3 x 3 = 81

1,4,9,16, …. are the answers to a number being squared so theyare called square numbers.

Pythagoras TheoremPythagoras Theorem

Square root means think what is multiplied by itself to make this number?

Answer these questions:14916

49

81

211001

12

347

9

11

10

Use your calculator to answer these questions:

8.54.25169400

1000

8100

225361

2.4085.040

132031.623

901519

Pythagoras TheoremPythagoras Theorem

c2

b2

a2a2+b2 = c2

b

a

c

In a right-angled triangle, the square on

the hypotenuse is equal to the sum of the

squares on the other two sides.

Hypotenuse

Pythagoras of Samos

(6 C BC)

b

a

Cut the squaresaway from the right angle triangle and cut up the segments of square ‘a’

Draw line segmentxy, parallel with the hypotenuse of the triangle

x

y

q

p

Draw line segmentpq, at right angles to Line segment xy.

Pythagoras TheoremPythagoras TheoremTo show how this works:

Now rearrange themto look like this.

You can see that theymake a square withlength of side ‘c’.This demonstrates thatthe areas of squares

a and badd up to be the area of square c

a2+b2 = c2

2 2 23 4x 2 23 4x

5 cmx

2 2 25 12x 2 25 12x

13 cmx

3 cm

4 cm

x1

5 cm

12 cm

x2

Pythagoras TheoremPythagoras Theorem

9 + 16x 25x

169x

Pythagoras TheoremPythagoras Theorem

2 2 25 6x 2 25 6x

7.8 cm (1 dp)x

2 2 24.6 9.8x 2 24.6 9.8x

10.8 cm (1 dp)x

5 cm

6 cm

x3

4.6 cm

9.8 cm

x4

Pythagoras TheoremPythagoras Theorem

Now do these:

x m9 m

11m

5

11 cm

23.8 cm6

x m

7.1 cm

x cm

3.4 cm7

8

25 m 7 m

x m

2 2 211 9x 2 211 9x

6.3 m (1 dp)x

2 2 223.8 11x 2 223.8 11x

21.1 cm (1 dp)x

2 2 27.1 3.4x 2 27.1 3.4x

7.9 cm (1 dp)x

2 2 225 7x 2 225 7x

24 mx

Pythagoras TheoremPythagoras Theorem

A boat sails due East from a Harbour (H), to a marker buoy (B),15 miles away. At B the boat turns due South and sails for 6.4 miles to a Lighthouse (L). It then returns to harbour. What is the total distance travelled by the boat?

2 2 215 6.4LH 2 215 6.4LH

16.3 milesLH

Total distance travelled = 21.4 + 16.4 = 37.7 miles

H B

L

15 miles

6.4 miles

Pythagoras TheoremPythagoras Theorem

12 ft9.5 ft

L

A 12 ft ladder rests against the side of a house. The top of the ladder is 9.5 ft from the floor. How far is the base of the ladder from the house?

2 2 212 9.5L 2 212 9.5L

7.3L ft