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Given that BDEF is a square and the area of triangle ABC is 1, find the number of square units in the area of the regular octagon.

 

 Nov 14, 2020
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Consider triangle ABC...... the interior angle of a regular octagon = (6/8) 180  = 135°

So angles BCA and BAC are supplemental to this =  45°

Thus sides BA = BC 

And ABC is a right angle...so BA and BC are legs

And calling the length of these  sides, S......we  have  that

 

area of ABC =  (1/2) (product of legs)  =   (1/2) S^2

1  = (1/2) S^2

2 = S^2

S = sqrt (2)

 

Then AC =  sqrt [ BA^2  + BC^2 ]  =  sqrt  [  (sqrt (2))^2  + (sqrt (2))^2 ]  = sqrt [ 2 + 2]  = sqrt 4  =  2

 

Then the side of the octagon =  2

 

A formula for the octagon's area =  2 ( 1 + sqrt 2 ) * S^2 =

 

2 ( 1 + sqrt 2) * 2^2  =

 

8 ( 1 + sqrt 2)  units ^2  ≈   19.3 units^2

 

 

cool cool cool

 Nov 14, 2020

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