The Hexagon ABCDEF is Regular. That Means All its Sides are the Same GMAT Problem Solving

Question: The hexagon ABCDEF is regular. That means all its sides are the same length and all its interior angles are the same size. Each side of the hexagon is 2 feet. What is the area of the rectangle BCEF?

(A) 4 square feet
(B) \(4\sqrt{3}\) square feet
(C) 8 square feet
(D) \(4+4\sqrt{3}\) square feet
(E) 12 square feet

Correct Answer: B

Solution and Explanation
Approach Solution 1:
There is only one approach to solve the problem. Hence, one Model Answer is provided.

Refer to the diagram below

First let’s calculate the width of Rectangle

now,

Connect sides BF and CE of the rectangle

All sides of the hexagon are 2 feet. Width of rectangle BCEF is 2 feet

Now let us calculate the length of Rectangle

Create 2 right triangles at angle D as shown in the diagram above

Now, Draw a perpendicular bisector from the angle D to opposite side of EC, refer diagram above

Note that , A regular hexagon always has internal angle measures of 120 degrees:
(n-2)(180) = 720 degrees total,
Divide it by 6 angles , each angle will be 120

Angle D is bisected into \(\frac{120}{2}\)= two angles of 60°, so

angle CDX = 60
Angle DXC = 90
Angle DCX = 30

--Either:

Angle DCB = 120. Rectangle vertex BCE = 90,
so (120 - 90) = 30;

--Two of the angles in Δ DCX total 150; (180- 150) = 30

Now there are 2 identical right 30-60-90 triangles left: Δ DEX and Δ DCX

30-60-90 right triangles have side lengths in ratio \(x\sqrt{3}:2x\)

The Side DE, opposite of the 90° angle, corresponds with 2x and = 22
The Side DX, opposite of the 30° angle, corresponds with x and therefore = 1
The Side EC, opposite of the 60° angle, corresponds with x√3 and therefore √=3

The two right 30-60-90 triangles are identical;

the length of the rectangle is the sum of their sides of √3

The Length of rectangle is : 2√3

Let’s find the Area of rectangle

Formula of Area of rectangle is (L∗W)

=(2 x 2√3)
=(4√3)

the area of the rectangle BCEF is (4√3) which is Answer B

“The hexagon ABCDEF is regular. That means all its sides are the same”- is a topic of the GMAT Quantitative reasoning section of GMAT. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. GMAT Quant practice papers improve the mathematical knowledge of the candidates as it represents multiple sorts of quantitative problems.

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The Hexagon ABCDEF is Regular. That Means All its Sides are the Same GMAT Problem Solving

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