A window is in the shape of a regular hexagon with each side of length

Question

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

gracie · Accepted Answer

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.

EMPOWERgmatRichC · Answer

Hi kkahuja91,

I'm going to give you a couple of hints so that you can attempt this question again on your own:

1) A regular hexagon has 6 equal sides and 6 equal angles (that are all 120 degrees).
2) You can 'cut' a regular hexagon into 6 equal EQUILATERAL triangles. If you know how to calculate the height of an equilateral triangle, then you should be able to answer this question.

GMAT assassins aren't born, they're made,
Rich

nalinnair · Answer

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.

nalinnair · Answer

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

Bunuel · Answer

______________

Done. Thank you.

kapru · Answer

Bunuel  I didn't get the below part. how does diagonal BE bisects angle ABC and DEF

catharsys · Answer

.

CyberStein · Answer

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

JeffTargetTestPrep · Answer

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

Sarjaria84 · Answer

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'.

MHIKER · Answer

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

GMATattempts212 · Answer

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?

EMPOWERgmatRichC · Answer

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).
 
GMAT assassins aren't born, they're made,
Rich

Elite097 · Answer

ThatDudeKnows   avigutman  how do we know that BE is bisecting the angle into 60 degree?

avigutman · Answer

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.

woohoo921 · Answer

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.

avigutman · Answer

Lots of good ways to solve this one, and yours is good as well,  woohoo921 .
My solution was more similar to  this one .

RenB · Answer

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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A window is in the shape of a regular hexagon with each side of length

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