The Vertices Of A Rectangle In The Standard (X,Y) Coordinate Place Are GMAT Problem Solving

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?

1. 0
2. 2/5
3. 4/7
4. 1
5. 7/4

“The vertices of a rectangle in the standard (x,y) coordinate place are ''- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. GMAT quant section examines the logical and rational skills of the students to solve quantitative problems. The students must choose the suitable option by doing proper calculations with mathematical understanding. The students must possess concepts of mathematical calculations to solve GMAT Problem Solving questions. The GMAT Quant topic in the problem-solving part cites calculative mathematical problems that need to be solved with suitable qualitative skills. The candidates can practice questions by answering from the book “GMAT Official Guide Quantitative Review”.

Solution and Explanation:

Approach Solution 1:

The problem statement informs that:

Given:

• The vertices of a rectangle in the standard (x,y) coordinate place are (0,0), (0,4), (7,0) and (7,4).
• A line passes through (2,2) partitions that divide the rectangle into 2 equal halves having the same area.

Find Out:

• The slope of this line.

The theorem of rectangle states that a rectangle when split into two equal areas by a line, and then the line passes through the centre of the rectangle. This centre point of the rectangle is the intersection of its two diagonals.
The point of intersection of the diagonals of the rectangle will be the midpoint of any diagonal.
Therefore, it will be the centre point of (0,0), and (7,4) or the centre point of (0,4) and (7,0).
That can be written as: either [(0+7)/2, (0+4)/2] or [(0+7)/2, (4+0)/2] = [3.5, 2]
Therefore, the slope of the line passing through the points (2,2) and (3.5,2) = (2-2)/(3.5-2) = 0

Correct Answer: (A)

Approach Solution 2:

The problem statement states that:

Given:

• The vertices of a rectangle in the standard (x,y) coordinate place are (0,0), (0,4), (7,0) and (7,4).
• A line passes through (2,2) partitions that divide the rectangle into 2 equal halves having the same area.

Find Out:

• The slope of this line.

To solve this problem, let’s have a look at the diagram below:

The line passes the rectangle in such a way that it divides the rectangle into two equal parts. Therefore, it can be inferred that the line that split the rectangle into two identical parts must be horizontal.
Thus as per the formula for a horizontal line, the slope of any horizontal line is always equal to zero.

Correct Answer: (A)

Approach Solution 3:

The problem statement frames that:

Given:

• The vertices of a rectangle in the standard (x,y) coordinate place are (0,0), (0,4), (7,0) and (7,4).
• A line passes through (2,2) partitions that divide the rectangle into 2 equal halves having the same area.

Find Out:

• The slope of this line.

A rectangle can be split into 2 parts with identical areas in nearly 3 ways:

1. The rectangle can be divided by a diagonal into 2 triangles with equal areas
2. The rectangle can be split by a vertical line drawn through the centre of its length that diverges the rectangle into 2 smaller rectangles
3. The rectangle can be divided by a horizontal line that goes through the middle of the width of the rectangle.

If the rectangle is drawn by using the vertices given on a coordinate plane, it can be analysed that (2,2) lies on the horizontal line. It passes through the centre of the width of the rectangle.

Therefore, this line separates the rectangle into two smaller rectangles with a length of 7 units and a width of 2 units and an identical area.
As per the formula for a horizontal line, the slope of any line parallel to the x-axis is always 0.
Therefore, we can say that the angle driven by this line with respect to the x-axis (in an anti-clockwise direction) is 0,
Therefore, the slope of any line is the tangent of this angle i.e Tan 0˚ = 0.

Correct Answer: (A)

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