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Three of the four vertices of a rectangle in the xy-coordinate plane

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Three of the four vertices of a rectangle in the xy-coordinate plane [#permalink] New post   10 Jun 2015, 04:19
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Three of the four vertices of a rectangle in the xy-coordinate plane are ( –5, 1), ( –4, 4), and (8, 0). What is the fourth vertex?

(A) (–4.5, 2.5)
(B) ( –4, 5)
(C) (6, –2 )
(D) (7, –3 )
(E) (10, 1)
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D
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Re: Three of the four vertices of a rectangle in the xy-coordinate plane [#permalink] New post   10 Jun 2015, 07:59
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Bunuel wrote:
Three of the four vertices of a rectangle in the xy-coordinate plane are ( –5, 1), ( –4, 4), and (8, 0). What is the fourth vertex?

(A) (–4.5, 2.5)
(B) ( –4, 5)
(C) (6, –2 )
(D) (7, –3 )
(E) (10, 1)

Kudos for a correct solution.


ALTERNATE

Length of Diagonals are same in a Rectangle

i.e. Distance Between ( –5, 1) and (8, 0) should be same as Distance between ( –4, 4) and 4th Vertex (x, y)

i.e. \((0-1)^2+(8+5)^2 = (y-4)^2+(x+4)^2\)

i.e. \(1+169 = (y-4)^2+(x+4)^2\)

Check Options:

4th Vertex can only be in either 1st or 4th Quadrant so Option A and B are already out of sync

Option (C) - (6, –2 ): (y-4)^2+(x+4)^2 = (-6)^2 + (10)^2 = 136 so INCORRECT
(D) (7, –3 ): (y-4)^2+(x+4)^2 = (-7)^2 + (11)^2 = 49+121 = 170 so CORRECT

Answer: Option
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D
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Re: Three of the four vertices of a rectangle in the xy-coordinate plane [#permalink] New post   15 Jun 2015, 02:46
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Bunuel wrote:
Three of the four vertices of a rectangle in the xy-coordinate plane are ( –5, 1), ( –4, 4), and (8, 0). What is the fourth vertex?

(A) (–4.5, 2.5)
(B) ( –4, 5)
(C) (6, –2 )
(D) (7, –3 )
(E) (10, 1)

Kudos for a correct solution.


MANHATTAN GMAT OFFICIAL SOLUTION:

Your GMAT scratchpad has a grid; use it to plot the diagram to scale.
Attachment:
2015-06-15_1342.png
2015-06-15_1342.png [ 13.23 KiB | Viewed 32275 times ]


“Eyeball” solution: The corner at ( –4, 4) looks like a right angle, so complete the rectangle with the dashed lines shown. The 4th point must be located approximately where the bigger dot is drawn. The closest answer choice is the point (7, –3 ). Alternatively, you could plot all of the answer choice points and see which one “works” with the three given points.

Alternatively, we could solve by checking the slopes of the solid lines we drew between the given points to prove those lines are perpendicular, that is, to prove we have drawn two sides of the rectangle correctly.

The slope of the long solid line = (4- 0)/(-4 - 8) = -1/3.
The slope of the short solid line = (4 - 1)/(-4 - (-5)) = 3.

The product of these slopes is -1/3*3 = -1, proving that the lines are perpendicular.

Compute the location of the 4th point, using the fact that the short sides have the same slope. The known short side connects the points ( –5, 1) and ( –4, 4). In other words, the bottom left corner is 1 to the left and 3 down from the top left corner. The unknown bottom right corner should therefore be 1 to the left and 3 down from the top right corner, or x = 8 – 1 = 7 and y = 0 – 3 = –3, corresponding to the point (7,–3).

The correct answer is D.
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Re: Three of the four vertices of a rectangle in the xy-coordinate plane [#permalink] New post   13 Jun 2015, 08:15
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Actually this is very easy to be solved if you draw on a paper.

