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Lines k and l intersect in the coordinate plane at point (3, [#permalink]

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30 Nov 2012, 15:57

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Lines k and l intersect in the coordinate plane at point (3, –2). Is the largest angle formed at the intersection between these two lines greater than 90°?

(1) Lines k and l have positive y-axis intercepts. (2) The distance between the y-axis intercepts of lines k and l is 5.

Line k and l intersect in the coordinate plane at point (3,-2).Is the largest angle formed at the intersection between these two lines greater than 90 degree? (1)Lines k and l have positive y- axis intercepts (2) The distance between the y-axis intercepts of lines k and l is 5

First of all, if I may make a couple suggestions. This problem, like any PS problem, should be posted in the Math/PS part of the forum, not in the General GMAT Questions and Strategies section. Also, you indicate neither a source nor the OA --- these are very helpful for folks who practice on GC.

I'm happy to solve this. I've attached a full solution with diagrams in the pdf.

Lines k and l intersect in the coordinate plane [#permalink]

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13 Dec 2012, 11:48

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Lines k and l intersect in the coordinate plane at point (3, –2 ). Is the largest angle formed at the intersection between these two lines greater than 90°?

(1) Lines k and l have positive y-axis intercepts. (2) The distance between the y-axis intercepts of lines k and l is 5. _________________

Line k and l intersect in the coordinate plane at point (3,-2).Is the largest angle formed at the intersection between these two lines greater than 90 degree? (1)Lines k and l have positive y- axis intercepts (2) The distance between the y-axis intercepts of lines k and l is 5 If anyone find my post helpful in gmat prep plz click kudos

It's a good question. We can solve it by visualizing the situation.

First of all, whenever two lines intersect, the four angles between them make up 360 degrees. Either the lines can be perpendicular in which case all angles are 90 degrees or the lines are not perpendicular in which case two angles are less than 90 and other two are more than 90. So all we have to figure out is whether the lines can be perpendicular (in which case there is no angle greater than 90). In every other case, there will be an angle greater than 90.

(1)Lines k and l have positive y- axis intercepts The black dotted lines show perpendicular lines. You can move them around as you want keeping the angle constant; there is no way both lines will have positive y intercept.

Attachment:

Ques4.jpg [ 12.79 KiB | Viewed 3506 times ]

Also, recall that the product of slopes of two perpendicular lines is -1. If one line has positive slope, the other must have negative slope. We can say that the lines cannot be perpendicular. Hence, there must be two angles greater than 90. Sufficient.

(2) The distance between the y-axis intercepts of lines k and l is 5 A little trickier. Let's try to figure out whether two perpendicular lines meeting at (3, -2) can have a distance of 5 between their y intercepts. Notice the purple and green lines in the figure above. The distance between the y intercepts is quite large. The distance will be smallest when the two line segments are of equal length.

Attachment:

Ques5.jpg [ 8.08 KiB | Viewed 3502 times ]

In the isosceles right triangle shown above, let's try to find x so that we know the minimum value of distance between the y intercepts.

\((1/2)*3*\sqrt{2}x = (1/2)*x*x\) (Area of the triangle using altitude = Area of triangle using two sides) \(x = 3\sqrt{2}\)

The distance between the y intercepts must be at least \(\sqrt{2}x\) i.e. 6.

Hence we can say the lines cannot be perpendicular. Sufficient.

Re: Lines k and l intersect in the coordinate plane at point (3, [#permalink]

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29 Dec 2012, 02:26

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Expert's post

Responding to a pm:

In statement 2, I am trying to figure out whether two perpendicular lines meeting at (3, -2) can have a distance of 5 between their points of y intercepts. I draw the perpendicular lines in various ways. I find that the distance between the points of y intercepts is quite large. Look at the dotted black lines. The horizontal line segment has the shortest length possible. The distance between the y intercepts is infinite (since the vertical line does not intersect the y axis) When you turn this pair towards the purple lines, the lengths of the line segments change. The distance between the y intercepts keeps getting smaller. The vertical line keeps getting shorter and the horizontal line keeps getting longer. At the other extreme is the pair of green lines. The black vertical line has become quite short (its now the green upper line) while the black horizontal line has become very long (its now the green lower line). What can you deduce about the minimum distance between the y intercepts? It must happen when both the line segments are of equal length (as shown in the second diagram). That is why we are considering the isosceles triangle. We find that this minimum distance between the points of y intercept must be 6. If that is the case, the distance between the points of y intercept cannot be 5 in case of perpendicular lines. Therefore, the lines must not be perpendicular.

