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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