JEE-Mains 2015 (18/30)

Geometry Level 3

The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices ( 0 , 0 ) (0,0) , ( 0 , 41 ) (0,41) and ( 41 , 0 ) (41,0) is :

820 901 780 861

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Sandeep Bhardwaj by Sandeep Bhardwaj , 28, India

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2 solutions

Srivardhan Srb
Jun 15, 2015

If we consider as a graphical model of 1block=1unit of area each, then (no of integer points=Area of the traingle - Hypotenuse length)

It's a right angled triangle => area=0.5bh= (41x41)/2 ~ 840 Hypotenuse=\sqrt{41^2 + 41^2} ~ 60

Ans=840-60=780

we're subtracting hypotenuse 'cause it crosses through the centers of the blocks of that side whose area is also calculated

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Yash Singh
Jun 14, 2015

it could be done easily by first making the triangle , then completing it in a square , calculating the number of integer coordinates as its area , dividing the result by 2 , subtracting the pints on the sides of the triangle

it will be an AP in the number of points 39+38+37+36+35...............................................1 39 terms Sn=39/2[(39x2) -(39-1)(-1)]=780

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