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Re: A window is in the shape of a regular hexagon with each side of length [#permalink]
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kkahuja91 wrote:
A window is in the shape of a regular hexagon with each side of length 80 Centimeters. If a diagonal through the center of the hexagon is W centimeters long, then w =?

(A) 80
(B) 120
(C) 150
(D) 160
(E) 240


Attachment:
Solution.png
Solution.png [ 8.02 KiB | Viewed 43920 times ]


Let us draw a regular hexagon ABCDEF, as in figure above. We need to find the length of the diagonal BE.
Given: It is a regular hexagon so, AB = BC = CD = DE = EF = AF = 80 cm and each angle is 120
Join BF and CE, making BCEF a rectangle. Angle BCE = 90
As the diagonal BE bisects angles ABC and DEF, angle CBE is 60. So, triangle BCE is a 30-60-90 triangle and the sides of this triangle is in the ratio - \(1:\sqrt{3}:2\)
Hence, BE = 160
Answer - D.
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Re: A window is in the shape of a regular hexagon with each side of length [#permalink]
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Dear Moderators,

The source of this question is mentioned as Manhattan GMAT. This is actually from Official Guide Quant Review 2016 (PS# 145). Request you to change it.

Thanks,
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Re: A window is in the shape of a regular hexagon with each side of length [#permalink]
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nalinnair wrote:
Dear Moderators,

The source of this question is mentioned as Manhattan GMAT. This is actually from Official Guide Quant Review 2016 (PS# 145). Request you to change it.

Thanks,
Nalin

______________

Done. Thank you.
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Re: A window is in the shape of a regular hexagon with each side of length [#permalink]
Bunuel I didn't get the below part. how does diagonal BE bisects angle ABC and DEF

Quote:
diagonal BE will bisects angles ABC and DEF, angle CBE is 60.
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A window is in the shape of a regular hexagon with each side of length [#permalink]
.

Originally posted by catharsys on 17 Aug 2016, 21:28.
Last edited by catharsys on 23 Oct 2020, 20:35, edited 1 time in total.
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A window is in the shape of a regular hexagon with each side of length [#permalink]
kkahuja91 wrote:
A window is in the shape of a regular hexagon with each side of length 80 Centimeters. If a diagonal through the center of the hexagon is W centimeters long, then w =?

(A) 80
(B) 120
(C) 150
(D) 160
(E) 240


Triangle in the hexagon is equilateral; you know it is equilateral because the degree of one of its angles is 60; you know it is 60, because the sum degrees of a hexagon is 720; 720/6 = 120 degrees per angle in hexagon; the triangle's sides splits that 120 degree hexagon angle in half, making 60 degrees.

The triangle is equilateral; all sides equal each other. therefore, BC = AB = 80. The question asks for diagonal, meaning the line cuts all the way through the hexagon - you have to double BC.

therefore, given BC = 80, 80 x 2 = 160 cm
Answer (D) 160
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Re: A window is in the shape of a regular hexagon with each side of length [#permalink]
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kkahuja91 wrote:
A window is in the shape of a regular hexagon with each side of length 80 Centimeters. If a diagonal through the center of the hexagon is W centimeters long, then w =?

(A) 80
(B) 120
(C) 150
(D) 160
(E) 240


When the hexagon is broken up by 6 diagonals, 6 equilateral triangles are created. Since each side of the hexagon is 80 cm and the length of 1 diagonal is 2 times the length of a side of the hexagon, we have 2 x 80 = 160. Thus, w = 160.

Answer: D
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Re: A window is in the shape of a regular hexagon with each side of length [#permalink]
kkahuja91 wrote:
A window is in the shape of a regular hexagon with each side of length 80 Centimeters. If a diagonal through the center of the hexagon is W centimeters long, then w =?

(A) 80
(B) 120
(C) 150
(D) 160
(E) 240


We are given a regular hexagon with each side = 80 cm.

If we visualize the hexagon in a circle we can see that it will be divided into 6 equilateral triangles with each side measuring 80 cms and equaling the radius of the circle.
Thus w = diameter of the radius = 2r = 2 x 80 = 160 cms.

i.e. option 'D'.
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Re: A window is in the shape of a regular hexagon with each side of length [#permalink]
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kkahuja91 wrote:
A window is in the shape of a regular hexagon with each side of length 80 Centimeters. If a diagonal through the center of the hexagon is W centimeters long, then w =?

(A) 80
(B) 120
(C) 150
(D) 160
(E) 240


If a regular hexagon is broken up by 6 diagonals, then 6 equilateral triangles are created.

The length of each side of the triangle = sides of the hexagon= 80 CM

The diagonal will be equal to the sum of the \(two \) inner sides \((80+80=160)\) of the triangle.

