Chapter 2
Pythagoras
Pythagoras
Pythagoras was a famous Greek Mathematician who discovered an amazing
connection between the three sides of any right angled triangle.
This relationship, which connects the 3 sides, means it is possible to
CALCULATE the length of one side of a right angled triangle as
long as you know the lengths of the other two.
Look at this right angled triangle with sides 3 cm, 4 cm and 5 cm.
5 cm
If you add the two smaller sides (3 cm and 4 cm) together,
do you get the longer side (5 cm) ? – NO.
Can you see that
3 cm
4 cm
•
32 = 9,
•
4 2 = 16,
• 5 2 = 25 ?
Can you also see that:-
32 + 4 2 =
9 + 16
= 25 = 5 2 ?
Pythagoras found that this connection between
the three sides of a right angled triangle
was true for every right angled triangle.
Introductory Exercise 2.0 (confirmation - possibly orally)
1.
The three sides of this right angled triangle are 6 cm, 8 cm and 10 cm.
(a)
Write down the values of 6 2 , 8 2 and 10 2 .
(b)
Find the value of 6 2 + 8 2 .
(c)
Check that 6 2 + 8 2 = 10 2 .
10 cm
8 cm
6 cm
2.
The three sides of this right angled triangle are 9 cm, 12 cm and 15 cm.
(a)
Write down the values of 9 2 , 12 2 and 15 2 .
(b)
Find the value of 9 2 + 12 2 .
(c)
Check that 9 2 + 12 2 = 15 2 .
15 cm
9 cm
12 cm
3.
The three sides of this right angled triangle are 5 cm, 12 cm and 13 cm.
(a)
Write down the values of 5 2 , 12 2 and 132 .
2
5 cm
2
(b)
Find the value of 5 + 12 .
(c)
Check that 5 2 + 12 2 = 132 .
Chapter 2
13 cm
12 cm
this is page 17
Pythagoras
Pythagoras Theorem
Pythagoras came up with a simple rule which shows the connection between
the three sides of any right angled triangle.
hypotenuse
c cm
The longest side of a right angled triangle is called the HYPOTENUSE.
If the three sides are a cm, b cm and c cm (the hypotenuse),
then Pythagoras’ Rule says :-
c 2 = a2 + b2
=>
b cm
a cm
We can use this rule to calculate the length of the hypotenuse of a right angled triangle if we know
the lengths of the two smaller sides.
Example 1 :- The two smaller sides of this right angled
triangle are 12 centimetres and 16 centimetres.
(c)
12 cm
(b)
To calculate the length of the hypotenuse,
use Pythagoras’ Rule.
use your “√” button
on the calculator
2
= a
2
+ b
16 cm
(a)
2
=>
c
=>
c 2 = 16 2 + 12 2
=>
c 2 = 256 + 144 = 400
=>
c =
400 = 20 cm.
This is how you set down the working.
Exercise 2·1
1.
Use Pythagoras’ Rule to calculate the length
of the hypotenuse in this triangle :-
3.
c 2 = a2 + b2
=>
c 2 = 15 2 + ...
=>
c 2 = 225 + ...
=>
c =
..... = ....
Use Pythagoras’ Rule (referred to as
PYTHAGORAS’ THEOREM) to calculate
the length of the hypotenuse in each of these
3 triangles :(a)
c
c
8 cm
15 cm
15 cm
20 cm
Copy and complete the working.
(b)
2.
Use Pythagoras’ Rule to calculate the length
of the hypotenuse in the
right angled
triangle
shown below.
c
15 cm
(c)
36 cm
Chapter 2
11·25 cm
6 cm
this is page 18
c
24 cm
7 cm
c
c
Pythagoras
In most cases, the 3 sides are not exact values.
Example 2 :-
c 2 = a2 + b2
=>
use your “√” button
on the calculator
2
2
= 11
c
7 cm
2
=>
c
+ 7
=>
c 2 = 121 + 49 = 170
=>
c =
170 = 13·0384048...
11 cm
= 13·04 cm
(to 2 decimal places),
(For the remainder of this exercise, give your answers correct to 2 decimal places).
4.
Use Pythagoras’ Theorem to calculate the
length of the hypotenuse in this triangle .
8.
Calculate the length of the hypotenuse
in this right angled triangle.
13·9 cm
c cm
6 cm
6·1 cm
x cm
10 cm
5.
Use Pythagoras’ Theorem to calculate the
length of the hypotenuse in the right angled
triangle shown .
9.
Sketch the following right angled triangles :Use Pythagoras’ Theorem to calculate the
length of the hypotenuse in each case.
