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

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The coordinates of points A and C are (0, -3) and (3, 3), respectively  [#permalink]

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New post 31 Jul 2011, 08:20
2
00:00
A
B
C
D
E

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  65% (hard)

Question Stats:

58% (01:47) correct 42% (02:15) wrong based on 78 sessions

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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, -√5)
B. (1, -1)
C. (2, 1)
D. (1.5, 0)
E. (√5, √5)

Spoiler: :: My take
No OA provided. I got the answer as B. The approach i adopted:

Found the distance between AC. Then split the distance in 1:2 ratio. Then using the distance formula solved for AB and BC, to find the coordinates of B.

Is this the best approach, or is there a better approach
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Re: The coordinates of points A and C are (0, -3) and (3, 3), respectively  [#permalink]

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New post 31 Jul 2011, 08:36
I used similar approach, but you don't really need to find the distance between points A and C. For coordinate geometry problems I usually find it useful to quickly draw the graphic so you realize fast how to solve it.
In this case, the distance on x is 3, so 1/3 between 0 and 3 will be 1. And the "height" between A and C is 6, so 1/3 is 2 from the bottom. -3+2=-1. The point is (1,-1).
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Re: The coordinates of points A and C are (0, -3) and (3, 3), respectively  [#permalink]

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New post 31 Jul 2011, 19:45
i got answer D

B is at the mid point on line AC, that is why 2BC = AC

find distance AD. Imagine a right triangle with hypotenuse = root base^2 + Altitude ^2

here base = distance on X axis = distance b/w X = 0 and X = 3. Base = 3
Altitude = distance on Y asis = distance b/w Y = - 3 and Y = 3. Altitude = 6

applying the formula we get AC = 3 root 5

now half of AC should be 3/2 root 5

the only answer that satisfy is D. ( calculate BC using the same approach above)
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Re: The coordinates of points A and C are (0, -3) and (3, 3), respectively  [#permalink]

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New post 01 Aug 2011, 06:56
1
manishgeorge,

there is no way D could be the answer. the stem of the question does not say that B is at the mid point on the line AC, and 2BC can never be = AC.

If AB = 2BC, it means that the AB segment is two times the length of BC. Or if you divide the segment AC in 3, two portions will correspond to AB, and only one to BC.

See the picture below. You don't need to know the distances on the hypotenuse, but rather the distances in the X axis. The blue portion means the distance AB, in the X axis from 0 to 2. The green portion is BC, in the X axis from 2 to 3. That way, two times BC will be the same as AB.

Now, graphically you can see that the point B is located in (2,1), then the correct answer is C (and I correct myself from my previous post).
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AB 2BC.jpg [ 10.14 KiB | Viewed 3329 times ]

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Re: The coordinates of points A and C are (0, -3) and (3, 3), respectively  [#permalink]

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New post 01 Aug 2011, 08:59
BTW, you don't need the distance formula to solve this. The base and height of the Pythagorean triangle formed by (0, -3), (3, -3) and (3, 3) are both divisible by 3. Two thirds of the base = 2 and two thirds of the height = 4. (0, -3) + (2, 4) = (2, 1). C.
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Re: The coordinates of points A and C are (0, -3) and (3, 3), respectively  [#permalink]

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New post 19 Aug 2011, 20:44
the x and y coordinates of a point which dived the line with endpoints (x1,y1 ) and (x2, y2 ) in the ratio r:s is
((rX2+sX2)/(r+s) , (rY2+sY1)/(r+s) )
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Re: The coordinates of points A and C are (0, -3) and (3, 3), respectively  [#permalink]

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New post 02 Sep 2011, 01:48
Point B divides AC in the ratio of 2:1 internally. We can use the section formula to get the coordinates of point B. The section formula says that if a point X divides the line joining A(x1,y1) and B(x2,y2) in the ratio m:n, then the coordinates of X are {m(x2)+n(x1)}/m+n, {m(y2)+n(y1)}/m+n

Therefore the coordinates of point B are {2(3) + 1(0)}/3, {2(3)+1(-3)}/3 = (2,1)
These are the coordinates of option (C), which is the correct answer.
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Re: The coordinates of points A and C are (0, -3) and (3, 3), respectively  [#permalink]

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New post 02 Sep 2011, 06:38
Hi GMAT Club! coordinate Geometry is Tough for me. Will you please suggest me the best way to crack Coordinate Geometry?
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Re: The coordinates of points A and C are (0, -3) and (3, 3), respectively  [#permalink]

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New post 02 Sep 2011, 07:35
bholakc wrote:
Hi GMAT Club! coordinate Geometry is Tough for me. Will you please suggest me the best way to crack Coordinate Geometry?


Please go through this at least:
math-coordinate-geometry-87652.html
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The coordinates of points A and C are (0, -3) and (3, 3), respectively  [#permalink]

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New post 31 Oct 2017, 17:36
ghostdude 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?

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


[/spoiler]


Plz not be bothered by my poor drawing :D
We can find the answer without actually calculating C coordination.
In the below picture, let's G be the point at (0,3), F (3,0) and E the interception between AC and Ox.
In triangle AGC, we have OG = OA = 3 and G is 90 degree
=> OE is the midsegment, and E should divide AC into 2 equal parts.
The midsegment is always half the length of its third side, so we have OE = (1/2)GC
As GC = OF, so OE = (1/2) OF = 3/2 = 1.5
OE = 1.5 mean E has coordination of (1.5, 0)
Now here's is the interesting part.
Compare:
AB = 2 BC
AE = EC
=> the x-coordinate of B should be between 1.5 and 3.
=> We can rule out option A, B and D.
Now we check C & E.
Option E: x-coordinate = y-coordinate = \sqrt{5} => B will lie on line segment y=x that go through origin.
Look, point C also has x-coordinate = y-coordinate, so it means O, C, B, A will be on the same straight line, which is obviously wrong.
So we are left with C as the final answer.
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Re: The coordinates of points A and C are (0, -3) and (3, 3), respectively  [#permalink]

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New post 31 Oct 2017, 20:46
IMO Option C
Point B divides AC in the ratio of 2:1 internally.
Therefore the coordinates of point B are {2(3) + 1(0)}/3, {2(3)+1(-3)}/3 = (2,1)
These are the coordinates of option (C), which is the correct answer.
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Re: The coordinates of points A and C are (0, -3) and (3, 3), respectively  [#permalink]

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New post 31 Oct 2017, 21:04
There is a section formula for this ..
(Mx2 + Nx1)/M+N where X1 and X2 is coordinate , ratio = M:N
(My2 + NY2)/M+N
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Re: The coordinates of points A and C are (0, -3) and (3, 3), respectively  [#permalink]

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