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

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Re: A rectangle is inscribed in a hexagon that has all sides of equal leng [#permalink]
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One shaded triangle = an equilateral triangle with side x

I.e. two shaded triangles = 2 equilateral triangles

Hexagon = 6 equilateral triangles

Posted from my mobile device
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Re: A rectangle is inscribed in a hexagon that has all sides of equal leng [#permalink]
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

Bunuel wrote:

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

Attachment:
1.png
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Re: A rectangle is inscribed in a hexagon that has all sides of equal leng [#permalink]
GMATinsight wrote:
One shaded triangle = an equilateral triangle with side x

I.e. two shaded triangles = 2 equilateral triangles

Hexagon = 6 equilateral triangles

Posted from my mobile device

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
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Re: A rectangle is inscribed in a hexagon that has all sides of equal leng [#permalink]
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ankurjuit wrote:
GMATinsight wrote:
One shaded triangle = an equilateral triangle with side x

I.e. two shaded triangles = 2 equilateral triangles

Hexagon = 6 equilateral triangles

Posted from my mobile device

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

Only a regular polygon have equal sides and equal angles so it can be determined from the question statement.
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Re: A rectangle is inscribed in a hexagon that has all sides of equal leng [#permalink]
GMATinsight wrote:
ankurjuit wrote:
GMATinsight wrote:
One shaded triangle = an equilateral triangle with side x

I.e. two shaded triangles = 2 equilateral triangles

Hexagon = 6 equilateral triangles

Posted from my mobile device

@ankurjuit
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

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

Got it, Thanks
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Re: A rectangle is inscribed in a hexagon that has all sides of equal leng [#permalink]
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Bunuel How are we supposed to solve this under 2 mins.
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Re: A rectangle is inscribed in a hexagon that has all sides of equal leng [#permalink]
GMATinsight wrote:
One shaded triangle = an equilateral triangle with side x

I.e. two shaded triangles = 2 equilateral triangles

Hexagon = 6 equilateral triangles

Posted from my mobile device

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
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Re: A rectangle is inscribed in a hexagon that has all sides of equal leng [#permalink]
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tejasvkalra wrote:
GMATinsight wrote:
One shaded triangle = an equilateral triangle with side x

I.e. two shaded triangles = 2 equilateral triangles

Hexagon = 6 equilateral triangles

Posted from my mobile device

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

tejasvkalra

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

You could check our course on Geometry which is affordable and enriching for this topic and will cover all parts that GMAT concerns

Subscribe Topic-wise UN-bundled Video course. CHECK FREE Sample Videos
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Re: A rectangle is inscribed in a hexagon that has all sides of equal leng [#permalink]
Very well explained +1 GMATinsight
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Re: A rectangle is inscribed in a hexagon that has all sides of equal leng [#permalink]
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GMATinsight wrote:
tejasvkalra wrote:
GMATinsight wrote:
One shaded triangle = an equilateral triangle with side x

I.e. two shaded triangles = 2 equilateral triangles

Hexagon = 6 equilateral triangles

Posted from my mobile device

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

tejasvkalra

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

You could check our course on Geometry which is affordable and enriching for this topic and will cover all parts that GMAT concerns

Subscribe Topic-wise UN-bundled Video course. CHECK FREE Sample Videos

Got it. Thanks
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Re: A rectangle is inscribed in a hexagon that has all sides of equal leng [#permalink]
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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

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Re: A rectangle is inscribed in a hexagon that has all sides of equal leng [#permalink]
BrentGMATPrepNow can you please explain this question. I am having trouble in visualizing the figure.

Thank you!
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Re: A rectangle is inscribed in a hexagon that has all sides of equal leng [#permalink]
GMATinsight wrote:
One shaded triangle = an equilateral triangle with side x

I.e. two shaded triangles = 2 equilateral triangles

Hexagon = 6 equilateral triangles

Posted from my mobile device

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 ??
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A rectangle is inscribed in a hexagon that has all sides of equal leng [#permalink]
Bunuel wrote:

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

Attachment:
The attachment 1.png is no longer available

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.

Attachments

20418b.jpg [ 16.71 KiB | Viewed 14777 times ]

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A rectangle is inscribed in a hexagon that has all sides of equal leng [#permalink]
Bunuel wrote:

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

Attachment:
1.png

BrentGMATPrepNow can you pls provide your solution
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Re: A rectangle is inscribed in a hexagon that has all sides of equal leng [#permalink]
Bunuel wrote:

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

Attachment:
1.png

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
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A rectangle is inscribed in a hexagon that has all sides of equal leng [#permalink]
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dave13 wrote:
Bunuel wrote:

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

Attachment:
1.png

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

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$$= x - \frac{2x}{3} = \frac{x}{3}$$
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