The coordinates of points A and C are (0, -3) and (3, 3)

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PGMAT: Senior | Next Rank: 100 Posts; Posts: 73; Joined: Fri Jul 23, 2010 1:30 pm

by PGMAT » Wed Jul 18, 2012 9:08 pm

The coordinates of points A and C are (0, -3) and (3, 3), respectively. If point B lies on line AC between points A and C, and if AB = 2BC, which of the following represents the coordinates of point B?

(1, -)
(1, -1)
(2, 1)
(1.5, 0)
(,)

what is the easiest way to solve this?

[spoiler]OA:C[/spoiler]

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Source: Beat The GMAT — Problem Solving | Wed Jul 18, 2012 9:08 pm

eagleeye: Legendary Member; Posts: 520; Joined: Sat Apr 28, 2012 9:12 pmFollowed by: 49 members; Thanked: 339 times; GMAT Score:770

by eagleeye » Wed Jul 18, 2012 9:28 pm

PGMAT wrote:The coordinates of points A and C are (0, -3) and (3, 3), respectively. If point B lies on line AC between points A and C, and if AB = 2BC, which of the following represents the coordinates of point B?

(1, -)
(1, -1)
(2, 1)
(1.5, 0)
(,)

what is the easiest way to solve this?

[spoiler]OA:C[/spoiler]

This is the fastest and easiest way to solve this:

since B divides A-C in ratio 2:1, B = (1*(A)+2*(C))/(1+2) =( 1*(0,-3) + 2*(3,3))/3
= ((0,-3)+(6,6))/3 = ((0+6),(-3+6))/3 = (6,3)/3 = (2,1)

Let me know if this helps

Last edited by eagleeye on Thu Jul 19, 2012 5:51 pm, edited 1 time in total.

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Thu Jul 19, 2012 12:30 pm

The coordinates of points A and C are (0, -3) and (3, 3), respectively. If point B lies on line AC between points A and C, and if AB = 2BC, which of the following represents the coordinates of point B?

A. (1, -Â�5)
B. (1, -1)
C. (2, 1)
D. (1.5, 0)
E. (Â�5, Â�5)

DRAW the figure and LOOK at the answer choices.

Since BC = (1/2)AC, point C must be closer to B than to A.
As the figure shows, point C must have a positive x-coordinate less than 3 and a positive y-coordinate less than 3.

The correct answer is C.

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PGMAT: Senior | Next Rank: 100 Posts; Posts: 73; Joined: Fri Jul 23, 2010 1:30 pm

by PGMAT » Thu Jul 19, 2012 3:54 pm

eagleeye wrote:
PGMAT wrote:The coordinates of points A and C are (0, -3) and (3, 3), respectively. If point B lies on line AC between points A and C, and if AB = 2BC, which of the following represents the coordinates of point B?

(1, -)
(1, -1)
(2, 1)
(1.5, 0)
(,)

what is the easiest way to solve this?

[spoiler]OA:C[/spoiler]
This is the fastest and easiest way to solve this:

since B divides A-C in ratio 1:2, B = (1*(A)+2*(C))/(1+2) =( 1*(0,-3) + 2*(3,3))/3
= ((0,-3)+(6,6))/3 = ((0+6),(-3+6))/3 = (6,3)/3 = (2,1)

Let me know if this helps

Hi, thank you for your response. If the question says AB = 2BC, doesn't this imply that point B is dividing the segment in 2:1? A----B--C

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eagleeye: Legendary Member; Posts: 520; Joined: Sat Apr 28, 2012 9:12 pmFollowed by: 49 members; Thanked: 339 times; GMAT Score:770

by eagleeye » Thu Jul 19, 2012 6:01 pm

PGMAT wrote:
eagleeye wrote:
PGMAT wrote:The coordinates of points A and C are (0, -3) and (3, 3), respectively. If point B lies on line AC between points A and C, and if AB = 2BC, which of the following represents the coordinates of point B?

(1, -)
(1, -1)
(2, 1)
(1.5, 0)
(,)

what is the easiest way to solve this?

[spoiler]OA:C[/spoiler]
This is the fastest and easiest way to solve this:

since B divides A-C in ratio 1:2, B = (1*(A)+2*(C))/(1+2) =( 1*(0,-3) + 2*(3,3))/3
= ((0,-3)+(6,6))/3 = ((0+6),(-3+6))/3 = (6,3)/3 = (2,1)

Let me know if this helps
Hi, thank you for your response. If the question says AB = 2BC, doesn't this imply that point B is dividing the segment in 2:1? A----B--C

You are right PGMAT: It was a typo. I meant to write 2:1.

Here's the general formula to do it.

If the line joining the point A(a1,a2) and B(b1,b2) is divided in the ratio m:n by point C, then coordinates of C are: (n*A+m*B)/(m+n)

Just to solidify the concept, if C divides the line-segment joining the points (8, 9) and (-7, 4) in the ratio 2 : 3,
coordinates of C are: (3*(8,9) + 2*(-7,4))/(2+3) = 1/5*((24,27)+(-14,8)) = 1/5*(24-14,27+8) = 1/5*(10,35) = (2,7).

If it makes it any easier, you could remember this too, for point C dividing two points into ratio m:n, C = (n*first point + m*second point)/(m+n).

Let me know if this helps

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