+1452 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!
+129933 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°
