VIP May 25

[ranjeet_1975]

25. In the xy-plane shown, how many points in the rectangular region, including the sides, have integer coordinates?
(A) 8
(B) 12
(C) 18
(D) 20
(E) 24

Official Answer is

SPOILER: D

What is the process to solve this question

[Shooter]

Perimeter = 2(3+4) = 14 [hence 14 integer points on the sides]

Let us see how many rectangles can we draw inside the original rectangle. We can just draw one more having (1,1) (1,2) (3,1) and (3,2) as it's edges.
perimeter of that rectangle = 2(2+1) = 6

Total = 20.

[12rk34]

If we draw vertical lines from each of the points on x-axis with integer coordinates (i.e. (1,0), (2,0), (3,0), (4,0)), total number of vertical lines including y-axis = 5
Similarly if we draw horizontal lines from (0,1), (0,2), (0,3) we will have total 4 horizontal lines including x-axis.
Now 5 vertical and 4 horizontal lies will intersect one another at 4 x 5 = 20 points.
These 20 points of intersection will have integer coordinates.
Hence D.

[genius_in_the_gene]

5 points along x axis
4 points along y axis

Total 5*4=20