Answer D
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Re: Three of the four vertices of a rectangle in the xy-coordinate plane [#permalink] New post   10 Jun 2015, 07:51
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Bunuel wrote:
Three of the four vertices of a rectangle in the xy-coordinate plane are ( –5, 1), ( –4, 4), and (8, 0). What is the fourth vertex?

(A) (–4.5, 2.5)
(B) ( –4, 5)
(C) (6, –2 )
(D) (7, –3 )
(E) (10, 1)

Kudos for a correct solution.


ALTERNATIVE

Slope of Line joining points A( –5, 1) and B( –4, 4) = (4-1) / [(-4)-(-5)] = 3/1 = m1

Slope of Line joining points B( –4, 4) and C( 8, 0) = (0-4) / [(8)-(-4)] = -4/12 = -1/3 = m2

This also suggests that 4th Point must be in Quadrant IV i.e Option C or D only can be true

i.e. AB and BC lines are perpendicular as m1 * m2 = -1

Slope of Line joining points A(–5, 1) and D( x, y) = (y-1) / (x+5) = m3

Slope of Line joining points C(8, 0) and D( x, y) = (y-0) / (x-8) = m4

Now m3 * m4 = -1

i.e. [(y-1) / (x+5)] * [(y-0) / (x-8)] = -1

i.e \(y(y-1) = -(x+5)*(x-8)\)

Let's Check Option C (6, -2)
LHS = (-2)(-2-1) = 6

RHS = (6+5)(6-8) = -22

INCORRECT

Let's Check Option D (7, -3)
LHS = (-3)(-3-1) = 12

RHS = -(7+5)(7-8) = 12

CORRECT

Answer: Option
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D
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Re: Three of the four vertices of a rectangle in the xy-coordinate plane [#permalink] New post   13 Jun 2015, 06:43
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Draw it on the paper, it takes a few seconds to "guess" the correct answer.

Then we could check the result with the formula for the Distance between two points.

Distance btw (-5;1) and (-4;4) is Sqrt(10)

then the correct point must be distant from (8;0) by Sqrt(10)
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Re: Three of the four vertices of a rectangle in the xy-coordinate plane [#permalink] New post   15 Oct 2016, 20:14
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Just derived - The correct formula is
x coordinate- ( x1+x3-x2) and y coordinate is (y1+y3-y2).
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Re: Three of the four vertices of a rectangle in the xy-coordinate plane [#permalink] New post   13 May 2018, 09:34
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As it’s a rectangle, opposite sides are equal and parallel.
So the slope of the opposites is going to be same m1=m2= 0-4/8+4 =-1/3
As we have the slope and (-5,1) eq of line obtained: 3y+2x=-2.
As it’s already clear that the point will lying in the fourth quadrant, so eliminate options and substitue it on the eq of the line.
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Re: Three of the four vertices of a rectangle in the xy-coordinate plane [#permalink] New post   06 Jul 2021, 06:50
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Using slope we can solve it. Let required point be (x,y)
As these are vertices of rectangle, the line connecting (-5,1) and (-4,4) is parallel to (8,0) and (x,y).So, the slope should be the same.
Now instead of calculating the slope, from the point (-5,1) and (-4,4) we know that for every 1 unit increase in x there is 3 units increase in y.
It also implies for every 1 unit decrease in x there will be 3 units decrease in y.
From (8,0) we get (7,-3) that is as we decrease 1 unit in x , 3 units in y is decreased.
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Three of the four vertices of a rectangle in the xy-coordinate plane [#permalink] New post   Updated on: 24 Aug 2023, 06:01
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Jaychoudhary wrote:
KarishmaB Bunuel
I have understood the solutions explained above but I tried to solve it by creating the line equation for (-5,1) (x,y) and (8,0)(x,y) and then putting the 2 equations equal .i.e, where they intersect which should give the unknowns as per my understanding. However , I am not getting the correct answer. Please help. Is it a wrong approach?


