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#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # Points J (3, 1) and K (– 1, – 3) are two vertices of an isosceles tria

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Magoosh GMAT Instructor G
Joined: 28 Dec 2011
Posts: 4487
Points J (3, 1) and K (– 1, – 3) are two vertices of an isosceles tria  [#permalink]

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3
10 00:00

Difficulty:   55% (hard)

Question Stats: 67% (02:46) correct 33% (03:09) wrong based on 140 sessions

### HideShow timer Statistics 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 _________________
Mike McGarry
Magoosh Test Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)
Magoosh GMAT Instructor G
Joined: 28 Dec 2011
Posts: 4487
Re: Points J (3, 1) and K (– 1, – 3) are two vertices of an isosceles tria  [#permalink]

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3
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 _________________
Mike McGarry
Magoosh Test Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)
##### General Discussion
Senior Manager  B
Joined: 28 Feb 2014
Posts: 294
Location: United States
Concentration: Strategy, General Management

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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

(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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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

(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
Senior Manager  B
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Concentration: Strategy, General Management
Points J (3, 1) and K (– 1, – 3) are two vertices of an isosceles tria  [#permalink]

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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

(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.
Manager  B
Joined: 16 Mar 2016
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Schools: Tuck '19
GMAT 1: 660 Q48 V33 GMAT 2: 710 Q50 V35 Re: Points J (3, 1) and K (– 1, – 3) are two vertices of an isosceles tria  [#permalink]

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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?
Intern  B
Joined: 02 Dec 2015
Posts: 1
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GMAT 1: 630 Q49 V27 Re: Points J (3, 1) and K (– 1, – 3) are two vertices of an isosceles tria  [#permalink]

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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.
Manager  B
Joined: 16 Mar 2016
Posts: 66
Schools: Tuck '19
GMAT 1: 660 Q48 V33 GMAT 2: 710 Q50 V35 Re: Points J (3, 1) and K (– 1, – 3) are two vertices of an isosceles tria  [#permalink]

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