Many a time, you would have been stumped by questions that ask you to calculate the number of points with integral coordinates inside an enclosed region. Let’s see how to solve these questions using a shorter method called the Pick’s theorem.

For an uninitiated candidates, the normal way of doing this is to plot the graph and then search for the points with integral coordinates. But, the method relies on observation and one is probably not in the best position to ‘solve’ something during the test. Let’s figure this out with the help of a question:

Number of points with integer coordinates which lie inside the triangle whose vertices are (0,0) , (0,21) and (21,0)?

Step 1: Find the area of the figure, triangle ABO in this case. You can easily find the area using the ½ * base * height as there is a right angle involved. If not, you can use the Shoelace theorem to find the area (A) easily

Step 2: Calculate the number of points on the sides of the closed figure. Generally, you will have at least one side on the X-axis or Y-axis to make life easier for you. The other side/s, you will have to form equations of the line/s and figure out the number of integral solutions to the same.

I would suggest noting down the vertices first to avoid double counting or missing out on a few points. Then you can find the number of integral solutions to the various lines and find the total number of border points (B)

Step 3: Calculate the number of points with integral coordinates that lie inside the figure (I) using the formula:

So, in the above question, we can proceed as follows:

Area = ½ * 21 * 21 = 220.5 square units

Vertices with integral coordinates = 3

Border points with integral coordinates on AO = 20

Border points with integral coordinates on BO = 20

To find number of border points on AB, we would need the equation of line AB which is given by x+y = 21

Number of integral solutions except (0,21) and (21,0) is 20

Total number of border points with integral coordinates is 3+20+20+20 = 63

So, we get:

If this type of question does appear in CAT, it would primarily be to differentiate among the very best of aspirants. Unless you have some time on your hands, it doesn’t make any sense to attempt this in the first go. However, once you have figured out all the sitters, you can definitely go for this type using the Pick’s theorem.

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