The answer is \(D\)
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A window is in the shape of a regular hexagon with each side of length [#permalink]
gracie wrote:
A regular hexagon consists of six identical equilateral triangles,
each of whose sides is 80 cm.
We know the triangles are equilateral because each of
the 3 diagonals that run through the center bisects the
regular hexagon's internal 120°∠ into two 60° ∠s.
Therefore, w=(2)(80)=160 cm.



How do we know the answer is not 160 * \sqrt{3} (assuming that was an option), i.e. the height of the equilateral triangle multiplied by two? Could a line not be formed by going straight through the shape?
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Re: A window is in the shape of a regular hexagon with each side of length [#permalink]
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Hi GMATattempts212,

In the context of this question, a 'diagonal' runs from one vertex to the 'opposite' vertex on the other side of the hexagon. If you use the 'heights' of two of the equilateral triangles, then you won't use any of the 6 vertices (your line will touch the middle of two opposite sides of the hexagon (meaning that you won't actually have a diagonal).

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Re: A window is in the shape of a regular hexagon with each side of length [#permalink]
ThatDudeKnows avigutman how do we know that BE is bisecting the angle into 60 degree?


nalinnair wrote:
kkahuja91 wrote:
A window is in the shape of a regular hexagon with each side of length 80 Centimeters. If a diagonal through the center of the hexagon is W centimeters long, then w =?

(A) 80
(B) 120
(C) 150
(D) 160
(E) 240


Attachment:
Solution.png


Let us draw a regular hexagon ABCDEF, as in figure above. We need to find the length of the diagonal BE.
Given: It is a regular hexagon so, AB = BC = CD = DE = EF = AF = 80 cm and each angle is 120
Join BF and CE, making BCEF a rectangle. Angle BCE = 90
As the diagonal BE bisects angles ABC and DEF, angle CBE is 60. So, triangle BCE is a 30-60-90 triangle and the sides of this triangle is in the ratio - \(1:\sqrt{3}:2\)
Hence, BE = 160
Answer - D.
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Re: A window is in the shape of a regular hexagon with each side of length [#permalink]
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Elite097 wrote:
ThatDudeKnows avigutman how do we know that BE is bisecting the angle into 60 degree?

Elite097 because BE bisects the regular hexagon into two identical equilateral trapezoids.
Here's a quote:
The long diagonal is the line between two opposite vertices. Long diagonals and bisecting lines coincide.
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A window is in the shape of a regular hexagon with each side of length [#permalink]
avigutman wrote:
Elite097 wrote:
ThatDudeKnows avigutman how do we know that BE is bisecting the angle into 60 degree?

Elite097 because BE bisects the regular hexagon into two identical equilateral trapezoids.
Here's a quote:
The long diagonal is the line between two opposite vertices. Long diagonals and bisecting lines coincide.


avigutman
I wanted to gently check with you to see if this was your approach based on the mention of the trapezoid.

Step 1: I split the hexagon into two trapezoids and solved for the longer base (because that longer base is the diagonal).
Step 2: I then realized that diagonal intersects the 120-degree angle in half, so if I drew a right triangle on both sides of the diagonal (because I am solving for 80 + 2x, with x being defined as the extra length of the diagonal compared to the other base on both sides)
Step 3: Then, I saw that I have a 30-60-90-degree triangle. So, 80=2x and therefore x=40.
--> 80 +2*40=160.

Thank you for all of your time and help.
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Re: A window is in the shape of a regular hexagon with each side of length [#permalink]
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woohoo921 wrote:
avigutman
I wanted to gently check with you to see if this was your approach based on the mention of the trapezoid.

Step 1: I split the hexagon into two trapezoids and solved for the longer base (because that longer base is the diagonal).
Step 2: I then realized that diagonal intersects the 120-degree angle in half, so if I drew a right triangle on both sides of the diagonal (because I am solving for 80 + 2x, with x being defined as the extra length of the diagonal compared to the other base on both sides)
Step 3: Then, I saw that I have a 30-60-90-degree triangle. So, 80=2x and therefore x=40.
--> 80 +2*40=160.

Lots of good ways to solve this one, and yours is good as well, woohoo921.
My solution was more similar to this one.
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Re: A window is in the shape of a regular hexagon with each side of length [#permalink]
kkahuja91 wrote:
A window is in the shape of a regular hexagon with each side of length 80 Centimeters. If a diagonal through the center of the hexagon is W centimeters long, then w =?

(A) 80
(B) 120
(C) 150
(D) 160
(E) 240


Used applied geometry approach-
If a circle is drawn in which the hexagon (window) is inscribed, all the diagonals will intersect at the centre. A regular hexagon is basically 6 equilateral triangles where each side equals the radius of the circle-
Thus diagonal= w= 2r= 80*2= 160
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Hexagon circle.png
Hexagon circle.png [ 14.95 KiB | Viewed 4804 times ]

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Re: A window is in the shape of a regular hexagon with each side of length [#permalink]
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