As for your point: "moreover cant we infer by observation that the only possible slopes as per statement two are - ive and in no case be positive"

Why should this be the case? Slope can be positive if the y intercept is negative. _________________

Re: Lines k and l intersect in the coordinate plane at point (3, [#permalink]

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29 Dec 2012, 02:53

VeritasPrepKarishma wrote:

Responding to a pm:

In statement 2, I am trying to figure out whether two perpendicular lines meeting at (3, -2) can have a distance of 5 between their points of y intercepts. I draw the perpendicular lines in various ways. I find that the distance between the points of y intercepts is quite large. Look at the dotted black lines. The horizontal line segment has the shortest length possible. The distance between the y intercepts is infinite (since the vertical line does not intersect the y axis) When you turn this pair towards the purple lines, the lengths of the line segments change. The distance between the y intercepts keeps getting smaller. The vertical line keeps getting shorter and the horizontal line keeps getting longer. At the other extreme is the pair of green lines. The black vertical line has become quite short (its now the green upper line) while the black horizontal line has become very long (its now the green lower line). What can you deduce about the minimum distance between the y intercepts? It must happen when both the line segments are of equal length (as shown in the second diagram). That is why we are considering the isosceles triangle. We find that this minimum distance between the points of y intercept must be 6. If that is the case, the distance between the points of y intercept cannot be 5 in case of perpendicular lines. Therefore, the lines must not be perpendicular.

As for your point: "moreover cant we infer by observation that the only possible slopes as per statement two are - ive and in no case be positive"

Why should this be the case? Slope can be positive if the y intercept is negative.

Hi Karishma

Thanx for prompt reply I have got a trailing question Excerpt from above explanation:-

"What can you deduce about the minimum distance between the y intercepts? It must happen when both the line segments are of equal length (as shown in the second diagram). That is why we are considering the isosceles triangle."

This is what i want to understand that Is it any property of Isosceles triangle that the unequal side must be of a minimum length???? Eg;- two sides are equal to 4 units than third side must be between 1 to 7 units than how can we assume that it will be less than 4 units as per the example presented by me. Is there any special property of Isosceles triangle that i am missing.

What i can understand from your explanation that you are trying to prove that distance 5 units as presented in statement is not possible hence statement 2 itself is insufficient and wrong. But when we are given a distance than why are you choosing the approach to prove that 5 cannot be the distance..Why cannot we use this distance to prove that the lines are not perpendicular... Hope I have clearly presented my doubt!!!!

Last edited by Archit143 on 04 Feb 2013, 18:45, edited 1 time in total.

Re: Lines k and l intersect in the coordinate plane at point (3, [#permalink]

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29 Dec 2012, 19:28

Expert's post

Archit143 wrote:

Excerpt from above explanation:-

"What can you deduce about the minimum distance between the y intercepts? It must happen when both the line segments are of equal length (as shown in the second diagram). That is why we are considering the isosceles triangle."

This is what i want to understand that Is it any property of Isosceles triangle that the unequal side must be of a minimum length???? Eg;- two sides are equal to 4 units than third side must be between 1 to 7 units than how can we assume that it will be less than 4 units as per the example presented by me. Is there any special property of Isosceles triangle that i am missing.

What i can understand from your explanation that you are trying to prove that distance 5 units as presented in statement is not possible hence statement 2 itself is insufficient and wrong. But when we are given a distance than why are you choosing the approach to prove that 5 cannot be the distance..Why cannot we use this distance to prove that the lines are not perpendicular... Hope I have clearly presented my doubt!!!!

I am not using any property of isosceles triangles. I am inferring that the triangle must be isosceles from the diagram. Look at the way things are at the extremes i.e. when k is parallel to y axis and turn the pair of lines till the other extreme i.e. when l is parallel to y axis. You can easily deduce that the intercept will be smallest when line segments are of equal length.

Any yes, I am assuming that the angle is 90 and then trying to find the distance between the points of y intercept. We know how to deal with right triangles. It is easy to find the distance between the y intercept points. Given length 5, its harder to find the angle.

