The vertices of a rectangle in the standard (x,y) coordinate place are

Question

The vertices of a rectangle in the standard (x,y) coordinate place are (0,0), (0,4), (7,0) and (7,4). If a line through (2,2) partitions the interior of this rectangle into 2 regions that have equal areas, what is the slope of this line?

A. 0
B. 2/5
C. 4/7
D. 1
E. 7/4

I got confused on this question. Can you show a good method of doing it?

A. 0
B. 2/5
C. 4/7
D. 1
E. 7/4

Bunuel · Accepted Answer

Look at the diagram below: Rectangle.png In order the line to divide the rectangle into two equal parts it must be horizontal. The slope of any horizontal line is zero. Answer: A. For more on Coordinate Geometry check here: math-coordinate-geometry-87652.html Hope it helps.

GMATinsight · Answer

CONCEPT  : The readers need to know that a rectangle can be divided into two equal area by a Straight line only when the straight line passes through the Centre of the Rectangle (Intersection of its two diagonals)   Draw a figure and know it for yourself.

The point of Intersections of the diagonals will be the midpoint of any diagonal i.e. Midpoint of (0,0), and (7,4) OR Midpoint of (0,4) and (7,0)

i.e. Either   OR   =

Slope of line passing through points (2,2) and (3.5,2) = (2-2)/(3.5-2) = 0

P.S. Line passing through (2,2) and slope =1 will also pass through origin and will divide the rectangle into One triangle and another Trapezium which will not have equal Areaa

saintforlife · Answer

First I assumed the line passes through the origin and is a diagonal of the rectangle making the slope 1. But then I realized that the slope can't be '1' because only a square would have a slope of 1. Since this is a rectangle, its slope has to be something else.

This is a good problem where the grid lines on the worksheet come in handy. Just need to make sure to draw the sketch to scale.

This is a good problem where the grid lines on the worksheet come in handy. Just need to make sure to draw the sketch to scale.

ronr34 · Answer

Why did you not check to see if it is the diagonal of the rectangle?
Is it not possible for the diagonal to split it into 2 equal shapes?

mau5 · Answer

It is not possible to have the point (2,2) on the diagonal. Had it been on the diagonals, the slope of this line would be :  \frac{4-0}{7-0} = \frac{2-0}{2-0}  which is obviously not the case as these 2 values are different.

ronr34 · Answer

Yes this is what I thought.
I just didn't understand if it was a given that we need to check it, or if there was
another way of knowing without making this equation and checking.

suhasancd · Answer

Why cant a line that pass through (2,2) and make 45 degrees(slope 1) with X axis and that also splits the rectangle into two quadrilaterals be assumed ?

Ashishsteag · Answer

Line passes through 2,2
Equation of line=y-y1=m(x-x1)
So y-2=m(x-2)

From the options I initially choose m=1
But it will intersect the line segment joining 0,4 and 7,4 at x=4,y,4
Two areas will be unequal

next chose m=0,
It will be a horizontal line

So,y=2 is the equation of the line and it will intersect line joining 7,4 and 7,0 at 7,2
Now we have two rectangles and both of them have areas=14.

So m=0 is the answer.

megha_2709 · Answer

I am unable to understand why cant it be a diagonal and why horizontal line. The rectangle has 1 pair of sides equal , the diagonal from origin passing thru 2,2 will divide into 2 parts that is equal. Please tell me why we are considering horizontal line and not the diagonal.

Regards
Megha

Regards
Megha

hdwnkr · Answer

A diagonal is not the only line that divides the area into two equal halves. Nor is the horizontal line. A vertical line could also divide a rectangle into two equal facts. In fact, if there is only one point given within a rectangle, we can draw a line that would not be horizontal, vertical or diagonal, that would divide a rectangle into two trapeziums of equal area. However, we need to identify which of the following points are on a line that divides the rectangle into two equal halves. A figure approach, as demonstrated by bunuel, helps.

However, if you wish to check through an extended version, find the equation of the line that would make the diagonal of the rectangle. You'd find that the point (2,2). does not lie on the diagonal. Also, it would not lie on a line that is vertical and that divides the rectangle into two equal halves. Another way of looking at the question is that the breadth of the rectangle is 4 and the length is 7. So a horizontal line with y co-ordinate 2 will for sure divide the rectangle into two equal halves. Also, a vertical line with x co-ordinate 3.5 would do the same job.

Hope this helps.

narendran1990 · Answer

I was initially confused that for splitting the triangle into 2 equal areas, diagonal might be required. But then, when the graph is plotted, the line that goes through (2,2) will go straight and have coordinates (0,2) & (7,2). The area formed is half of the rectangle and the slope would be 0.  .

rbhan91 · Answer

In the above graph as shown by Bunuel

the line through (2,2) is parallel to both axis of rectangle rectangle

We know the product of slopes for parallel lines

so the slope of the line through (2,2) is 0 (since slopes of other lines can not be zero)

I hope it's clear

Note: You can think of a 45-degree solution, but answer not in option list.

omereg · Answer

M=0 (horizental line) can be obvious once we draw the rectangle and point (2,2).

In case we didn't have M=0 as an optional answer we should have looked for line that goes through (2,2) and devide the rectangle to two equal-sized trapezoids.

I wonder, what is the 'safest' and quickest way to find this line?

Thanks for the answers

dcummins · Answer

If you drag the diagram out on your scratch pad accurately then this should be very straight forward as the only line that could possibly divide this rectangle into two equal areas is a horizontal line through 2,2; thus, the slope must be 0.

Kali123 · Answer

Query
Hi Bunuel,

"In order the line to divide the rectangle into two equal parts it must be horizontal. The slope of any horizontal line is zero" Why is this always true?

veerberry123 · Answer

Thatll make it a square..

Basshead · Answer

Recall that a horizontal slope = 0; a vertical slope = undefined.

Given the coordinates (2,2), the only way to split the rectangle evenly is to have a horizontal line going through the rectange.

Therefore we can conclude the slope = 0.

dmcquiter · Answer

This is how I got it, once I realized the line was through point 2,2 I knew it couldn't be vertical or through the vertices because both parts weren't even. As Bunuel stated, if the point was (3.5, 2), then the line could be in a few different directions with different slopes because that's dead middle of the rectangle.

Brindac2 · Answer

I have a question. Can this kind of geometry question come up in GMAT focus edition?

The vertices of a rectangle in the standard (x,y) coordinate place are

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The vertices of a rectangle in the standard (x,y) coordinate place are

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