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Points J (3, 1) and K (– 1, – 3) are two vertices of an isosceles tria

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Points J (3, 1) and K (– 1, – 3) are two vertices of an isosceles tria  [#permalink]

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New post 08 Dec 2014, 15:56
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Points J (3, 1) and K (– 1, – 3) are two vertices of an isosceles triangle. If L is the third vertex and has a y-coordinate of 6, what is the x-coordinate of L?
(A) –3
(B) –4
(C) –5
(D) –6
(E) –7


For a collection of coordinate geometry practice problems, as well as the OE of this problem, see:
https://magoosh.com/gmat/2014/gmat-prac ... -geometry/

Mike :-)

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Re: Points J (3, 1) and K (– 1, – 3) are two vertices of an isosceles tria  [#permalink]

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New post 17 Aug 2016, 14:34
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3
sudhirgupta93 wrote:
Can someone please explain why we consider JK as the base and not a side that could be equal to other side?
Also, do we have to plug in all answer choices to see which is the correct one by using distance formula or do we get some quadratic equation in terms of x if we equate the two distances KL and JL?

Dear sudhirgupta93,

I'm happy to respond. :-) One important thing to appreciate about GMAT math is that there is more to understand than simply the mathematics itself--there's also the mind of the test maker. GMAT questions, especially harder ones, are designed specifically so that if one sees one particular shortcut, that unfolds very quickly to a solution. The insightful way takes less than 30 seconds. The brute force calculations, such as you are discussing, would take 10+ minutes. That's always the wrong choice on the GMAT.

The insight to this problem is the fact that any two points of the form (a, b) and (-b, -a) are reflections of each other over the line y = - x. A mirror line is a perpendicular bisector of any segment connecting a point with its image. This very naturally makes this two points the endpoints of the base of an isosceles triangle, because the vertex could be anywhere on the mirror line. See:
GMAT Math: Special Properties of the Line y = x

Without that insight, we have no idea, and we would have to spend 10-15 minutes doing a ton of algebra. That approach is a strategic disaster and will not prepare you for doing math on the GMAT. Most harder GMAT Quant questions are about having the right insight, the insight that simplifies the problem.

See the blog article:
How to do GMAT Math Faster

Does all this make sense?
Mike :-)
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Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)
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New post 10 Jun 2015, 18:54
1
Use the distance formula.
The line between (-1,-3) and (3,1) forms the "base" of the isosceles triangle. We just need to find the distance to the apex of the triangle. The two legs, or distances will be equal, so set them equal to each other.
d=sqrt((3-x)^2+(1-6)^2)=sqrt((-1-x)^2+(-3-6)^2)
x=-6

Answer:
(D) –6
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Re: Points J (3, 1) and K (– 1, – 3) are two vertices of an isosceles tria  [#permalink]

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New post 13 Jun 2015, 07:11
1
peachfuzz wrote:
Use the distance formula.
The line between (-1,-3) and (3,1) forms the "base" of the isosceles triangle. We just need to find the distance to the apex of the triangle. The two legs, or distances will be equal, so set them equal to each other.
d=sqrt((3-x)^2+(1-6)^2)=sqrt((-1-x)^2+(-3-6)^2)
x=-6

Answer:
(D) –6


I've handled it the same way and got the same result.

But how do we know that J and K form the base ? My guess is that if it doesn't, we could have other solutions
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Points J (3, 1) and K (– 1, – 3) are two vertices of an isosceles tria  [#permalink]

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New post 13 Jun 2015, 08:38
mike34170 wrote:
peachfuzz wrote:
Use the distance formula.
The line between (-1,-3) and (3,1) forms the "base" of the isosceles triangle. We just need to find the distance to the apex of the triangle. The two legs, or distances will be equal, so set them equal to each other.
d=sqrt((3-x)^2+(1-6)^2)=sqrt((-1-x)^2+(-3-6)^2)
x=-6

Answer:
(D) –6


I've handled it the same way and got the same result.