Yes, you can form equations of the lines on which the fourth point will lie and find their point of intersection to get the fourth point.

But how do you form the equation of the lines? Note that knowing only one point on which it lies does not give us an equation of the line. We either need two points or we need one point and slope.

One line will pass through (8, 0) and will have the same slope as the line passing through the two points (-5, 1) and (-4, 4). So you first find the slope of the line passing through (-5, 1) and (-4, 4) and then use that slope to find the equation of the line passing through (8, 0).

The other line will pass through (-5, 1) and will have the same slope as the line passing through (-4, 4) and (8, 0).So you first find the slope of the line passing through (-4, 4) and (8, 0) and then use that slope to find the equation of the line passing through (-5, 1).

This will give you the equations of the two lines. Then you can find their point of intersection.

Originally posted by KarishmaB on 11 Jan 2022, 23:11.
Last edited by KarishmaB on 24 Aug 2023, 06:01, edited 1 time in total.
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Re: Three of the four vertices of a rectangle in the xy-coordinate plane [#permalink] New post   28 Jun 2015, 12:19
Bunuel wrote:
Bunuel wrote:
Three of the four vertices of a rectangle in the xy-coordinate plane are ( –5, 1), ( –4, 4), and (8, 0). What is the fourth vertex?

(A) (–4.5, 2.5)
(B) ( –4, 5)
(C) (6, –2 )
(D) (7, –3 )
(E) (10, 1)

Kudos for a correct solution.


MANHATTAN GMAT OFFICIAL SOLUTION:

Your GMAT scratchpad has a grid; use it to plot the diagram to scale.
Attachment:
2015-06-15_1342.png


“Eyeball” solution: The corner at ( –4, 4) looks like a right angle, so complete the rectangle with the dashed lines shown. The 4th point must be located approximately where the bigger dot is drawn. The closest answer choice is the point (7, –3 ). Alternatively, you could plot all of the answer choice points and see which one “works” with the three given points.

Alternatively, we could solve by checking the slopes of the solid lines we drew between the given points to prove those lines are perpendicular, that is, to prove we have drawn two sides of the rectangle correctly.

The slope of the long solid line = (4- 0)/(-4 - 8) = -1/3.
The slope of the short solid line = (4 - 1)/(-4 - (-5)) = 3.

The product of these slopes is -1/3*3 = -1, proving that the lines are perpendicular.

Compute the location of the 4th point, using the fact that the short sides have the same slope. The known short side connects the points ( –5, 1) and ( –4, 4). In other words, the bottom left corner is 1 to the left and 3 down from the top left corner. The unknown bottom right corner should therefore be 1 to the left and 3 down from the top right corner, or x = 8 – 1 = 7 and y = 0 – 3 = –3, corresponding to the point (7,–3).

The correct answer is D.



Hi Bunuel, please explain why the rectangle cant be such that one side joins the points (-5,1) and (8,0), and the opposite side joins the points (-4,4) and the unknown? Thats what I did initially, but it led to a wrong answer. What (apparently obvious) clues did I miss? :(

My approach was this: two parallel lines have equal slopes so slope for the line joining (-5,1) and (8,0) = slope for the line joining the points (-4,4) and the unknown. For the latter part (slope involving unknown point), x coordinate would be one unit more than (8,0), since the difference b/w x coordinates is 1 unit here: (-5,1) and (-4,4). Then solved for y (the only unknown variable left now).

TIA!
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Re: Three of the four vertices of a rectangle in the xy-coordinate plane [#permalink] New post   28 Jun 2015, 21:59
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mrish7 wrote:
Bunuel wrote:
Bunuel wrote:
Three of the four vertices of a rectangle in the xy-coordinate plane are ( –5, 1), ( –4, 4), and (8, 0). What is the fourth vertex?

