Pythagoras
Theorum
Math 314
Pythagorean Triples
 Can you think of 3 natural numbers that would
work in a right angled triangle?
 The easiest is (3,4,5). Is this true?
 If c² = a² + b² Verify your answer given the #5
must be the largest value or c²
 5²= 3² + 4²
 25 = 9 + 16
 25=25 True 3,4,5 are Pythagorean triples
Label the Triangle
 Which of these numbers (3,4,5) must be
the hypotenuse?
5
3
4
 Does the placement of the 3, 4 or 5 make
a difference?
Creating other Pythagurus
Triples. Your turn!
 Create 3 on your
own and ask a friend
to guess what the
other one is?
 Label two out of the
three legs and / or
triangle.
 Explain to them.
Make it a decimal
(always two places)
Pythagorean Triples with
Fractions – Consecutive
Fraction Method
 Consider 11 and 13
 11 and 13 are consecutive odd numbers
 1+1
11 13
 Multiply denominators by each other (11 *
13)
 Answer is 143. Therefore…
Pythagorean Triples
Fractions

13 + 11
143
 24 (DO NOT REDUCE EVEN IF YOU CAN)
143
 Pythagorean triple is 24, 143 and 145
 Pythagorean triple is numerator,
denominator and denominator + 2.
 Prove it or verify it.
Verify
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Is 24, 143 and 145 Pythagorean triples?
c² = a² + b²
145² = 24² + 143²
21025 = 576 + 20449
21025 = 21025 It works!
Example #2
 2 and 4
 1 + 1
2
4
 4+2
8
 6
8
 Pythagorean triple is…
 (6, 8, 10)
Even Odd Method (Faster)
 You get 2 consecutive even or odd
numbers; for example 7 & 9
 Add them (7 + 9) = 16
 Multiply them (7 * 9) = 63
 Multiply them add 2 = 7 * 9 + 2 = 65
 Triple is 16, 63, 65
Other Examples
 Generate a Pythagorean triple using the
even – odd seed method.
 4, 6
 Answer: (10,24,26)
 8,10
 Answer (18,80,82)
 11,13
 Answer (24,143,145)
Another Method –
Equation Method
 Pick two natural numbers A + B such that
A>B
 A and B must be positive
 1) a² - b²
 2) 2ab
 3) a² + b²
Equation Method to
Calculate Pythagorean Triple
A = 11; B = 3
a² - b²
11² - 3²
112
2ab
2 (11)(3)
66
a² + b²
121 + 9
130
Examples – Formula
Method
 Generate a Pythagorean triple using the
formula method
A = 6; B = 1
 Remember A²-B² 2AB A²+B²
A² - B² = 36-1 = 35
 2AB = 2 (6) (1) = 12
 A²+B² = 6² + 1² = 37
 The numbers are (12, 35, 37)
More Examples
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

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
A=6;B=2
Solution (24,32,40)
A=6;B=3
Solution (27,36,45)
A = 12 ; B = 1
Solution ( 24, 143, 145)
Definitions
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Equilateral Triangle: All sides are equal
Isosceles Triangle: Two sides are equal
Scalene: All sides are different
What will you do when asked to calculate
Perimeter of Triangle?
Add up all the sides
Area of Triangle?
Base x Height / 2
Algebra and Pythagoras
 How would you express the relationship
between measures of the sides of the
following right triangle
5r
3p
4q
25r²= 9p² + 16 q²
R=?
R = 9p² + 16 q²
25
Calculating Area of an
Isosceles Triangle
12
12
Cut triangle in half to calculate height
c² = a² + b²




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
12² = 5² + a² (half of 10)
144 = 25 + a²
119 = a²
10
a= 10.91
Area of isosceles triangle = base x height / 2
10 x 10.91 / 2 = 54.55
Finding x with two missing
variables
 Triangle has different lengths
x
9
7
5
Before calculating the x, find height
Therefore, do 2 Pythagoras's – double the fun!
Calculating Height
 We have two right angle triangles but we
cannot get to the one with x directly so
we need a middle step
 1st step is to find out missing value of x…
to figure that out use Pythagoras
 x² = height² + 7²
 You also know that 9² = height² and 5²
Finding Height or k
x
9
k
7
5
81 = k² + 25
56 = k²
k = 7.48
Finding x
x
9
7.48
7
x² = 7.48² + 7²
x² = 104.95
X = 10.24
5
Practice – Word Problems
 Both a chair lift and a gondola are used
to transport skiers to the top of a ski hill.
The length of the gondola cable is twice
the length of the chair lift cable. The
situation is represented by
Word Problem
chair lift cable
gondola cable
400
500
If the gondola travels at 5m per second,
how long with the gondola ride take?
Word Problem
chair lift cable
gondola cable
400
500
c²
= 400² + 500² (find out c, then double to get g)
c² = 410000
C = 640 .31
Solution


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Gondonla or G = 2c
G = 2 (640.31)
G = 1280.62
1280.62 / 5 = 256.12 seconds
Word Problems - Ladder
 A ladder is leaning against a wall 8.4m
above the ground and extends 3m past
the top of the wall. The foot of the ladder
is 3.5m from the wall.
 Find the length of the ladder to the
nearest tenth.
 How many decimal places is tenth?
hundredth, thousandth?
Diagram of Ladder

3m
8.4m
3.5m
Ladder Solution
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c² = a² + b²
c² = 8.4² + 3.5²
c² = 70.56 + 12.25
c² = 82.81
C = 9.1
What do you do now?
9.1 + 3 = 12.1m is the length of the
ladder.
Rational Numbers
 All rational numbers can be written in the form
of fractions. For example;
 14 = 14/1
 0.72 = 72/100
 1.76 = 176/100
 These numbers have a zero or a group of
digits that repeat indefinitely. i.e.
 1) 14
 2) 17.626262 or 17.62
 3) 3.6666 or 3.6
Irrational Numbers
 Irrational numbers have non –
terminating, non repeating decimals.
After the decimal, no pattern of numbers
will repeat. Examples are…
 Pie & square root of 2.