Transcript Who wants to be a Millionaire?

Who wants to be a Millionaire?
Pythagorean Triads.
Video Clip courtesy of YouTube:
Please view the video clip, from the TV quiz show “Who wants to be a Millionaire?”, at
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Pythagorean Triads
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2
2
2
3 + 4 = 9 + 16 = 25 = 5
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Pythagoras’ Theorem
• Pythagoras’ Theorem
states that, in a rightangled triangle, the
square of the
hypotenuse is equal to
the sum of the squares
of the other two sides,
2
a
+
2
b
=
2
c.
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Proof of Pythagoras’ Theorem
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The 3-4-5 Triangle
• It is not surprising that there are some right-angled triangles
where all three sides are whole numbers called
Pythagorean Triangles. The three whole number sidelengths are called a Pythagorean triple or triad. The most
famous example is a = 3, b = 4 and c = 5, called "the 3-4-5
triangle". We can check it as follows:
32 + 42 = 9 + 16 = 25 = 52 so a2 + b2 = c2.
• This triple was known to the Babylonians (who lived
in the area of present-day Iraq and Iran) even as long
as 5 000 years ago!
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More Pythagorean Triads
• Is the 3-4-5 the only Pythagorean Triad? No, because we
can double the length of the sides of the 3-4-5 triangle and
still have a right-angled triangle: its sides will be 6-8-10 and
we can check that 102 = 62 + 82. Continuing this process by
tripling 3-4-5 and quadrupling and so on, we have an infinite
number of Pythagorean triads:
3
6
9
12
15
18
...
4
8
12
16
20
24
5
10
15
20
25
30
All of these will have the same shape
(have the same angles) but differ in size
- the mathematical term is that they are
all similar triangles. If they were the
same size but in different positions or
orientations, the triangles are called
congruent.
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Are there any other differently-shaped rightangled triangles with whole number sides?
Yes; one is 5, 12, 13 & another is 7, 24, 25.
We can check that they have right angles
by using Pythagoras' Theorem that the
squares of the two smaller sides sum to
the square of the longest side.
52 + 122 = 25 + 144 = 169 = 132
72 + 242 = 49 + 576 = 625 = 252
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Which of the following is not a
Pythagorean Triad?
3:4:5
9 : 40 : 41
14 : 48 : 50 20 : 48 : 52
5 : 12 : 13
10 : 24 : 26 15 : 20 : 25 21 : 28 : 35
6 : 8 : 10
11 : 60 : 61 15 : 36 : 39 21 : 72 : 75
7 : 24 : 25
12 : 16 : 20 17 : 90 : 93 24 : 32 : 40
9 : 12 : 15
13 : 84 : 85 18 : 24 : 30 25 : 60 : 65
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Which of the following is not a
Pythagorean Triad?
3:4:5
9 : 40 : 41
14 : 48 : 50 20 : 48 : 52
5 : 12 : 13
10 : 24 : 26 15 : 20 : 25 21 : 28 : 35
6 : 8 : 10
11 : 60 : 61 15 : 36 : 39 21 : 72 : 75
7 : 24 : 25
12 : 16 : 20 17 : 90 : 93 24 : 32 : 40
9 : 12 : 15
13 : 84 : 85 18 : 24 : 30 25 : 60 : 65
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What is the total length a ?
3.6
4.8
6.0
6.4
7.2
9.6
10.0
a
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What is the total length a ?
3.6
4.8
6.0
6.4
7.2
9.6
10.0
a
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What is the length b ?
3.6
4.8
6.0
6.4
7.2
9.6
10.0
b
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What is the length b ?
3.6
4.8
6.0
6.4
7.2
9.6
10.0
b
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What is the length c ?
3.6
4.8
6.0
6.4
7.2
9.6
10.0
c
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What is the length c ?
3.6
4.8
6.0
6.4
7.2
9.6
10.0
c
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References / Sources:
http://www.jaconline.com.au/mathsquestnsw/mq8nsw/investigations/05_pythag_triads.pdf
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Pythag/pythag.html#345
http://www.education.vic.gov.au/studentlearning/teachingresources/maths/mathscontinuum/mcd/
M57501P.htm#a2
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