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A triangle is formed drawn on the XY plane by three vertices (0,20), (

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A triangle is formed drawn on the XY plane by three vertices (0,20), (  [#permalink]

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New post 05 Mar 2019, 07:22
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A triangle is formed drawn on the XY plane by three vertices (0,20), (0,0) and (20,0). The number of points with integer coordinates inside the triangle (excluding all the points on the boundary) are?

A) 171
B) 190
C) 210
D) 420
E) 240
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Re: A triangle is formed drawn on the XY plane by three vertices (0,20), (  [#permalink]

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New post 05 Mar 2019, 09:56
We have to exclude the boundaries, then we should start with (1,1) to (1,19), i.e, 18 points.
next is (2,1) to (2,18), i.e, 17 points.
In doing so, we'll get 18+17+16+......+1
sum of n consecutive integers is given by n(n+1)/2 = 18(19)/2 = 19*9 = 171

A is the answer.
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Re: A triangle is formed drawn on the XY plane by three vertices (0,20)  [#permalink]

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New post 05 Mar 2019, 21:14
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mangamma wrote:
A triangle is formed drawn on the XY plane by three vertices (0,20), (0,0) and (20,0). The number of points with integer coordinates inside the triangle (excluding all the points on the boundary) are?

A. 171
B. 190
C. 210
D. 420
E. 240


Here x+y<20 with x>0, y>0.
Since we have to find out the integer co-ordinates, hence max(x)=18, min(x)=1, max(y)=18, min(y)=1

1) x+y=19; possible combinations of (x,y): (1,18), (2,17),...........(18,1) [Total pairs=18]
2) x+y=18; possible combinations of (x,y): (1,17), (2,16),...........(17,1) [Total pairs=17]
.
.
.
18) x+y=2; possible combinations of (x,y): (1,1) [Total pairs=1]

Therefore, the total no of integer co-ordinate points(excluding the boundaries):

18+17+............+1
\(=\frac{18(18+1)}{2}\)
\(=\frac{18*19}{2}\)
=9*19
=171

Ans. (A)
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Re: A triangle is formed drawn on the XY plane by three vertices (0,20)   [#permalink] 05 Mar 2019, 21:14
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