A rectangle is inscribed in a hexagon that has all sides of equal leng

Question

https://gmatclub.com/forum/download/file.php?id=54694 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 A. x/6 B. x/4 C. x/3 D. x/2 E. 2x/3 PS20418 1.png

BrentGMATPrepNow · Accepted Answer

Key property: A regular hexagon consists of 6 equilateral triangles

So we can divide the given hexagon into 6 equilateral triangles as follows: 
 https://i.imgur.com/MahwQ2m.png 
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:
 https://i.imgur.com/FpYGdeV.png

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) 
= 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

Answer: C

GMATinsight · Answer

Two doubts:
1. The question does not state is a regular hexagon, why do you assume 6 equilateral triangles
 My response: The polygon is called regular polygon when all sides are equal as well as all angles are equal and question states that.

2. 6 Equilateral Triangles are formed when a point from the center joins the 6 vertices. Here triangles are different
 My response: We need to join the vertices with center of hexagon and then see how the shaded figure is being drawn. SHaded triangle = 2*Half of equilateral

Please check the figure

GMATinsight · Answer

One shaded triangle = an equilateral triangle with side x

I.e. two shaded triangles = 2 equilateral triangles

Hexagon = 6 equilateral triangles

Shaded = hexagon/3 = x/3

Answer Option C

Posted from my mobile device

Archit3110 · Answer

let side of hexagon = 4
so total area of hexagon ; 6*4*4*√3/4 = 41.56( x)
and shaded part is nothing but area of two equilateral ∆ ; 2*16*√3/4 ; 13.8

put value of x in each option in C ; we get 41.56/3 ; 13.8
OPTION C

NitishJain · Answer

Two doubts:
1. The question does not state is a regular hexagon, why do you assume 6 equilateral triangles
2. 6 Equilateral Triangles are formed when a point from the center joins the 6 vertices. Here triangles are different

yashikaaggarwal · Answer

Only a regular polygon have equal sides and equal angles so it can be determined from the question statement.

NitishJain · Answer

Got it, Thanks

psnogi · Answer

Bunuel  How are we supposed to solve this under 2 mins.

tejasvkalra · Answer

I didn't understand your solution. Could you please explain how an hexagon contains 6 equilateral triangles ? Shouldn't it be 4 ? 2 at the top and bottom and 2 after dividing the rectangle into 2

GMATinsight · Answer

tejasvkalra

Please check the video (from our Video course on geometry) attached here which explains how a hexagon is equivalent to 6 equilateral triangles

https://www.youtube.com/watch?v=v7FYALsKt-w

RS81 · Answer

Very well explained +1  GMATinsight

tejasvkalra · Answer

Got it. Thanks

Fdambro294 · Answer

Rule: You can Divide a Regular Hexagon by using 3 Vertex to Opposite Vertex Diagonals into 6 Equilateral Triangles all with the Same Side as the Regular Hexagon.

If you FIRST Split the Regular Hexagon up into these 6 Equilateral Triangles and THEN add in the 2 Lines forming the Shaded Regions, you will see that the 2 Diagonals that form the Shaded Rectangles are Each (2) * (Height) of any 1 of the Equilateral Triangles.

Examining the picture, you see that the Top Shaded Portion is 1 of the Equilateral Triangles. Also, the Bottom Shaded Portion is 1 of the Equilateral Triangles.

Therefore, the Shaded Portion is Exactly 2 out of the 6 Equilateral Triangles that make up the Area of the Regular Hexagon.

If the Area of the Entire Regular Hexagon = X

Area of the 2 Shaded Regions = (2/6) * X = X/3

Answer C

Rocknrolla21 · Answer

BrentGMATPrepNow  can you please explain this question. I am having trouble in visualizing the figure.

Thank you!

aakash13111989 · Answer

Hi Gmat Insight ,

I understand that Regular Hexagon is made up of 6 equilateral triangles. But how we know the shaded portion is 3 parts of it ??

Basshead · Answer

We can divide the hexagon into 6 equilateral triangles, with each shaded triangle representing 1 equilateral triangle.

Each equilateral represents  \frac{1}{6}x

Therefore 2 equilateral triangles represent  \frac{x}{3} .

Properties of a Regular Hexagon:  
- It has six sides and six angles.
- Lengths of all the sides and the measurement of all the angles are equal.
- The total number of diagonals in a regular hexagon is 9.
- The sum of all interior angles is equal to 720 degrees, where each interior angle measures 120 degrees.

Answer is C.

dave13 · Answer

BrentGMATPrepNow  can you pls provide your solution

dave13 · Answer

KarishmaB how about an alternative solution where you subtract an area of rectangle from an area of hexagon i did this but smth went wrong

\frac{3*x^2 \sqrt{3} }{2 }- 2x^2\sqrt{3} area of rectangle i got this way: each shaded region represents isosceles triangle, so if i draw perpendicular line i get 30:60:90 trangle hence length of rectangle is 2x\sqrt{3} and width x

KarishmaB · Answer

I would much rather use the symmetry of a regular hexagon and the triangles I can carve out of it as done above. 
But if you wanted to use geometry as suggested by you, note that x is the area of the hexagon, not a side.

Since a hexagon is made up of 6 equilateral triangles, its area will be  6 * \frac{\sqrt{3} * a^2}{4} = x  ... (I)

If side of the hexagon is a, in the 30-60-90 shaded triangle (half of each shaded triangle), the side opposite to 60 is  \frac{\sqrt{3}a}{2}  which means the width of the rectangle is  \sqrt{3}a .

Area of rectangle is  a*\sqrt{3}a = \sqrt{3}a^2 = \frac{2x}{3}  (from (I) above)

Shaded area  = x - \frac{2x}{3} = \frac{x}{3}

A rectangle is inscribed in a hexagon that has all sides of equal leng

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A rectangle is inscribed in a hexagon that has all sides of equal leng

Prep Toolkit

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