But how do we know that J and K form the base ? My guess is that if it doesn't, we could have other solutions


Yeah, its because the y coord is fixed at 6. Any other value for y would not result in an isosceles triangle.
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Re: Points J (3, 1) and K (– 1, – 3) are two vertices of an isosceles tria  [#permalink]

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New post 17 Aug 2016, 11:30
Can someone please explain why we consider JK as the base and not a side that could be equal to other side?
Also, do we have to plug in all answer choices to see which is the correct one by using distance formula or do we get some quadratic equation in terms of x if we equate the two distances KL and JL?
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Re: Points J (3, 1) and K (– 1, – 3) are two vertices of an isosceles tria  [#permalink]

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New post 18 Aug 2016, 00:01
Lets assume JK is not considered as base. Then, JL=JK
then, \sqrt{((3-x)^2)+25} = \sqrt{(16+16)}
this doesn't give us integral values for x which is not one of the options.
So, we can consider JL=KL.
The rest is known.
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Re: Points J (3, 1) and K (– 1, – 3) are two vertices of an isosceles tria  [#permalink]

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New post 18 Aug 2016, 02:02
1
mikemcgarry wrote:
sudhirgupta93 wrote:
Can someone please explain why we consider JK as the base and not a side that could be equal to other side?
Also, do we have to plug in all answer choices to see which is the correct one by using distance formula or do we get some quadratic equation in terms of x if we equate the two distances KL and JL?

Dear sudhirgupta93,

I'm happy to respond. :-) One important thing to appreciate about GMAT math is that there is more to understand than simply the mathematics itself--there's also the mind of the test maker. GMAT questions, especially harder ones, are designed specifically so that if one sees one particular shortcut, that unfolds very quickly to a solution. The insightful way takes less than 30 seconds. The brute force calculations, such as you are discussing, would take 10+ minutes. That's always the wrong choice on the GMAT.

The insight to this problem is the fact that any two points of the form (a, b) and (-b, -a) are reflections of each other over the line y = - x. A mirror line is a perpendicular bisector of any segment connecting a point with its image. This very naturally makes this two points the endpoints of the base of an isosceles triangle, because the vertex could be anywhere on the mirror line. See:
GMAT Math: Special Properties of the Line y = x

Without that insight, we have no idea, and we would have to spend 10-15 minutes doing a ton of algebra. That approach is a strategic disaster and will not prepare you for doing math on the GMAT. Most harder GMAT Quant questions are about having the right insight, the insight that simplifies the problem.

See the blog article:
How to do GMAT Math Faster

Does all this make sense?
Mike :-)



Thank you Mr Mike. That was a great insight. I wish we were taught coordinate geometry like that in our school..
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Re: Points J (3, 1) and K (– 1, – 3) are two vertices of an isosceles tria  [#permalink]

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New post 03 Mar 2019, 19:12
mikemcgarry wrote:
sudhirgupta93 wrote:
Can someone please explain why we consider JK as the base and not a side that could be equal to other side?
Also, do we have to plug in all answer choices to see which is the correct one by using distance formula or do we get some quadratic equation in terms of x if we equate the two distances KL and JL?

Dear sudhirgupta93,

I'm happy to respond. :-) One important thing to appreciate about GMAT math is that there is more to understand than simply the mathematics itself--there's also the mind of the test maker. GMAT questions, especially harder ones, are designed specifically so that if one sees one particular shortcut, that unfolds very quickly to a solution. The insightful way takes less than 30 seconds. The brute force calculations, such as you are discussing, would take 10+ minutes. That's always the wrong choice on the GMAT.

The insight to this problem is the fact that any two points of the form (a, b) and (-b, -a) are reflections of each other over the line y = - x. A mirror line is a perpendicular bisector of any segment connecting a point with its image. This very naturally makes this two points the endpoints of the base of an isosceles triangle, because the vertex could be anywhere on the mirror line. See:
GMAT Math: Special Properties of the Line y = x

Without that insight, we have no idea, and we would have to spend 10-15 minutes doing a ton of algebra. That approach is a strategic disaster and will not prepare you for doing math on the GMAT. Most harder GMAT Quant questions are about having the right insight, the insight that simplifies the problem.

See the blog article:
How to do GMAT Math Faster

Does all this make sense?
Mike :-)


mikemcgarry i just read through the blogpost and feel like someone just gave me a key to solve these isosceles questions in coordinate geometry. What a brilliant blogpost and kudos for the easy language.
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Re: Points J (3, 1) and K (– 1, – 3) are two vertices of an isosceles tria   [#permalink] 03 Mar 2019, 19:12
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