If the Equation |x|+|y|= 5 Encloses a Certain Region on the Graph, What is the Area of that Region?

Question: If the equation |x|+|y|= 5 encloses a certain region on the graph, what is the area of that region?

1. 5
2. 10
3. 25
4. 50
5. 100

This topic is a part of GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "McGraw-Hill's Conquering the GMAT Math" published in the year 2007. GMAT Quant section consists a total of 31 questions. This is a GMAT Problem Solving question that allows the candidates to select the correct answer from the five answer choices provided. The total time allotted for this part is 62 minutes which allows the candidate 2 minutes to answer each question.

Solution and Explanation:

Approach Solution 1

As we know that the Modulus Function ‘|x|’ is defined as
x, x 0
= -x, x < 0

Make the list of different combinations of x, y 0 and x, y < 0.

According to the above table, Plot the graph of |x| + |y| = 5. The graph will look like:

From this graph, we can see that four right- angled triangles are formed.
Area of one triangle = ½ * Base * Height
Here, in the figure: Base = Height = 5
We have four triangles, so the area of the whole figure is:
The area of the above figure is = 4 * Area of all small triangles
Area of small triangles= 4* ½*5*5 = 50

Correct option: D

Approach Solution 2

From the graph, we have seen that X and Y intercepts are (0, 5), (0, -5), (5, 0), and (-5, 0) (just make x = 0 and find y then make y = 0 and find x). Now join all these points and plot a graph:

From the plotted graph, we can seen that the diagonals of the quadrilateral are (10, 10), and also the perpendicular bisectors of each other. So we can say that this figure must be a square.
The area of the figure can be calculated as:
Area of square = (Diagonal^2)/ 2
Area of square = (10^2)/ 2
Area of square = 50

Hence, the correct option is D.

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