# Maths Investigation: Number of Sides

The numbers 3, 4 and 5 satisfy the condition 32+42=52,

Because 32= 3x3 =9

42= 4x4 =16

52= 5x5 =25

And so... 32+42=9+16=25=52

I now have to find out if the following sets of numbers satisfy a similar condition of (smallest number) 2+ (middle number) 2= (largest number) 2.

a) 5, 12, 13

52+122 = 25+144 = 169 = 132.

b) 7, 24, 25

72+242 = 49+576 = 625 +252

Here is a table containing the results:

I looked at the table and noticed that there was only a difference of 1 between the length of the middle side and the length of the longest side.

I already know that the (smallest number) 2+ (middle number) 2= (largest number) 2. So I know that there will be a connection between the numbers written above. The problem is that it is obviously not:

(Middle number) 2+ (largest number) 2= (smallest number) 2

Because, 122 + 132 = 144+169 = 313

52 = 25

The difference between 25 and 313 is 288 which is far to big, so this means that the equation I want has nothing to do with 3 sides squared. I will now try 2 sides squared.

(Middle)2 + Largest number = (smallest number)2

= 122 + 13 = 52

= 144 + 13 = 25

= 157 = 25

This does not work and neither will 132, because it is larger than 122. There is also no point in squaring the largest and the smallest or the middle number and the largest number. I will now try 1 side squared.

22 + 13 = 5

This couldn´t work because 122 is already larger than 5, this also goes for 132. The only number now I can try squaring is the smallest number.

12 + 13 = 52

25 = 25

This works with 5 being the smallest number/side but I need to know if it works with the other 2 triangles I know.

4 +5 = 32

9 = 9

And...

24 + 25 = 72

49 = 49

It works with both of my other triangles. So...

Middle number + Largest number = Smallest number2

If I now work backwards, I should be able to work out some other odd numbers.

E.g. 92 = Middle number + Largest number

81 = Middle number + Largest number

I know that there will be only a difference of one between the middle number and the largest number. So, the easiest way to get 2 numbers with only 1 between them is to divide 81 by 2 and then using the upper and lower bound of this number. So.

81 = 40.5

2

Lower bound = 40, Upper bound = 41.

Middle side = 40, Largest side = 41.

Key

Longest/Largest Side = Length of Longest Side.

Middle Side = Length of Middle Side.

Shortest Side = Length of Shortest Side.

Let´s see if this works for a triangle that I already know.

72 = Middle number + Largest number

49 = Middle number + Largest number

49 = 24.5

2

Lower bound = 24, Upper bound = 25.

Middle Side = 24, Largest Side =25.

This matches the answers I already have with 7 being the shortest side, so I think that this equation works. I now believe I can fill out a table containing the Shortest, Middle and longest sides, by using the odd numbers starting from 3. I already know that the middle and longest side with the shortest length being 3, 5,7 or 9. So I will start with the shortest side being 11.

112 = Middle number + Largest number

121 = Middle number + Largest number

121 = 60.5

2

Lower bound = 60, Upper bound = 61.

Middle Side = 60, Largest Side =61.

32 = Middle number + Largest number

169 = Middle number + Largest number

169 = 84.5

2

Lower bound = 84, Upper bound = 85.

Middle Side = 84, Largest Side =85.

152 = Middle number + Largest number

225 = Middle number + Largest number

225 = 112.5

2

Lower bound = 112, Upper bound = 113.

Middle Side = 112, Largest Side =113.

172 = Middle number + Largest number

289 = Middle number + Largest number

289 = 144.5
...