Problem Solving

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


OA: A

bml1105 wrote:The vertices of a rectangle in the standard (x,y) coordinate plane 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


OA: A

A rectangle can be divided into 2 regions with equal areas in 3 ways : either by a diagonal that divides rectangle into 2 triangles with equal areas or
by a vertical line that passes through the mid point of its length that divides rectangle into 2 smaller rectangles or
by a horizontal line that passes through the mid point of the width of the rectangle.

If you draw the rectangle using the given vertices on a coordinate plane, you will find that (2,2) lies on the horizontal line that passes through the mid point of the width of rectangle. This line divides the rectangle into 2 smaller rectangles with length 7 units and width 2 units and same area. Slope of any line parallel to x axis (i.e a horizontal line) is always 0. This can be stated in other ways also, such as the angle made by this line with respect to x axis (in anti clockwise direction) is 0, slope of any line is the tangent of this angle and Tan 0 = 0.

So the correct answer is A

Regards
Sukriti

I'm sure I am over-thinking this, but technically couldn't a rectangle be divided equally into two trapezoids?

Example: A rectangle has a width of 4 feet and a length of 7 feet. You could take 1 foot from the bottom left corner and draw a diagonal through the 1 foot mark on the top right corner. The two trapezoids would still have equal areas. Right?

[spoiler]{A}[/spoiler]

bml1105 wrote:I'm sure I am over-thinking this, but technically couldn't a rectangle be divided equally into two trapezoids?

Example: A rectangle has a width of 4 feet and a length of 7 feet. You could take 1 foot from the bottom left corner and draw a diagonal through the 1 foot mark on the top right corner. The two trapezoids would still have equal areas. Right?

Did you try to draw? If no, then do try.. You will yourself answer your question; just keep in mind that we need "same areas" of two pieces..

If the question were, 3x3 square and 2,2 is the point ... then I guess your observation would have played.

Is it generally safe to assume that on the GMAT, if they give us only one point, don't tell us any part of where it intersects and ask what the slope is, it's going to be something like this?

I just couldn't figure out how to find a slope, without assuming one of the three that sukriti mentioned.