Name of Lecturer: Mr. J.Agius
Course: HVAC 2
Lesson 33
Chapter 4: Trigonometry
Pythagoras' Theorem
Years ago, a man named Pythagoras found an amazing fact about
triangles:
If the triangle had a right angle (90°) ...
... and you made a square on each of the three sides, then ...
... the biggest square had the exact same area as the other two squares put together!
It is called "Pythagoras' Theorem" and can be written in one
short equation:
a2 + b2 = c2
Note:


c is the longest side of the triangle
a and b are the other two sides
Definition
The longest side of the triangle is called the "hypotenuse", so the formal
definition is:
In a right angled triangle:
the square of the hypotenuse is equal to
the sum of the squares of the other two sides.
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Name of Lecturer: Mr. J.Agius
Course: HVAC 2
Sure ... ?
Let's see if it really works using an example.
Example: A "3,4,5" triangle has a right angle in it.
Let's check if the areas are the same:
3 2 + 42 = 52
Calculating this becomes:
9 + 16 = 25
It works ... like Magic!
Why Is This Useful?
If we know the lengths of two sides of a right angled triangle, we can
find the length of the third side. (But remember it only works on right
angled triangles!)
How Do I Use it?
Write it down as an equation:
a2 + b2 = c2
Now you can use algebra to find any missing value, as in the following
examples:
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Name of Lecturer: Mr. J.Agius
Example:
Course: HVAC 2
Solve this triangle.
a2 + b2 = c2
52 + 122 = c2
25 + 144 = c2
169 = c2
c2 = 169
c = √169
c = 13
You can also read about Squares and Square Roots to find out why √169 = 13
Example:
Solve this triangle.
a2 + b2 = c2
92 + b2 = 152
81 + b2 = 225
Take 81 from both sides:
Example:
b2 = 144
b = √144
b = 12
What is the diagonal distance across a square of
size 1?
a2 + b2 = c2
12 + 12 = c2
1 + 1 = c2
2 = c2
c2 = 2
c = √2 = 1.4142...
It works the other way around, too: when the three sides of a triangle
make
a2 + b2 = c2, then the triangle is right angled.
Example: Does this triangle have a Right Angle?
Does a2 + b2 = c2 ?


a2 + b2 = 102 + 242 = 100 + 576 = 676
c2 = 262 = 676
They are equal, so ...
Yes, it does have a Right Angle!
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Name of Lecturer: Mr. J.Agius
Example:


