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

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

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New post 18 Feb 2012, 17:31
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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)

How this can be solved guys?
[Reveal] Spoiler: OA

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

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enigma123 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)

How this can be solved guys?


Even though it's possible to set an algebraic equation to get the answer, it would be much easier to draw the line segment AC and you will literally see the answer:
Attachment:
The coordinates.PNG
The coordinates.PNG [ 14.83 KiB | Viewed 28238 times ]
Since AB is twice the length of BC then the only acceptable choices is B (2, 1). Two points (1, -5) and (5, 5) does not lie on AC at all, (1.-1) is closer to A than to C and (1.5, 0) divides AC in half.

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

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New post 18 Feb 2012, 18:01
Thanks Bunuel - I didn't get it sorry. How did you arrive at 2 as the co-ordinate for B? Sorry again. Also, I was trying to solve this by using the distance formula for AC.
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Re: The coordinates of points A and C are (0, -3) and (3, 3) [#permalink]

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New post 18 Feb 2012, 18:05
enigma123 wrote:
Thanks Bunuel - I didn't get it sorry. How did you arrive at 2 as the co-ordinate for B? Sorry again. Also, I was trying to solve this by using the distance formula for AC.


I just put all five points on a plane and saw that the only acceptable answer is C (2, 1). Look at other 4 points (blue) on the diagram and read my explanation: neither of them can divided AC into ratio 2:1.
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Re: The coordinates of points A and C are (0, -3) and (3, 3) [#permalink]

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New post 03 Sep 2013, 10:24
Bunuel wrote:
enigma123 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)

How this can be solved guys?


Even though it's possible to set an algebraic equation to get the answer, it would be much easier to draw the line segment AC and you will literally see the answer:
Attachment:
The coordinates.PNG
Since AB is twice the length of BC then the only acceptable choices is C (2, 1). Two points (1, -5) and (5, 5) does not lie on AC at all, (1.-1) is closer to A than to C and (1.5 0) divides AC in half.

Answer: C.



question says AB=2AC and B is the in between point of AC. it can be inferred from that B is the middle point of AC. So, all it need to find out the midpoint of AC.

M1= 3-0/2 = 1.5
M2= 3-3/2 = 0

So, B (1.5, 0)

If I am wrong please correct me.

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

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New post 03 Sep 2013, 10:30
zachowdhury wrote:
Bunuel wrote:
enigma123 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)

How this can be solved guys?


Even though it's possible to set an algebraic equation to get the answer, it would be much easier to draw the line segment AC and you will literally see the answer:
Attachment:
The coordinates.PNG
Since AB is twice the length of BC then the only acceptable choices is C (2, 1). Two points (1, -5) and (5, 5) does not lie on AC at all, (1.-1) is closer to A than to C and (1.5 0) divides AC in half.

Answer: C.



question says AB=2AC and B is the in between point of AC. it can be inferred from that B is the middle point of AC. So, all it need to find out the midpoint of AC.

M1= 3-0/2 = 1.5
M2= 3-3/2 = 0

So, B (1.5, 0)

If I am wrong please correct me.


The questions says that AB = 2BC, not that AB=2AC.
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Re: The coordinates of points A and C are (0, -3) and (3, 3) [#permalink]

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New post 03 Sep 2013, 16:51
How would you find the coordinates algebraically?

I've done problems where AB=3BC, so you can easily find the average of two points, then again average the midpoint and C to find the answer, but that is not possible for a problem like this where the ratio of distance is 2:1.

Thanks

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

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New post 03 Sep 2013, 21:14
enigma123 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)

How this can be solved guys?



For the Graph,please look at Bunuel's post.

We can find distance between points A & C using the distance formula : \sqrt{(x2-x1)^2 +(y2-y1)^2}

We get \sqrt{(3-0)^2+(3-(-3))^2} ------> \sqrt{45}

Now at point B ---> AB= 2 BC ------> AB+BC =\sqrt{45} or 3BC =\sqrt{45} ----> BC =\sqrt{5}
So distance from C to B will be \sqrt{5}

Now from answer choice, we can see only option C gives as \sqrt{5} as the distance between Point B and C

:)

Bunuel's reply is more crisp and time saving.
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Re: The coordinates of points A and C are (0, -3) and (3, 3) [#permalink]

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Refer the first case (i.e. Formula for INTERNAL DIVISION)

In our case \(\frac{m}{n} = \frac{2}{1}\) AND (x1 y1) = (0,-3) (x2 y2) = (3, 3)

X Coordinate = \(\frac{2(3)+0}{(1+2)}\) ----------> \(\frac{6}{3}\) ---------> 2

y Coordinate = \(\frac{2(3)+(-3)}{(1+2)}\) ----------> \(\frac{3}{3}\) ---------> 1

Hence B (x y) = 2, 1

Refer my this article for brief understanding of this concept.
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Untitled.png [ 53.89 KiB | Viewed 27415 times ]


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

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New post 21 Oct 2013, 09:10
Bunuel wrote:
AccipiterQ 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, -\(\sqrt{5}\))
B (1, -1)
C (2, 1)
D (1.5, 0)
E (\(\sqrt{5}\),\(\sqrt{5}\))

could not get this one for the life of me; I calculated what the length of AC was, but after that didn't know how to solve

[Reveal] Spoiler: OE
Point B is on line AC, two-thirds of the way between Point A and Point C. To find the coordinates of point B, it is helpful to imagine that you are a point traveling along line AC.

