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

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The vertices of a rectangle in the standard (x,y) coordinate [#permalink]  29 Sep 2013, 18:43
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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?
[Reveal] Spoiler: OA

Last edited by Bunuel on 29 Sep 2013, 22:50, edited 1 time in total.
Edited the question.
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Re: The vertices of a rectangle in the standard (x,y) coordinate [#permalink]  29 Sep 2013, 22:52
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teeva wrote:
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?

Look at the diagram below:
Attachment:

Rectangle.png [ 7.17 KiB | Viewed 7365 times ]
In order the line to divide the rectangle into two equal parts it must be horizontal. The slope of any horizontal line is zero.

For more on Coordinate Geometry check here: math-coordinate-geometry-87652.html

Hope it helps.
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Re: The vertices of a rectangle in the standard (x,y) coordinate [#permalink]  21 Oct 2013, 22:03
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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.
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Re: The vertices of a rectangle in the standard (x,y) coordinate [#permalink]  02 Nov 2013, 09:59
Bunuel wrote:
teeva wrote:
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?

Look at the diagram below:
Attachment:
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.

For more on Coordinate Geometry check here: math-coordinate-geometry-87652.html

Hope it helps.

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?
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Re: The vertices of a rectangle in the standard (x,y) coordinate [#permalink]  02 Nov 2013, 21:43
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ronr34 wrote:
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?

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.
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Re: The vertices of a rectangle in the standard (x,y) coordinate [#permalink]  03 Nov 2013, 00:31
mau5 wrote:
ronr34 wrote:
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?

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.

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.
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Re: The vertices of a rectangle in the standard (x,y) coordinate [#permalink]  15 Jan 2015, 09:28
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Re: The vertices of a rectangle in the standard (x,y) coordinate [#permalink]  20 Jun 2015, 05:14
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 ?
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The vertices of a rectangle in the standard (x,y) coordinate [#permalink]  20 Jun 2015, 08:06
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suhasancd wrote:
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

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 ?

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 [(0+7)/2, (0+4)/2] OR [(0+7)/2, (4+0)/2] = [3.5, 2]

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
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The vertices of a rectangle in the standard (x,y) coordinate   [#permalink] 20 Jun 2015, 08:06
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