(A) (–4.5, 2.5)
(B) ( –4, 5)
(C) (6, –2 )
(D) (7, –3 )
(E) (10, 1)

Kudos for a correct solution.


MANHATTAN GMAT OFFICIAL SOLUTION:

Your GMAT scratchpad has a grid; use it to plot the diagram to scale.
Attachment:
2015-06-15_1342.png


“Eyeball” solution: The corner at ( –4, 4) looks like a right angle, so complete the rectangle with the dashed lines shown. The 4th point must be located approximately where the bigger dot is drawn. The closest answer choice is the point (7, –3 ). Alternatively, you could plot all of the answer choice points and see which one “works” with the three given points.

Alternatively, we could solve by checking the slopes of the solid lines we drew between the given points to prove those lines are perpendicular, that is, to prove we have drawn two sides of the rectangle correctly.

The slope of the long solid line = (4- 0)/(-4 - 8) = -1/3.
The slope of the short solid line = (4 - 1)/(-4 - (-5)) = 3.

The product of these slopes is -1/3*3 = -1, proving that the lines are perpendicular.

Compute the location of the 4th point, using the fact that the short sides have the same slope. The known short side connects the points ( –5, 1) and ( –4, 4). In other words, the bottom left corner is 1 to the left and 3 down from the top left corner. The unknown bottom right corner should therefore be 1 to the left and 3 down from the top right corner, or x = 8 – 1 = 7 and y = 0 – 3 = –3, corresponding to the point (7,–3).

The correct answer is D.



Hi Bunuel, please explain why the rectangle cant be such that one side joins the points (-5,1) and (8,0), and the opposite side joins the points (-4,4) and the unknown? Thats what I did initially, but it led to a wrong answer. What (apparently obvious) clues did I miss? :(

My approach was this: two parallel lines have equal slopes so slope for the line joining (-5,1) and (8,0) = slope for the line joining the points (-4,4) and the unknown. For the latter part (slope involving unknown point), x coordinate would be one unit more than (8,0), since the difference b/w x coordinates is 1 unit here: (-5,1) and (-4,4). Then solved for y (the only unknown variable left now).

TIA!


Hi mrish7,

Considering your argument, the slope of line joining points (-5,1) and (8,0) = -1/13

Now the third point (-4,4) must be joined with (-5, 1) or with (8, 0) to make the other side of rectangle and the new line must have slope of 13 because Product of slopes of two perpendicular lines = -1

Case 1: (-4,4)is joined with (-5, 1), the slope of this line = 3/1

Case 2: (-4,4)is joined with (8, 0) the slope of this line = 4/(-12) = -1/3

Both cases rejected. Therefore this assumption is INCORRECT

I hope it helps!
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Re: Three of the four vertices of a rectangle in the xy-coordinate plane [#permalink] New post   15 Oct 2016, 20:09
Isnt there is a formula to do the same?
x coordinate (x1+x3-x2) and y coordinate (y1+y2-y3)??
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Re: Three of the four vertices of a rectangle in the xy-coordinate plane [#permalink] New post   15 Oct 2016, 20:49
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AmritaSarkar89 wrote:
Just derived - The correct formula is
x coordinate- ( x1+x3-x2) and y coordinate is (y1+y3-y2).


Although your previous score is already pretty decent but if you are considering taking GMAT again to improve it further then a small piece of advice is "Do away with formulas" as much as possible.

A person getting the score as your profile shows can't be illogical and logical people should understand maths logically. It rewards them with greater score always

In Coordinate geometry, ALWAYS DRAW THE FIGURE while solving questions... It makes your understanding very smooth and easy
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Re: Three of the four vertices of a rectangle in the xy-coordinate plane [#permalink] New post   15 Oct 2016, 21:18
GMATinsight wrote:
AmritaSarkar89 wrote:
Just derived - The correct formula is
x coordinate- ( x1+x3-x2) and y coordinate is (y1+y3-y2).


