REARRANGING
The order I put the letters in are: G , C , F , A , D , H , E , B
For this particular investigation, there is no correct order for the letters to be in. Here is my explanation.
1) a + b + c = 12
Because it states that the perimeter is 12 therefore all 3 sides of the right-angled triangle will add up to 12.
2) a + b = 12 - c
The same equation, simply rearranged.
3) By Pythagoras' Theorem, a² + b² = c²
So...
4) Square both sides: a² + 2ab + b² = 144 - 24c + c²
(a times a) + ((a times b) + (a times b)) + (b times b) = (12 times 12) - ((12 times c) + (12 times c)) + (c times c), we square both sides because of Pythagoras' Theorem.
5) So 2ab = 144 - 24c
We already know that a² + b² = c², so we can remove that from the last equation to simplify it.
6) Divide by 2: ab = 72 - 12c
To find ab alone.
7) Area of triangle: ab/2
This rule explains why we do the next step.
8) So the area of the triangle = 36 - 6c
This is the final answer.
METHOD ADAPTION
Simply change the numbers in the method explained above to prove that when the perimeter is 30, the area is 225 - 15c.
1) a + b + c = 30
2) a + b = 30 - c
3) a² + b² = c²
4) a² + 2ab + b² = 900 - 30c + c²
5) 2ab = 900 - 60c
6) ab = 450 - 30c
7) Area of the triangle = ab/2
8) Area of the triangle = 255 - 15c