Pythagoras
The History of Pythagoras…..
He lived from about 569BC to about 475BC
and spent most his life in Italy.
He was a Greek philosopher who made important
developments in astronomy, mathematics and music.
He never married, but had a group of
followers called the Pythagoreans.
Unfortunately, very little is
known about Pythagoras
because none of his writings
have survived.
He is best known for the
Pythagorean Theorem which
was known to the Babylonians
1000 years earlier but he may
have been the first to prove it.
Pythagoras’ Theorem
• “For any right-angled triangle, the square of the
hypotenuse is equal to the sum of the squares of the
other two sides.”
a2 + b2 = h2
What is a right-angled triangle?
What does hypotenuse mean?
What does squaring a number mean?
Sum of two numbers means?
In a right angled triangle, the
longest side is opposite the right
angle and is called the
hypotenuse (h).
Which is the hypotenuse – x, y or z
x
z
x
z
y
y
x
x
y
y
z
z
Proof:
a² + b² = c²
Finding the length of the hypotenuse
Examples 1:
Find the value of x correct to 2dp.
h2
= a 2 + b2
h2
= 72 + 52
= 49 + 25
7
= 74
h = √74
= 8.60 (2dp)
x
5
Example 2
Calculate the length of the hypotenuse
5² + 8² = h²
25 + 64 = h²
89 = h²
h = 89
h = 9.43 (2dp)
5m
8m
Riddle sheet
Example 3:
Find the length of the ladder correct
to 2dp.
h2 = a2 + b2
h2 = 42 + 62
= 16 + 36
h = √52
= 7.21m (2dp)
Length of ladder = 7.21m (2dp)
x
6m
4m
Exercise 18.2
pg 259
Testing out the Theory with a 3, 4, 5 triangle
a 2 + b2 = h 2
5
3
Check: 3² + 4²
= 9 + 16
= 25
=5²
4
A Pythagorean Triple is a triple of
natural numbers (a,b,c) such that
a2 + b2 = c2
e.g. 3, 4, 5
Can you find another
Pythagorean triple
where a, b and c are
less than 100?
Do Now:
Find the length of side c
c
3.1m
8.5m
3.1² + 8.5² = c²
9.61 + 72.25 = c²
81.86 = c²
c = 9.045 (3dp)
Finding a side which isn’t the
hypotenuse
Find the length of x to 2dp.
h2 = a2 + b2
x2 + 22 = 52
x2 + 4= 25
x2 = 25 – 4
x2 = 21
x = √21
= 4.58 (2dp)
x
5
2
Find the length x that a 10m
ladder would go up the wall
if the foot of the ladder is
2m from the wall.
h2 = a2 + b2
x2 + 22 = 102
x2 = 100 – 4
x = √96
= 9.80 (2dp)
Diagonal distance = 9.80m
(2dp)
10m
x
2m
Do Now
h  a b
2
2
2
5252 = 4502 + b2
73125 = b2
b = 270.42km
Applications
1. Draw a diagram of a right-angled triangle.
2. Put the measurements on the diagram, placing the unknown
as x.
3. Solve for x leaving your answer in sentence form.
1.6m
x
4·2m
Riddles – What do two bullets have when they get married?
- What did Lancelot say to the beautiful Ellen?
- What happened at the milking contest?
6.2m
4.6m
If you stand 4.2m from a 6.2m high
tree and look diagonally at the top,
how far is it from the top of your head
to the top of the tree if you are 1.6m
tall? Give your answer to 1dp.
h2 = a2 + b2
x2 = 4.62 + 4.22
= 21.16 + 17.64
= 38.8
x = √38.8
= 6.2 (2dp)
The diagonal distance is 6.2m (1dp).
Do Now:
• Use Pythagoras’ Theorem to find the length of the
side labelled x.
1.
2.
x
8m
5.8cm
x
x = 6.17cm
2.1cm
10m
x = 6m
An Air NZ plane takes off from Dunedin airport heading to Nelson. The
plane climbs at an angle of 35° from the time it takes off and flies at
this angle for 10km.
Unfortunately after flying for 10km at an angle of 35° disaster strikes
and the plane is struck by a huge lightning bolt, causing the plane to
fall from the sky.
If the plane was at a height less than 6.5km, then there will be
survivors. But if the plane was higher than 6.5km when it was hit, then
there will not be any survivors.
I happened to be with a group of people directly under where the plane
was hit by lightning and someone shouted out ‘Are there any
mathematicians here, does anyone know how to do trigonometry to
work out how high the plane was?’
Luckily I was there to use my trigonometry skills so I could work out
how high the plane was when it hit and I was able to tell the people
whether we needed to call an ambulance for the survivors or if we
needed body bags.
Do Now:
1.
42.2 m (1dp)
2. For each of these triangles write down which side is the
hypotenuse, opposite and adjacent.
e
p
d
f
q
r
Labelling sides of triangles
• Hypotenuse = the longest
side opposite the right angle.
• Opposite = opposite the
marked angle.
• Adjacent = side that joins the
right angle to the marked
angle.
Adjacent (A)

