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Let B,A,D be three consecutive vertices of a regular 18-gon.  A regular heptagon is constructed on AB, with a vertex C next to A. Find BCD in degrees. 

 

In the diagram below, each side of convex quadrilateral ABCD is trisected. The area of ABCD is 180. Find the area of the shaded region. Thanks, pls help quick!

Noori

 Jul 4, 2020
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First problem: By coordinates 

 

A = (1,0)

B = (0.924328, 0.471208)

C = (1.222783, -1.147542)

D = (0.924328, -0.471208)

 

==> angle BCD = 75 degrees

 

Second problem: 

 

Triangle(BPS) is similar to Triangle(BAC)     --->     Area(BPS) / Area(BAC)  =  BP2 / BA2  =  22 / 32  =  4/9

     Consequently, Area(APSC)  =  5/9 · Area(ABC)

 

Similarly, Area(DWT) / Area(DAC)  =  4/9     --->     Area(WACT)  =  5/9 · Area(DAC)

 

Therefore, Area(APSCTW)  =  5/9 · Area(ABCD)     --->     Area(APSCTW)  =  5/9 · 180  =  20

 Jul 4, 2020

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