14 cm
(a)
(b)
h cm
8 cm
5 cm
c cm
h cm
18 cm
9 cm
6.
Calculate the length of the hypotenuse
marked p cm.
11 cm
4·5 cm
(c)
(d)
15 cm
12·3 m
8·5 cm
p cm
h cm 5·2 m
h m
10 cm
7.
Calculate the length of the line marked q cm.
(e)
(f)
h mm
17 cm
h cm
22 cm
33 mm
25 mm
16 cm
q cm
Chapter 2
32 cm
this is page 19
Pythagoras
Problems involving Pythagoras’ Theorem
Pythagoras
c2 = a2 + b2
Whenever you come across a problem involving finding a
missing side in a right angled triangle, you should always
consider using Pythagoras’ Theorem to calculate its length.
Exercise 2·2
(The triangles in these questions are right-angled)
1.
4.
A strong wire is used to support
a pole while the cement, holding
it at its base, dries.
wire
Calculate the length of
the wire.
A cable is used to help ferry supplies onto a
yacht from the top of a nearby cliff.
7·5 m
37 m
4m
2.
54 m
A ramp is used to help push wheelchairs
into the back of an ambulance.
Calculate the length of the cable used.
5.
This wooden door wedge is 12·5 cm long.
and 3·1 cm high.
ramp
3·1 cm
0·9 m
5·2 m
Calculate the length of the ramp.
12·5 cm
Calculate the length of the sloping face.
3.
A plane left from Erin Isle airport.
The pilot flew 175 kilometres West.
He then flew 115 kilometres due North.
6.
This trapezium shape has a line of symmetry
shown dotted on the figure.
4·2 cm
North
115 km
3·9 cm
175 km
3·6 cm
Erin Isle
Calculate how far away the plane then was
from Erin Isle.
Chapter 2
Calculate the length of one of the sloping
edges and hence calculate the perimeter of
the trapezium.
this is page 20
Pythagoras
7.
A triangular corner unit (shown in yellow),
is built to house a TV set.
11.
Rhombus PQRS has its 2 diagonals, PR and QS,
crossing at its centre C.
122 cm
Q
133 cm
x cm
Calculate the length
of the long edge
of the unit (x).
8.
20 cm P
R
S
25 cm
A lawn in Edinburgh’s Princes Street is in the
shape of a rectangle 26 metres long by 14·5
metres wide.
Calculate the PERIMETER of the rhombus.
12.
14·5 m
path
C
Two wires are used to support a tree in danger
of falling down after a recent storm.
26 m
A path runs diagonally through the lawn.
12·6 m
Calculate the length of the path.
sloping
roof
9.
The picture shows
the side view of a
conservatory.
10·9 m
13·3 m
Calculate the total length of the support wires.
2·7 m
1·9 m Calculate the length
of the sloping roof.
13.
3·2 m
10.
Calculate the PERIMETER of these 2 triangles.
(a)
The roof of a garage is in the shape of an
isosceles triangle.
1·2 m
9 cm
?m
40 cm
4·6 m
(b)
10·5 cm
28 cm
Calculate the length of one side of the
sloping roof.
Chapter 2
this is page 21
Pythagoras
Calculating the Length of one of the Smaller Sides
You can use Pythagoras’ Theorem to calculate one of the smaller sides of a right angled triangle.
This time, you are asked to find the length of the smaller side (a) :=>
a2 = c 2 – b2
=>
a 2 = 25 2 – 15 2
=>
a 2 = 625 – 225 = 400
=>
a =
can you see why
the “–” sign ?
15 cm
400 = 20 cm
a cm
Exercise 2·3
1.
5.
Calculate the length of the side of this right
angled triangle marked with a t.
This isosceles triangle has a base of 96 cm
and a sloping edge of 52 cm.
52 cm
a2 = c 2 – b2
45 cm
t cm
27 cm
2
2
=>
t
= 45 – 27
=>
t 2 = 2025 – 729
=>
t 2 = ........
=>
t = .... cm
96 cm
Calculate the area of the triangle.
Shown is the side view of a wooden bread tin.
Calculate the size of each of the smaller sides
in the following right angled triangles.
(a)
H
2
6.
2.
25 cm
36 cm
(b)
32 cm
15 cm
8 cm
26 cm
38 cm
f cm
x cm
e cm
3.
A wheelchair ramp
has a sloping side
8·2 m long and a
horizontal base
7·1 m long.
Calculate the length (x) of the base of the bin.
17 cm
8·2 m
?
7.