When you travel all the way from point A to point C, your x-coordinate changes 3 units (from x = 0 to x = 3). Two-thirds of the way there, at point B, your x-coordinate will have changed 2/3 of this amount, i.e. 2 units. The x-coordinate of B is therefore x = 0 + 2 = 2.
When you travel all the way from point A to point C, your y-coordinate changes 6 units (from y = -3 to y = 3). Two-thirds of the way there, at point B, your y-coordinate will have changed 2/3 of this amount, i.e. 4 units. The y-coordinate of B is therefore y = -3 + 4 = 1.

Thus, the coordinates of point B are (2,1).

The correct answer is C.


Merging similar topics. Please refer to the solutions above.


Similar question to practice:
in-the-rectangular-coordinate-system-above-the-line-y-x-144774.html
in-the-xy-coordinate-plane-is-point-r-equidistant-from-143502.html
in-the-rectangular-coordinate-system-the-line-y-x-is-the-132646.html
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Hope it helps.
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Re: The coordinates of points A and C are (0, -3) and (3, 3) [#permalink]

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New post 24 Jan 2014, 03:47
y = mx + n

n = -3 (from point A)
3 = 3m -3 (from point C)
m=2
Equation of line AC: y = 2x - 3

Now put in the given answers. None works except from C and D. Both points lie on the line. From the question we know that AB = 2BC. So lets calculate the middle of the line AC. (Xa+Xb)/2 = 0+3 / 2 = 1.5 and (Ya+Yb)/2 = -3+3 / 2 = 0

SO we have M = (1.5;0) which is answer D. Hence answer C is correct.

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

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New post 22 Mar 2015, 08:52
enigma123 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)

How this can be solved guys?



I think this question can be solved without pen and paper .
if B were at mid of A and C the coordinates of B were (1.5,0) but we are told that B is closer to C so definitely X coordinate will be more than 1.5 , only 1 point in list of options makes sense . option D . Ignore Option E as it lies outside the line segment AC as we are told that 'point B lies on line AC between points A and C'

noticed that there are edits to the option E from (5,5) to ( \(\sqrt{5},\sqrt{5}\)), so possible options are C and E .
Slope of line AC=\(\frac{2}{1}\) slope of AB should be same as slope of AC
slope of AC = \((\sqrt{5}+3)/(\sqrt{5} - 0)\) this cannot be equal to 2.
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The coordinates of points A and C are (0, -3) and (3, 3) [#permalink]

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Want a really long way to solve this? No fancy formulas no estimation on paper.

We have from the known coordinates that the big triangle has sides 3, 6, 3BC.

So, \(3^2+6^2=3BC^2\)

We get that \(bc=\sqrt{5}\)

Note that now we have a smaller triangle

Then, using the proportions given, so AB=2BC, we can determine that the h of the small triangle is 2.
We can also calculate the other side since we know the hypotenuse (\(\sqrt{5}\))

So: \(\sqrt{5}=2^2+x^2\) X=1

Then we can use the same concept of proportions AB=2BC to determine that\(\sqrt{5}\) segment corresponds
to 1\3rd of the big triangle. So, the distance from origin on X-axis to the side (base) of the small triangle is 2. And that means that C (2,y). We got our X.

Now get the linear equation of the hypotenuse of the big triangle. Using the given points A and B, we get that slope=6/3 or 2 and y-intercept is -3

So y=2x-3

Now plug-in the X=2 to get that Y=1

Answer C. Logical deduction + Pythagoras Theorem only.
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Re: The coordinates of points A and C are (0, -3) and (3, 3) [#permalink]

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New post 13 Jul 2016, 22:01
First of all point is closer to C. to get exact coordinates
how much distance from y axis will give us X coordinate and distance from x axis will give us Y coordinate
difference of Y coordinates of two points 3-0= 3
now this distance will be divided in such a way that it will give us AB :BC 2:1 ratio. X coordinate will be two.
similarly Y ccordinate will be 3-(-3) = 6
2:1 is 4 is to 2; from -3 it will 4 points that is 1
(2,1)

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

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New post 11 Jan 2017, 11:09
Let co-ordinate of B = \((x_1, y_1)\)
Therefore, \(x_1 - 0 = 2(3 - x_1)\) ==>\(3(x_1) = 6\) => \(x_1 = 2\)
\(y_1 - (-3) = 2 (3 - y_1)\)==> \(3(y_1) = 6 -3\) ==> \(y_1 = 1\)
Ans.: C

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

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New post 14 Jun 2017, 21:18
enigma123 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)

How this can be solved guys?


For coordinate geometry questions its important to have dry erase grid broad- these questions can be more difficult than they need to be and you can end up doing more math then you really need to without a coordinate plane. An dry erase board with a coordinate plane on it can be purchased here

https://www.amazon.com/Manhattan-GMAT-S ... %2F+Marker

Secondly, the key word in this question is "coordinate." There is no need to find the distance between point a and c and there is no need to draw out similar triangles- with a coordinate plan we could simply draw this on the board- now, if AB=2BC then that simply means point B would 2/3 of the height and thus

C

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

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New post 16 Sep 2017, 06:48
enigma123 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)

How this can be solved guys?



The length of AC = √((x1-x2)^2 )+(y1-y2)^2 = √((0-3)^2 )+(-3-3)^2 = √45 = 3√5
From the question AB = 2BC, this means that AB = 2√5 and BC = √5
By applying the length formula In the multiple choices, the correct answer should give us the right lengths of AB and BC.
Choice C does that:
AB = √((x1-x2)^2 )+(y1-y2)^2 = √((0-2)^2 )+(-3-1)^2 = √20 = 2√5
BC = = √((x1-x2)^2 )+(y1-y2)^2 = √((3-2)^2 )+(3-1)^2 = √5

Kudos [?]: 2 [0], given: 13

The coordinates of points A and C are (0, -3) and (3, 3)   [#permalink] 16 Sep 2017, 06:48
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The coordinates of points A and C are (0, -3) and (3, 3)

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