Beyond Pythagoras - Mathematical Investigation
1) Do both 5, 12, 13 and 7, 24, 25 satisfy a similar condition of :
(Smallest number)² + (Middle Number)² = (Largest Number) ² ?
5, 12, 13
Smallest number 5² = 5 x 5 = 25
Middle Number 12² = 12 x 12 = 144+
169
Largest Number 13² = 13 x 13 = 169
7, 24, 25
Smallest number 7² = 7 x 7 = 49
Middle Number 24² = 24 x 24 = 576+
625
Largest Number 25² = 25 x 25 = 625
Yes, each set of numbers does satisfy the condition.
They are both Pythagorean triples.
Area = 12 x 5
2
Area = 6 x 5
Area = 30
Perimeter = 5
12
13+
30
13
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5
12
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Area = 24 x 7
2
Area = 12 x 7
Area = 84
Perimeter = 7
24
25+
56
7
24
Area = 40 x 9
2
Area = 20 x 9
Area = 180
Perimeter = 9
40
41+
90
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41
40
9
b)
3)
From my data I can construct a table to make the identification of
patterns easier.
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So far I have observed the following patterns:
· The smallest side length advances by two each time.
· All the middle numbers are even
· The middle length is always a multiple of four.
· The longest length is always one unit more than the middle length.
· Both the shortest and longest side lengths are always odd.
· Both the area and perimeter are always even numbers.
I can immediately see the formula to derive the shortest side length
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