I have made a python program to allow the user to input the size of the grid and it prints out all the possibilities of the area the triangle in different required coordinates. The result can be seen on the right and the code on the bottom. on the left section are answers to questions asked and on the top left, is the method i used to workout the area of the triangle. In my conclusion I found that (5x+5y-xy)/2 is going to give me the area of the triangle. Replacing the 5 to any number can change the grid size. (ax+ay-xy)/2.
Answers:
Yellow Triangle area = 10.5cm^2
Blue Triangle area = 11.5cm^2
Green Triangle area = 11cm^2
Purple Triangle area = 10.5cm^2
Therefore, The Blue Triangle has the largest area.
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Smallest area the triangle can have in a 5 by 5 grid: 2.5cm^2 = (0,0), (0,1), (5,5)
Largest area the triangle can have in a 5 by 5 grid: 12.5cm^2 = (5,0), (0,5), (5,5)
To work out the area of triangle in 5*5 grid: (5x+5y-xy)/2
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Smallest area, 6 by 6 grid: 3cm^2 = (0,0), (0,1), (6,6)
Largest area, 6 by 6 grid: 18cm^2 = (6,0), (0,6), (6,6)
To work out the area of triangle in 6*6 grid: (6x+6y-xy)/2
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Smallest area, 7 by 7 grid: 3.5cm^2 = (0,0), (0,1), (7,7)
Largest area, 7 by 7 grid: 24.5cm^2 = (7,0), (0,7), (7,7)
To work out the area of triangle in 7*7 grid: (7x+7y-xy)/2
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Smallest area, 100 by 100 grid: 50cm^2 = (0,0), (0,1), (100,100)
Largest area, 100 by 100 grid: 5000cm^2 = (100,0), (0,100), (100,100)
To work out the area of triangle in 100*100 grid: (100x+100y-xy)/2
Thank you for reading, I hope you enjoyed my answer to the problem!