Although your previous score is already pretty decent but if you are considering taking GMAT again to improve it further then a small piece of advice is "Do away with formulas" as much as possible.

A person getting the score as your profile shows can't be illogical and logical people should understand maths logically. It rewards them with greater score always

In Coordinate geometry, ALWAYS DRAW THE FIGURE while solving questions... It makes your understanding very smooth and easy



Thanks for your piece of advice.
Yes I am planning on retaking to hit something close to 750. The problem is since our childhood we had been pushed into memorizing formulas and till date I am struggling to get away with that canopy on my thought process.
Honestly yes, I am trying to logically breakdown every problem :)
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Re: Three of the four vertices of a rectangle in the xy-coordinate plane [#permalink] New post   29 Apr 2018, 10:47
Three of the four vertices of a rectangle in the xy-coordinate plane are ( –5, 1), ( –4, 4), and (8, 0). What is the fourth vertex?

(A) (–4.5, 2.5)
(B) ( –4, 5)
(C) (6, –2 )
(D) (7, –3 )
(E) (10, 1)

Since diagonals of rectangles are Equal ,we can calculate the distance between the diagonals and other two points of vertex must also satisfy that distance .
Points ( –5, 1), and (8, 0) have a distance of sq root of 170 and similarly point ( –4, 4) third vertex and fourth vertex points mentioned in the option must satisfy this distance which is option D(7,-3).
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Re: Three of the four vertices of a rectangle in the xy-coordinate plane [#permalink] New post   13 May 2018, 14:57
Without solving, we have y = 1, 4, 0. 1-4 = -3; hence, we need |3| as the difference in y which implies that the last y must be 3 or -3. Only D makes this true so D.
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Re: Three of the four vertices of a rectangle in the xy-coordinate plane [#permalink] New post   28 Oct 2020, 23:38
(-5 , 1) and (-4 , 4) compose the Left Hand Width of the Rectangle

and

Point (8 , 0) is 1 of the 2 Vertices on the Right Hand Width of the Rectangle

Opposite Sides in a Rectangle must be Parallel to Each Other and Opposite Sides must be EQUAL in Length.


Thus, if we count the Same Horizontal Distance and Same Vertical Distance that exists between Points (-5 , 1) and (-4 , 4) we will have an Opposite Side that is both Parallel and Equal in Length


From (-5 , 1) ------> (-4 , 4)

Horizontal Distance = 1 Unit to the Right

Vertical Distance = 3 Units UP


Therefore, the 4th Vertex will either be at:

Starting from Point (8,0):

(1st Possibility)
Count 1 Horizontal Unit to the RIGHT: X Coord = 9
and
Count 3 Vertical Units UP: Y Coord = 3

(9 , 3)


OR


(2nd Possibility)
just REVERSE the Directions for Each to keep the Slope the Same

Count 1 Horizontal Unit to the LEFT: X Coord = 7
and
Count 3 Vertical Units DOWN: Y Cood = (-)3


(7 , -3)


-D- is the Correct Answer
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Re: Three of the four vertices of a rectangle in the xy-coordinate plane [#permalink] New post   24 Dec 2021, 04:09
using mid point logic works without having to deal with quadratics

let (x,y) be the 4rth vertex
by roughly plotting the 3 points we can see that the mid point of (-5,1) and (8,0) needs to be equal to mid point of (-4,4) and (x,y)
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Three of the four vertices of a rectangle in the xy-coordinate plane [#permalink] New post   11 Jan 2022, 08:45
KarishmaB Bunuel
I have understood the solutions explained above but I tried to solve it by creating the line equation for (-5,1) (x,y) and (8,0)(x,y) and then putting the 2 equations equal .i.e, where they intersect which should give the unknowns as per my understanding. However , I am not getting the correct answer. Please help. Is it a wrong approach?
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Three of the four vertices of a rectangle in the xy-coordinate plane [#permalink]
11 Jan 2022, 08:45
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