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$ABCDEFGH$ is a regular octagon of side 12cm. Find the area in square centimeters of trapezoid $BCDE$. Express your answer in simplest radical form.

 Apr 15, 2018
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Call the intersection of BE with CH point X. Then the triangle BCX is a right isosceles right triangle.

This means that angle CBE is a 45 degree angle. 

And, both BX and CX are equal to 6·sqrt(2).

CX is a height of the trapezoid and equals 6·sqrt(2).

CD is one base of the trapezoid and equals 12.

BE is the other base of the trapezoid and equals 6·sqrt(2) + 12 + 6·sqrt(2) = 12 + 12·sqrt(2).

 

The area of the trapezoid is (1/2) · height · (base #1 + base #2)

                                       =    (1/2) · 6·sqrt(2) · ( 12 + 12·sqrt(2) )

                                       =    3 ·sqrt(2) · ( 12 + 12·sqrt(2) )

                                       =  36·sqrt(2) + 72

 Apr 15, 2018

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