Course: HVAC 2
Does an 8, 15, 16 triangle have a Right Angle?
Does 82 + 152 = 162 ?
82 + 152 = 64 + 225 = 289,
but 162 = 256
So, NO, it does not have a Right Angle
Example:
Does this triangle have a Right Angle?
Does a2 + b2 = c2 ?
Does (√3)2 + (√5)2 = (√8)2 ?
Does 3 + 5 = 8 ?
Yes, it does!
So this is a right-angled triangle
Exercise 1
Use the information given in the diagrams to find the required lengths, giving your answers
correct to three significant figures.
1) Find AC.
2)
Find DF.
E
C
13 cm
19 cm
5m
D
A
9m
B
F
3) Find BC
4)
Find LN
N
C
1.6 cm
81 cm
A
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2.8 cm
B
L
62 cm
M
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Name of Lecturer: Mr. J.Agius
Course: HVAC 2
In ABC, B = 90o, AC = 3.24 cm and BC = 1.6 cm. Find
AB.
Example
Answer
Before working it is important to draw this information given, so that you would know what
you are doing.
AC2
= AB2 + BC2 (Pythagoras’ theorem)
C
3.242 = AB2 + 1.62
Hypotenuse
1.6 cm
2
10.498 = AB + 2.56
7.938 = AB2
3.24 cm
(Putting 2.56 on
A
the other side)
AB
B
7.938
=
= 2.82
(to 3 s.f.)
Exercise 2
Use the information given in the diagrams to find the required lengths, giving your answers
correct to three significant figures.
1) Find AB
2)
Find PQ
R
C
9.3 cm
14 cm
6 cm
4.8 cm
P
A
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Q
B
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Name of Lecturer: Mr. J.Agius
Course: HVAC 2
Exercise 3 Problems
In DEF, F = 90o, DF = 2.8 cm and DE = 4.2 cm. Find EF.
1)
2)
In the figure, ADC = 180o,
AB = 4 cm, AD = 3.2 cm and
BC = 2.8 cm.
B
4 cm
2.8 cm
Find a) BD
A
3.2 cm
C
D
b) AC.
c) Is ABC a right angle?
Give a reason for your answer.
E
5 cm
3)
Calculate AC, AD and AE.
D
5 cm
C
5 cm
A
4)
B
5 cm
Calculate the lengths marked with letters.
3cm
a)
2cm
x
y
6cm
b)
6cm
a
8cm
b
c)
5cm
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13cm
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Name of Lecturer: Mr. J.Agius
Course: HVAC 2
5)
A ladder is 5 metres long. It leans against a wall with one end on the ground 1 metre
from the wall. The other end of the ladder just reaches a windowsill. Calculate the
height of the windowsill above the ground.
6)
Rowena is in her helicopter above the Canary Wharf tower. She flies due west for 4.6
miles and then due north for 1 mile. She is then above the lake. Calculate the direct
distance, in miles, of the Canary Wharf tower from the lake.
1 mile
4.6 miles
7)
An isosceles triangle ABC has AB = 9.5 cm and BC = 9.5 CM. The height of the
triangle is BD and BD = 7.2 cm. Work out the length of AC.
A
9.5cm
7.2cm
D
B
9.5cm
C
8)
An isosceles triangle XYZ has XY = 26 cm, ZX = 26 cm and ZY = 18 cm. Calculate
the height of triangle XYZ.
9)
ABC is a right-angled triangle. AB is of length 4m and BC is of length 13m. Calculate
the length of AC.
C
13m
A
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4m
B
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Name of Lecturer: Mr. J.Agius
10)
Course: HVAC 2
Work out the length, in metres, of side AB of the triangle.
B
20m
A
11)
30m
C
In a circle with centre O and radius 7cm, there is a chord 11cm long. How far is the
chord from the centre?
11cm
12)
The diagram shows the cross-section of a workshop. Find the length of each sloping
edge of the roof.
B
A
C
5.4 m
4m
E
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3.7 m
3m
12.5 m
D
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Name of Lecturer: Mr. J.Agius
13)
14)
Course: HVAC 2
A mirror is in the shape of an octagon. It is formed by starting with a square of 90 cm
and cutting an isosceles triangle from each corner. The two equal sides of each
triangle are of length 30 cm.
a)
Find the lengths of the sides of the octagon.
b)
Find the distance of each vertex from the centre of the octagon.
A chord AB of a circle with centre O, is 10cm long. The midpoint of AB is the point
C and OC is of length 3 cm. What is the radius of the circle?
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Name of Lecturer: Mr. J.Agius
Course: HVAC 2
15)
The diagram shows a circle with centre O and radius 10 cm. AB and CD are two
parallel chords with midpoints M and N. The lengths of AB and CD are two parallel
chords with midpoints M and N. The lengths of AB and CD are 12 cm and 16 cm
respectively. Find the distance between M and N.
16)
Find the missing lengths
a)
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b)
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Name of Lecturer: Mr. J.Agius
17)
Course: HVAC 2
A children’s game involves catching a ball
in a hollow cone. This diagram shows a
section through the centre of the ball and
cone. The sloping sides of the cone are
tangents to the ball as shown.
Find
a) the angle marked yo at the vertex of the
cone
b) the slant height, x cm of the cone.
18)
A scaffold pole PTQ rests against an oil drum with centre O as shown. Find:
a)
TOA
b)
TCA
c)
PB
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