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Senior Manager  V
Joined: 25 Dec 2018
Posts: 485
Location: India
Concentration: General Management, Finance
GMAT Date: 02-18-2019
GPA: 3.4
WE: Engineering (Consulting)
A triangle is formed drawn on the XY plane by three vertices (0,20), (  [#permalink]

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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
NUS School Moderator V
Joined: 18 Jul 2018
Posts: 1022
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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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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VP  D
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Joined: 01 Oct 2017
Posts: 1017
WE: Supply Chain Management (Energy and Utilities)
Re: A triangle is formed drawn on the XY plane by three vertices (0,20)  [#permalink]

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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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Rise above the storm, you will find the sunshine 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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