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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.

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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.

May 29, 2018

#1
+18754
+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

May 29, 2018
#2
+988
+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

May 29, 2018
edited by GYanggg  May 29, 2018
#3
+102383
+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 = √ [300]  ≈  17.32  units

May 29, 2018