Calculate the
perimeter of
this right angled
triangle.
41 cm
7·1 m
40 cm
Calculate the height of the ramp.
8.
4.
hm
85 m
47 m
Chapter 2
A helium balloon is
tethered by a rope to
the ground as shown
opposite.
Calculate the height
of the balloon.
Shown is a right
angled isosceles
triangle ABC.
Calculate the value
of t.
this is page 22
C
t cm
A
30 cm
t cm
B
Pythagoras
Mixed Examples
In the following exercise, if you are asked to find :the hypotenuse
a shorter side
—>
c
use c 2 = a 2 + b 2 .
—>
2
2
b
2
use a = c – b .
a
You must decide which formula you have to use.
Example 1 :-
Example 2 :-
17 cm
15 cm
x cm
12 cm
y cm
7 cm
(here, you are
looking for a
short side)
y2 = 15 2 + 72
x2 = 17 2 – 12 2
x2 = 289 – 144
y2 = 225 + 49
note
x2 = 145
y2 = 274
x
y =
=
(here, you are
looking for the
long side)
145 = 12·04 cm
note
274 = 16·55 cm
Exercise 2·4
1.
Use the appropriate formula to find the value of x each time :–
(a)
(b)
(c)
20 cm
x cm
9 cm
4 cm
x cm
8 cm
(d)
33 mm
11·5 m
123 cm
(e)
x mm
7·6 m
(f)
142 cm
41 mm
xm
7m
5·1 m
xm
x cm
2.
Andy was answering this question in a
class test.
x2 = 14 2 + 9 2
His working was set down as shown :–
Why should Andy have known
that his answer had to be wrong,
by just comparing it to the length
of the hypotenuse of the triangle ?
Chapter 2
x cm
14 cm
x2 = 196 + 81
x2 = 277
9 cm
this is page 23
x =
277 = 16·6 cm
Pythagoras
3.
ONE of the following two answers is known
to be the correct value for y in this question.
7.
This warning sign is
in the shape of an
isosceles triangle.
48 cm
y cm
7·2 cm
y = 15·7 cm ?
14·0 cm
DANGER
Calculate the height
of the sign.
y = 9·6 cm ?
44 cm
Without actually doing the calculation, say
which one it must be and why the other is
obviously wrong.
8.
A ladder was leaning against a wall. It began
to slide away from the wall, but it stopped when
its base came to rest against a smaller wall.
before
4.
The tip of this pencil
is in the shape of an
isosceles triangle.
12·5 mm
11·3 mm
after
7·2 m
H
7·2 m
h
w
Calculate the width
of the pencil (w).
6·1 m
2·9 m
(a) Calculate the original height (H) of the
top of the ladder above the ground.
5.
(b) Calculate the new height (h) of the top
of the ladder.
This Scottish Flag is 2·35 metres long and 1·86
metres wide.
(c) By how many metres had the top of the
ladder slipped ?
1·86 m
9.
An orienteering competition was held over a
triangular course.
2nd checkpoint
•
2·35 m
What length must each diagonal strip be?
6.
1·9 km
•
Start
Finish
A cannon ball was fired and flew in a straight
line for 450 metres where it exploded 85
metres above the enemy lines.
3·2 km
•
1st checkpoint
From the start, the participants walk East to the
1st checkpoint, North to the 2nd one and then
race back to the finishing line.
Calculate the overall distance of the event.
85 m
450 m
10.
dm
6·3 m
Calculate the distance (d m) from the cannon
to the enemy soldiers.
3·4 m
Chapter 2
this is page 24
A lamppost fell over
during a storm and came
to rest with its top resting
wall ? m
against the top of a wall.
Calculate the height of
the wall.
Pythagoras
Distances Between Coordinate Points
y
•B
Consider the two coordinate points A(–3, –1) and B(5, 5).
They are plotted on the coordinate diagram opposite.
To calculate the distance from A to B :-
•
A
•
draw in the 2 dotted lines to make a right
angled triangle APB.
•
write down the lengths of the two sides AP and BP.
•
use Pythagoras’ Theorem to calculate length of AB.
x
P
AB2 = AP2 + PB2
AB2 = 82 + 62
AB2 = 64 + 36 = 100
AB = √100 = 10 boxes
Exercise 2·5
1.
(a) Make a copy of this coordinate diagram,
showing the 2 points P(1, –1) and Q(4, 3).
4.
For each pair of points below, calculate the
length of the line joining them, giving your
answer to 2 decimal places each time.
(a) F(2, – 4) and G(–1, 5)
y
(b) U(6, –2) and V(0, 4).
