pythagoras
DESCRIPTION
Pythagoras LessonsTRANSCRIPT
Investigate the relationship
PythagorasPythagoras was a famous Greek Mathematician who discovered a connection between the 3 sides of any right-angled triangle
570BC - 495BC
1
23
4 5
Area 3 = Area 1 + Area 2
c2 = a2 + b2therefore
PythagorasThe Theorem of Pythagoras
c
a b
Area 1
Area 2
Area 3
We can know calculate one sideof a right-angled triangle if we
know the lengths of the other two
PythagorasNaming the sides of a triangle
Hypoten
use
The longest side of a right-angled triangle is called the hypotenuse. Its always opposite the right angle
c
a
b
PythagorasLets look at how to use Pythagoras' Rule
8cm
6cmc
a
bThe two smaller sides of the right angled triangle are
8cm and 6cm
To calculate the length of the hypotenuse
c2 = a2 + b2
c2 = 82 + 62
c2 = 100c = √100 = 10cm
We can also use Pythagoras to calculate one of the smaller sides
Pythagoras
12cm7cm
xTo calculate the length of a smaller side
c2 = a2 + b2
x2 = 122 - 72
x2 = 95x = √95 = 9.75cm (2 d.p.)
Subtract
Hint
c2 = a2 + b2
r2 = 8.22 + 1.62
r2 = 69.8r = √69.8 = 8.4m
Whenever a problem involves trying to find the missing side of a right-angled triangle consider Pythagoras' Theorem
PythagorasProblem solving using Pythagoras Theorem
8.2m
1.6mramp
Shown above is a design for a stunt bike ramp
Calculate the length of the ramp
x2 = 502 + 652
x2 = 6725x = √6725 = 82cmFrom Pythagoras the door
must be right-angled
PythagorasProblem solving using Pythagoras Theorem
Can I prove this door has right-angles at the corners?
50cm
65cm82cm
The CONVERSE of Pythagoras
We can use Pythagoras Theorem in reverse
Prove that triangle ABC is rightangled
A
B
C6.8cm 5.1cm
8.5cm
AB = 6.8
BC = 5.1
AC = 8.5
AB2 = 46.24
BC2 = 26.01
AC2 = 72.25
AB2 + BC2 = 72.25 = AC2
By the converse of pythagoras' theorem the triange is rightangled at B
Calculate the length of the roof
c2 = a2 + b2
r2 = 4.32 + 2.52
r2 = 24.74r = √24.74 = 5m
PythagorasProblem solving using Pythagoras Theorem
roof
4.3m
5.1m
7.6m
Working2.5m
PythagorasHarder Problem solving using Pythagoras Theorem
Gold chains are displayed diagonally on a square boardof side 20inches. Calulate the length of the longest chain.
20 inches
c2 = a2 + b2
c2 = 202 + 202
c2 = 800c = √800 = 28.3inches
PythagorasHarder Problem solving using Pythagoras Theorem
P
Q
R
S 16cm
7cm
PQRS is a rhombusCalculate the length of side PQ
c2 = a2 + b2
PQ2 = 3.52 + 82
PQ2 = 76.25PQ = √76.25 = 8.7cm
3.5cm
8cm
P
Q
(Round answer to 1 d.p.)
PythagorasHarder Problem solving using Pythagoras Theorem
Find the distance between point A and point B
X
y
0A(2,3)
B(8,7)
73
82
4
6
x2 = 62 42
x2 = 20x = √20 x = 4.47
Pythagoras (in 3D)
7cm7cm
7cm
A
B C
D
E
F G
H
Using pythagoras can we find the distance form F to D?
GF
H
7cm
7cm
HF
D
9.9cm
7cm
FG2 = 72 + 72
FG2 = 98
FG = √98 = 9.9cm
FD2 = 72 + 9.92
FD2 = 147.01
FD = √147.01FD = 12.12cm
GF
H
7cm
7cm
HF
D
9.9cm