Mike has taken the approach you are looking for in his solution above. He says that since the distance between the intercepts is 5, the angle cannot be 90. _________________

Re: Lines k and l intersect in the coordinate plane at point (3, [#permalink]

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13 May 2013, 20:36

supri1234 wrote:

Lines k and l intersect in the coordinate plane at point (3, –2 ). Is the largest angle formed at the intersection between these two lines greater than 90°?

(1) Lines k and l have positive y-axis intercepts. (2) The distance between the y-axis intercepts of lines k and l is 5.

I see where mike and karishma are coming from. However, for me the question is asking two things:

1) Are the lines perpendicular and 2)they are not overlapping lines. ( Now i knw their can be arguments about this but it is possible that both the lines overlap each other)

If that's the case i can't see why A give the answer as both the lines can be over-lapping.However B clearly states that lines are not overlapping and can't be perpendicular. Hence B.

Re: Lines k and l intersect in the coordinate plane at point (3, [#permalink]

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13 May 2013, 21:35

supri1234 wrote:

Lines k and l intersect in the coordinate plane at point (3, –2 ). Is the largest angle formed at the intersection between these two lines greater than 90°?

(1) Lines k and l have positive y-axis intercepts. (2) The distance between the y-axis intercepts of lines k and l is 5.

The question is just asking if the two lines are perpendicular.

statement1:- The two lines have postive intercepts ie they intercept above x -axis. As you move the intercept away from the origin the acute angle between the lines will continously increase .this will become maximum if one intercept is at alomst origin and another at infinity.In theis case the acute angle will be TAN inverse(3/2) which is almost 45 so the maximum acute angle possible is 45 .Clearly we cannot get 90 .so this statement is alone enough

statement 2:-Distance between the y-axis intercept of the two lines is 5.

Now lets take a triangle with the distance between the y-axis intercepts of the two triangles as the base .It will only be a right triangle if the t other two sides are 3 and 4 .But the height of this triangle is 3.This cannot be a right triangle .So this alone is sufficient

The answer is D _________________

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Re: Lines k and l intersect in the coordinate plane at point (3, [#permalink]

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13 May 2013, 21:49

Bluelagoon wrote:

supri1234 wrote:

Lines k and l intersect in the coordinate plane at point (3, –2 ). Is the largest angle formed at the intersection between these two lines greater than 90°?

(1) Lines k and l have positive y-axis intercepts. (2) The distance between the y-axis intercepts of lines k and l is 5.

I see where mike and karishma are coming from. However, for me the question is asking two things:

1) Are the lines perpendicular and 2)they are not overlapping lines. ( Now i knw their can be arguments about this but it is possible that both the lines overlap each other)

If that's the case i can't see why A give the answer as both the lines can be over-lapping.However B clearly states that lines are not overlapping and can't be perpendicular. Hence B.

Experts please advise.

Thanks!

If both lines are over lapping then the maximum angle between the lines is 180 degrees. So the answer is yes.If both lines are not overlapping then the maximum angle is greater than 90 .So the answer is yes. so statement A is sufficient. _________________

"Kudos" will help me a lot!!!!!!Please donate some!!!

Completed Official Quant Review OG - Quant

In Progress Official Verbal Review OG 13th ed MGMAT IR AWA Structure

Yet to do 100 700+ SC questions MR Verbal MR Quant

Re: Lines k and l intersect in the coordinate plane at point (3, [#permalink]

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14 May 2013, 09:11

Expert's post

Bluelagoon wrote:

supri1234 wrote:

Lines k and l intersect in the coordinate plane at point (3, –2 ). Is the largest angle formed at the intersection between these two lines greater than 90°?

(1) Lines k and l have positive y-axis intercepts. (2) The distance between the y-axis intercepts of lines k and l is 5.

I see where mike and karishma are coming from. However, for me the question is asking two things:

1) Are the lines perpendicular and 2)they are not overlapping lines. ( Now i knw their can be arguments about this but it is possible that both the lines overlap each other)

If that's the case i can't see why A give the answer as both the lines can be over-lapping.However B clearly states that lines are not overlapping and can't be perpendicular. Hence B.

Experts please advise.

Thanks!

You are given that k and l intersect at a point. It means they are not overlapping. _________________

Re: Lines k and l intersect in the coordinate plane at point (3, [#permalink]

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