Question

An integral point means that both coordinates of the point are integers. The number of integral points exactly in the interior of the triangle with vertices (0, 0), (0, 21) and (21, 0) (see Fig) is

(A)  133

(B)  190

(C)  233

(D)  105

Answer 1

Correct option  (b)  190

Explanation :

The integral points must be on the vertical lines x = 1, 2, 3, …, 20. The number of integral points on x = 1 inside the triangle are (1, 1), (1, 2), (1, 3), …, (1, 19) (total number is 19). Similarly, the number of points on x = 2 is 18, on x = 3 is 17, etc.

Finally, the number of points on x = 19 is 1 and on x = 20 is 0. 

Therefore, the total number of integral points inside the triangle is

19 + 18 + 17 + .... + 1 + 0 = 19 x 20/2 = 190