Opposite (O)
Opposite (O)

Adjacent (A)
Sohcahtoa
The three trig ratios are
o

• sin =
h
• cos =
h
o
a
h
o
a
a
• tan =
They allow you to find a side or angle of a right angled triangle if
you know an angle or one side or 2 angles.
Sometimes this is written in the form below
o
sin
o
a
h
cos
h
tan
a
Quick Quiz
1.
x
3
9.5 (1dp)
9
4. What is the ratio for the
sine of an angle?
x
2.
14.89 (2dp)
3.5
5. What is the ratio for the
cosine of an angle?
15.3
3. Which is the hypotenuse, opposite,
adjacent?
f
d
e
6. What is the ratio for the
tangent of an angle?
Finding a side
1. When given a side length and an angle, you
can not use Pythagoras’ Theorem.
2. Identify which side you have and which you
need.
3. Decide which rule to use.
4. Substitute values and rearrange to solve.
Example
Find the length of x.
Have o and h so use
soh
o
sin
x
a
h
cos
o
o
h
tan
14cm
h
a
60°
Want o so cover o and get
x = sin 60 x 14
= 12cm (nearest cm)
Find the length of x.
Have o and h so use
soh
o
sin
h
Want h so cover h and get
4
x=
sin 40
= 6cm (nearest cm)
o
a
cos
h
h
x
tan
a
4cm
o
40
Find the length of x.
Have a and h so use
Cah
o
sin
a
o
a
h
cos
h
Want a so cover a to get
x = cos45 x 10
= 7cm (nearest cm)
tan
10
a
h
45
x
Find the length of x.
Have a and o so use
toa
cos
x a
o
a
h
tan
a
35
6cm
Want a so cover a to get
6
x=
tan 35
= 9cm (nearest cm)
o
Do Now:
• A ladder, 5 metres long, leans against a wall and
makes an angle of 70° with the ground. How far
is:
a) The top of the ladder from the ground?
b) The foot of the ladder from the wall?
(Hint: draw a diagram first!)
a) 4.7m (1dp)
b) 1.71m (2dp)
• Ex 20.7 pg 294 - Riddle
• Worksheet
• Ex 20.3 pg 285 - Applications
Do Now
1. Find x
Use soh
o = sin 74 x 4
= 3.85km
o
a
sin
h
cos
74°
4kmo
h
tan
a
y km
2. Find y
Use cah o
A = cos
74 xh4
sin
= 1.10km
o
a
cos
h
3. Draw the trig ratio triangles
in your book
a
tan
o
sin
x km
o
a
h
cos
h
tan
a
Finding an Angle
When given a problem where you have to find
the angle :
1.Identify which two sides are given.
2.Decide which rule to use (SOH-CAH-TOA).
3.Substitute values and solve.
Inverse sin, cos, tan
• When we use the trig triangles to find the
angle, we find sinθ, cosθ or tanθ.
• So to find the angle θ, we need to use the
inverse button (shift sin, cos or tan)
• Find θ if
– Sin θ = 0.5
– Cos θ = 0.688
– tan θ = 1.34
30o
46.5o
53.3o
Find the angle
Have o and h so use
soh
o
sin
o
a
h
cos
tan h a
10
h
Want angle so cover sin to get
5
Sinθ =
o

10
= 0.5
θ = sin-1 0.5
= 30o
5
Shift sin
Find the angle
Have a and h so use
Cah
o
sin
o
a
h
cos
h
tan
a
12
h
Want angle so cover cos
Cos θ = 4
12
4
θ = cos-1 ( )
12
= 70.5o (1dp)
a
4
n
o
Find the angle
Have a and o so use
a
Toa
h
cos
h
o
tan
12 o
a
Want angle so cover tan
tan θ = 12
4
a
4
θ = tan-1 ( 3 )
= 71.6o (1dp)
Ex 21.3 pg 297
Ex 21.4 pg 298
Find x
• Find x
Use soh
o = sin 62 x 5
= 4.41km
62
o
sin
Find θ
Sin θ = 0.4
Cos θ = 3
tan θ = 4
5
8
5km
o
a
h
23.6o
41.4o
32o
cos
h
tan
x km
a
DO Now
6km
Find the angle
• Use soh
o
sin
o
a
h
cos
h
12km
• Sin-1 (
)
• = 30o
Find the other side
Could use Pythagoras, cos or tan!
10.4km
6
12
tan
a
A good game
Applications
Find
o the height
a of the
building
sin
h
cos h
Have o and a
So use toa
Want o so cover o to get
Height = tan 39 x 46.0
= 37.3m
o
tan
a
Find the angle of the
road
Have o and h
So use soh
o
Want angle so cover sin to get
Sin θ = 0.530
sin
a
h
cos
h
3.00
θ =sin-1 0.1767
= 10.18o
• Ex 21.4 pg 298
• When finished get Riddle sheet