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If ABC is an equilateral triangle, AC lies on the x-axis, and A is clo

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If ABC is an equilateral triangle, AC lies on the x-axis, and A is clo  [#permalink]

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New post 10 May 2015, 05:08
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A
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E

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  55% (hard)

Question Stats:

70% (01:51) correct 30% (01:45) wrong based on 148 sessions

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If ABC is an equilateral triangle, AC lies on the x-axis, and A is closer to the origin than C, what are the coordinates of point A?
Image two should just be consultated after you solved the question.

(1) The coordinates of point C are (6,0).
(2) The coordinates of point B are (4,2√3).

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If ABC is an equilateral triangle, AC lies on the x-axis, and A is clo  [#permalink]

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New post 10 May 2015, 17:09
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1
Great geometry question, and one that has a takeaway bigger than the problem itself. Statement 1 is very insufficient here, because knowing coordinate C on the x-axis gives us absolutely no insight into the values of the other coordinates. However, statement 2 subtly gives us all insight into the value of the other coordinates, because not only do we know the coordinates of point B, but we also know that the triangle has a base along the x-axis (from the question stem). This means that statement 2 gives us the triangle's height.

If you break the equilateral triangle up into 2 30:60:90 triangles (a very common construction with equilateral triangles, make sure that you are comfortable with this ratio) then you can utilize the ratio x:x*sqrt3:2x ratio of the sides opposite the 30:60:90 angles. This would make the height (which is opposite the 60 degree angle) equal to x*sqrt3, meaning that x=2, meaning that the distance from the center of the triangle's base to point A (opposite the 30 degree angle) is 2. Because we already know that A is on the x-axis (and therefore has a Y value of 0), and we know that point B has an X value of 4, we therefore know that point A is at (2,0). This makes statement 2 sufficient on its own and thus the answer B.

A big takeaway from this problem - both of the statements together are clearly sufficient. If they tell you that the triangle is equilateral, and then give you two coordinates, nearly everyone will know that you can solve that. Meaning...that is too easy, and most likely a trap. For questions like this, Veritas does a great job of teaching data sufficiency strategy to help test takers avoid falling into these kinds of common traps, and instead dig deeper into the more complicated and probably independently sufficient statement. We have an entire book dedicated to avoiding just these kinds of traps.

I hope this helps!
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Re: If ABC is an equilateral triangle, AC lies on the x-axis, and A is clo  [#permalink]

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New post 11 May 2015, 22:54
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There are two information given in the question which make the question easily solvable:

1.ABC is an equilateral triangle- Its far easier to work with an equilateral triangle. We know all the angles of the triangle and we know the special properties of its altitudes,medians and angle bisectors.

2. Point A lies on x -axis- We just need to find the x-coordinate of the point A as the y-coordinate is 0

Analyze the Given Info
The question tells us about an equilateral triangle ABC one of whose sides AC lie on the x-axis. We are told that point A is closer to the origin than point C and we are asked to find the coordinates of point A. As discussed above we just need to find the x-coordinate of point A. There are two easy visible ways by which we can find the x-coordinate of point A:

a. We know the coordinate of one of the points of triangle ABC and the length of the side of the triangle. As we know the y coordinate of point A, we can use the distance formula to calculate the x coordinate of point A.

b. We know any point on x-axis and its distance from point A.


Let's see if the statements provide us with any of the information.

Analyze statement-I independently
St-I gives us the coordinates of vertex C of triangle ABC. However we don't know the length of the side of the triangle, hence we will not be able to find out the coordinates of point A.

Statement-I is not sufficient to answer the question.

Analyze statement-II independently
St-II tells us about the coordinates of vertex B. Again it is very easy to deduce that since we don't know the length of side of the triangle, we will be unable to find out the coordinates of point A.

This is the point where the properties of equilateral triangle help us get to our answer. Since we know the coordinates of point B, we can find the x-coordinate of the altitude from vertex B to base AC i.e. D. Also we know that angle BAC= 60° and length of altitude BD = y-coordinate of vertex B. Hence in triangle ABD, we can use trigonometric ratios to find out the length of AD. Once we know the length of AD and coordinates of point D, we can find the coordinates of point A.

Image

Alternatively, if we remember the formula for length of the altitude of an equilateral triangle = (√3a)/2, we can calculate the value of 'a' which is the side of the equilateral triangle. Now we know the coordinates of one of the vertices of the triangle and the length of the side of the triangle. These data points can be used to find the coordinates of point A.

Hope its clear!

Regards
Harsh
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Re: If ABC is an equilateral triangle, AC lies on the x-axis, and A is clo  [#permalink]

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New post 12 Apr 2016, 14:53
This question is the perfect trap answer c question
if we analyze closely,then statement 1 gives us no vital information
whereas statement 2 gives us the height of equilateral triangle indirectly from which we can easily find out the co-ordinates of point A which after calculation comes out to be (2,0)
hence statement is clearly sufficient
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Re: If ABC is an equilateral triangle, AC lies on the x-axis, and A is clo  [#permalink]

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Re: If ABC is an equilateral triangle, AC lies on the x-axis, and A is clo   [#permalink] 04 Jul 2017, 06:42
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