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

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Joined: 25 Dec 2018
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A triangle is formed drawn on the XY plane by three vertices (0,20), (  [#permalink]

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05 Mar 2019, 07:22
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5
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Difficulty:

55% (hard)

Question Stats:

53% (02:14) correct 47% (01:29) wrong based on 17 sessions

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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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WE: Engineering (Energy and Utilities)
Re: A triangle is formed drawn on the XY plane by three vertices (0,20), (  [#permalink]

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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

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

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05 Mar 2019, 21:14
2
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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