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Find the number of units in the length of diagonal DA of the regular hexagon shown. Express your answer in simplest radical form.

 

 

 

 

Guest May 29, 2018
 #1
avatar+12510 
+2

For a polygon, the SUM of the internal angles is  (n-2) * 180    (n=6 for hexagon)

(6-2)*180 = 720     Thus EACH angle is :   720/6 = 120 degrees

 

So the large angle of the triangle is 120 degrees    the other two are equal to each other

  there are 180 degrees in any triangle:   

   (180-120) / 2 = 30 degrees for each of the smaller angles.

 

Now use the law of sines

 

10/sin30 = x/(sin120)

10/(.5)  * (sqrt3 )/ 2 = x

10 sqrt 3

 

 

10/sin30  *  sin 120  =  17.32

ElectricPavlov  May 29, 2018
 #2
avatar+793 
+2

I already drew a diagram, but don't know how to upload it from my computer.

 

First you connect the point A to the two points that construct the line segment labeled 10. 

 

AD is congruent to the line segment perpendicular to the side labelled 10. 

 

You have constructed a 30 - 60 - 90 triangle. The hypotenuse is 20.

 

The leg is the square root of 20^2 - 10^2.

 

Can someone tell me how to upload an image, then this solution would be much clearer. 

 

Thanks

GYanggg  May 29, 2018
edited by GYanggg  May 29, 2018
 #3
avatar+86859 
+1

As an alternative,  we can use the Law of Cosines to solve this

 

The interior angle of a hexagon = 120°....so....

 

AD^2  = 10^2 + 10^2  - 2 (10 *10) cos (120°)

 

AD^2  = 200  - 200 (- 1 / 2)

 

AD^2  =  200 + 100

 

AD = √ [300]  ≈  17.32  units

 

 

cool cool cool

CPhill  May 29, 2018

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