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

Question

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)

GMATinsight · Accepted Answer

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  D

Bunuel · Answer

MANHATTAN GMAT OFFICIAL SOLUTION: Your GMAT scratchpad has a grid ; use it to plot the diagram to scale. 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.

GMATinsight · Answer

ALTERNATIVE

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

Slope of Line joining points B( –4, 4) and C( 8, 0) = (0-4) /   = -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.   *   = -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  D

mike34170 · Answer

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)

douglasvg · Answer

Actually this is very easy to be solved if you draw on a paper.

Answer D

mrish7 · Answer

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!

GMATinsight · Answer

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!

AmritaSarkar89 · Answer

Just derived - The correct formula is 
x coordinate- ( x1+x3-x2) and y coordinate is (y1+y3-y2).

GMATinsight · Answer

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

AmritaSarkar89 · Answer

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

Jayiet89 · Answer

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).

lincy93 · Answer

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.

ColumbiaEMBA · Answer

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.

Fdambro294 · Answer

(-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

vidyasagar151 · Answer

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.

Knightclaw · Answer

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)

Jaychoudhary · Answer

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?

KarishmaB · Answer

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.

Noida · Answer

An easy solution will be to make use of the fact that the diagonals bisect each other.

So, point of intersection of diagonal formed by the points (-5,1) and (8,0) will be 3/2 and 1/2.

This point (3/2,1/2) will be the mid point of the diagonal formed by the points (-4,4) and (x,y).

So,  \frac{(x + (-4))}{2}  =  \frac{3}{2}

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

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