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# Polygon Questions

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1: Let ABCDEF be a regular hexagon, and let G,H,I be the midpoints of sides AB,CD,EF respectively. If the area of triangle GHI is 225, what is the area of hexagon ABCDEF?

2: In pentagon MATHS, angle \(M \cong \angle T \cong \angle H\) and angle A is supplementary to angle S. How many degrees are in the measure of angle H?

Thanks!

#1
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1.  We can generalize the true side lengths of GHI  and the hexagon from the following pic, ACG GHI  will form an equilateral triangle....call the side, S

Its  area  =   (1/2) S^2  sin ( 60) =   (1/2)S^2√3/2   =   √3/4S^2

So....we can find the length of one side of GHI, thusly

225  = √3/4 S^2

225 * 4 / √3   = S^2

900 / √3   =  S^2

( 30 /4√3  )  = S

Look at the pic.....triangle CJB  comprises 1/6 of the hexagon's area....and it is an equilateral triangle with a side of 2

And note that in the pic....its side will be (2/3)  that  of the true side length of triangle GHI   = 20/4√3

So....the area of the hexagon is just

(6) (1/2) (20/4√3)^2 sin 60  =

3 (400/√3)√3/2   =

(3/2)*400  =  600 units^2

Second one  ;

M + T + H  + A  + S  = 540

M + T + H + A  + (180-A)  = 540

Since M = T = H  we have that

3H + A  + 180 - A  = 540

3H + 180  = 540

3H  = 360

H  = 120°   Apr 2, 2018
#2
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Thanks so much CPhill!