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

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31 Jul 2011, 08:20

00:00

A

B

C

D

E

Difficulty:

85% (hard)

Question Stats:

52% (01:41) correct 48% (01:53) wrong based on 52 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)

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

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

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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).

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

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01 Aug 2011, 06:56

1

This post received KUDOS

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).

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

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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.

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

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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.
_________________

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

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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.

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

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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.