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Pythagoras Theorem
Reminder of square numbers:
12 =
22 =
32 =
42 =
1x1=
2x2=
3x3=
4x4=
1
4
9
16
Index number
32
Base number
The index number tells us how
many times the base number
is 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 they
are called square numbers.
Pythagoras Theorem
means think what is multiplied by itself to
make this number?
Use your calculator to answer
Answer these questions:
these questions:
1 1
5.8  2.408
4 2
25.4  5.040
9 3
169  13
16  4
400  20
49  7
1000  31.623
Square root
81  9
8100  90
121  11
225  15
100  10
361  19
Pythagoras Theorem
In a right-angled triangle,
the square on the
hypotenuse is equal to the
sum of the squares on the
other two sides.
b
2
c
c
b
a
2
Pythagoras of Samos
Hypotenuse
(6 C BC)
a +b
2
a
2
2
=c
2
Pythagoras Theorem
To show how this works:
b
Cut the squares
away from the right
angle triangle and cut
up the segments
q
of square ‘a’
a
Draw line segment x
xy, parallel with the
hypotenuse of the
triangle
Draw line segment
pq, at right angles to
Line segment xy.
y
p
Now rearrange them
to look like this.
You can see that they
make a square with
length of side ‘c’.
This demonstrates that
the areas of squares
a and b
add up to be the
area of square c
2
2
2
a +b
=c
Pythagoras Theorem
x 2  32  4 2
1
3 cm
x
x  32  42
x  9 + 16
x  25
x  5 cm
4 cm
x 2  52  122
2
x
5 cm
x  5  12
2
x 
12 cm
169
x  13 cm
2
Pythagoras Theorem
3
5 cm
x 2  52  6 2
x
x  52  62
x  7.8 cm (1 dp)
6 cm
x 2  4.62  9.82
4
x
4.6
cm
9.8 cm
x  4.6  9.8
x  10.8 cm (1 dp)
2
2
x 2  252  72
Pythagoras Theorem
x  252  72
x  11  9
2
Now do these:
5
11m
2
x  11  9
2
2
xx m24 m
8
x  6.3 m (1 dp)
xm
9m
6
11 cm
2
x  23.8  11
2
23.8 cm
3.4 cm
7.1 cm
x cm
2
2
x  23.82  112
x  21.1 cm (1 dp)
xm
7
25 m
x 2  7.12  3.4 2
x  7.12  3.42
x  7.9 cm (1 dp)
7m
Pythagoras 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?
H
15 miles
B
LH 2  152  6.42
6.4 miles
LH  152  6.42
LH  16.3 miles
Total distance travelled = 21.4 + 16.4 = 37.7 miles
L
Pythagoras Theorem
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?
L2  122  9.52
L  122  9.52
12 ft
9.5 ft
L  7.3ft
L