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

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Manager
Joined: 25 Dec 2018
Posts: 129
Location: India
Concentration: General Management, Finance
GMAT 1: 590 Q48 V23
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WE: Engineering (Consulting)
A triangle is formed drawn on the XY plane by three vertices (0,20), (  [#permalink]

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05 Mar 2019, 07:22
3
00:00

Difficulty:

45% (medium)

Question Stats:

56% (01:54) correct 44% (01:38) wrong based on 9 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
Director
Joined: 18 Jul 2018
Posts: 752
Location: India
Concentration: Finance, Marketing
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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Director
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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
1
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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