A Rectangle is Inscribed in a Hexagon That Has All Sides of Equal Length GMAT Problem Solving

Question: A rectangle is inscribed in a hexagon that has all sides of equal length and all angles of equal measure, as shown in the figure above. If the total area of the hexagon is x, then the sum of the areas of the two shaded regions is

1. x/6
2. x/4
3. x/3
4. x/2
5. 2x/3

“A rectangle is inscribed in a hexagon that has all sides of equal length”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation:

Approach Solution 1:

There is only one approach to solve the problem statement.
So we can divide the given hexagon into 6 equilateral triangles as follows:

This means, the area of the entire hexagon = the area of 6 equilateral triangles
We'll now label the four portions that comprise the shaded area as follows:

So.…
The area of region A = 1/2 of an equilateral triangle
The area of region B = 1/2 of an equilateral triangle
The area of region C = 1/2 of an equilateral triangle
The area of region D = 1/2 of an equilateral triangle
So, the total shaded area = (1/2) + (1/2) + (1/2) + (1/2) = 1+1
= 2
So, the shaded area is equal to the area of 2 equilateral triangles.
Since the hexagon consists of 6 equilateral triangles, the shaded portion = 2/6 of the entire hexagon.
So, if the hexagon has an area of x, the area of the shaded portion = 2/6 of x = 1/3 of x = x/3

Correct Answer: C

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