•Q
•
P
5.
Terry thinks triangle AST below is isosceles.
To prove it is, he has to find the lengths of the
2 lines AS and AT and show they are equal.
x
y
•A
(b) By drawing in the 2 dotted lines, create
a right angled triangle and use it to
calculate the length of the line PQ.
2.
S
•
(a) Draw a new coordinate diagram and plot
the 2 points M(– 4, 2) and N(8, 7)
(b) Create a right angle triangle in your figure
and determine the length of the line MN.
•T
x
(a) Write down the length of the line AT.
(b) Calculate the length of the line AS.
3.
Calculate the distance between the 2 points :R(–2, 0) and S(5, 4),
giving your answer correct to 2 decimal places.
Chapter 2
(c) Was Terry correct ?
6.
Prove that triangle LMN is isosceles where
L(–2, 2), M(6, 8) and N(4, –6).
this is page 25
Pythagoras
The CONVERSE of Pythagoras’ Theorem
Pythagoras’ Theorem only works on a right angled triangle.
We can use Pythagoras Theorem “in reverse” to actually prove that a triangle is right angled.
A
Example :-
5·2 cm
3·9 cm
Look at triangle ABC opposite
B
We can prove it is right angled as follows :• Write down the 3 sides :- AB = 5·2,
• Square each side :-
C
6·5 cm
AC = 3·9,
BC = 6·5.
AB2 = 27·04, AC2 = 15·21,
BC2 = 42·25.
• Add the two smaller squares together :- AB2 + AC2 = 27·04 + 15·21 = 42·25.
• Check if this is the same value as the largest square :- AB2 + AC2 = 42·25 = BC2 .
• We say that, by the CONVERSE of Pythagoras’ Theorem, the triangle is proven
to be right angled at A.
Exercise 2·6
1.
3.
Check if this
triangle is
right angled at Q.
18 cm
Q
Copy and complete :-
(a)
7·5 cm
P
19·5 cm
Decide which of the following is/are right
angled triangles, and which is/are not :(b)
9·6 m
35 mm
84 mm
18·0 m
R
20·4 m
91 mm
• PQ2 = 182 = 324,
• QR2 = 7·52 = .....
4.
• PR2 = ....2 = ....
A groundsman wishes to make sure the football
pitch is “square” (its corners are at 90°).
• PQ2 + QR2 = 324 + .... = ...... = PR2
105 m
63 m
• by the Converse of Pythagoras’ Theorem,
triangle PQR must be r....... a...... at Q
2.
W
6·6 cm
84 m
To check, he measures the diagonal length.
Is the pitch “square” ?
Show that this triangle
8·8 cm
is NOT right angled.
U
5.
11·1 cm
V
i.e. (Show that UW2 + VW2 ≠ UV2 )
Has this flagpole been
erected correctly,
13·5 m
so that it is
vertical ?
10·8 m
8·1 m
Chapter 2
this is page 26
Pythagoras
Remember Remember..... ?
1.
Calculate the lengths of the missing sides in the
following right angled triangles :-
6.
Topic in a
Nutshell
This shape consists of a rectangle with an
isosceles triangle attached to its end.
18 cm
35 cm
22 cm
x cm
7 cm
y cm
24 cm
23 cm
7 cm
L cm
5·6 m
(a) Calculate the total length (L) of the figure.
3·1 mm
2·9 mm
(b) Now calculate its area.
zm
6·8 m
2.
w mm
7.
Shown are the points F(–5, –2) and G(4, 3).
y
Shown is an isosceles triangle.
(a) Calculate the height
of the triangle.
(b) Now calculate its area.
•
Calculate the area of the following rectangle :Draw a coordinate diagram, plot the two points
and calculate the length of the line FG.
50 cm
8.
Draw a new set of axes, plot the 2 points
M(–2, 6) and N(5, –3) and calculate the length
of the line MN.
9.
Prove that one of the following IS a right
angled triangle and the other is NOT.
48 cm
4.
x
F
30 cm
3.
•G
39 cm
Calculate the perimeter of this right angled
triangle :9 cm
6·8 cm
19·2 cm
40 cm
5.
xm
A
Calculate the value of
x, which indicates the
length of the sloping
side of this trapezium.
5·1 cm
8·0 cm
20·8 cm
10. Prove that PQRS is a rectangle :-
13·5 m
P
8·7 m
Q
18·7 cm
S
6·2 m
Chapter 2
B
8·3 cm
this is page 27
16·5 cm
8·8 cm
R
Pythagoras