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# qweschun

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The convex pentagon \$ABCDE\$ has \$\angle A = \angle B = 120^\circ\$, \$EA = AB = BC = 2\$ and \$CD = DE = 4\$. What is the area of \$ABCDE\$?

Atroshus  Aug 29, 2017
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#1
+38
0

I got 11sqrt3 but it wasn't correct

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half hexagon at the bottom = 3sqrt3

((3sqrt3/2)*2^2)/2

top triangle = 8sqrt3

edge length 2= long diagonal 4 << length from E to C

draw point halfway E to C to create point F

EF=CF=2

pythagorean theorem: DF= 2sqrt3

area of top triangle = DF*4 = 8sqrt3

Atroshus  Aug 29, 2017
#2
+76160
+2

I get something a little different, atroshus.....

Refer to the following image :

Note that the area of triangle EAB  = (1/2)(2)(2)sin120  = 2 * √3 / 2  =  √3

And this triangle is isosceles......so angle ABE  = 30°

But since angle ABC  = 120°.....then triangle EBC is a right triangle with angle EBC = 90°

And since EB is a transversal cutting parallel segments AB and EC....then angle ABE  = angle BEC = 30°....then triangle EBC is a 30 - 60 - 90 right triangle with angle BCE = 60°

And since BC  = 2 and is opposite the 30° angle......then the hypotenuse EC is twice this = 4

So......the area of triangle EBC  = (1/2)(2)(4)sin 60  = 4 * √3/2  =  2 √3

And triangle CDE is equilateral with an area = (1/2)(4)(4) sin60  = 8 √3 / 2  = 4√3

So the total area of ABCDE  =  [ √3 + 2√3 + 4√3 ] =  7√3 units^2

CPhill  Aug 29, 2017

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