A Sail Shaped Quadrilateral

Geometry Level 2

In the image bellow we have a A B C D ABCD quadrilateral whose shape resembles a sail from a boat. The green angle measures 22 5 o 225^o degrees and all the orange angles measure 4 5 o 45^o degrees. If the segment that connects points A A and C C measure 4 4 cm, what is the area of the A B C D ABCD quadrilateral (in cm²)?

12 8 6 10

Victor Paes Plinio by Victor Paes Plinio , 21, Brazil

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1 solution

Edwin Gray
Aug 26, 2018

Drop a perpendicular from C to AD; call the intersection E. Let CE = k, and BC = m. Then triangle BEA is a 45,45, 90 isosceles triangle, so that AE = sqrt(16 - k^2) = BE = m + k. then the area of quadrilateral ABCD = area of Triangle ABE + area of triangle CED = (1/2) sqrt(16 - k^2) sqrt(16 - k^2) + (1/2)(k)^2 = 8. Ed Gray

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