SOLUTION: In the given figure, the two rectangles EFGH and ACDE share a common corner at E and overlap so that BC = 7. What is the area of the shaded region ABGFEA? Image is in the link:

Algebra ->  Finance -> SOLUTION: In the given figure, the two rectangles EFGH and ACDE share a common corner at E and overlap so that BC = 7. What is the area of the shaded region ABGFEA? Image is in the link:       Log On


   

Question 1061892: In the given figure, the two rectangles EFGH and ACDE share a common corner at E and overlap so that BC = 7. What is the area of the shaded region ABGFEA?
Image is in the link: http://prntscr.com/dkmau9
Answer by ikleyn(52794) About Me  (Show Source):
You can put this solution on YOUR website!
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In the given figure, the two rectangles EFGH and ACDE share a common corner at E and overlap so that BC = 7.
What is the area of the shaded region ABGFEA?
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1.  Consider the quadrilateral ABHE (which is the intersection area of the original rectangles).


2.  Draw the diagonal BE in this quadrilateral.

    The length of the diagonal BE is  |BE| = sqrt%28abs%28AE%29%5E2%2Babs%28AB%29%5E2%29 = sqrt%2812%5E2%2B1%5E2%29 = sqrt%28145%29.


3.  Therefore, the length of the segment BH is  |BH| = sqrt%28abs%28BE%29%5E2-abs%28HE%29%5E2%29 = sqrt%28145-8%5E2%29 = sqrt%2881%29 = 9.


4.  The area of the triangle ABE is %281%2F2%29%2Aabs%28AB%29%2Aabs%28AE%29 = %281%2F2%29%2A12%2A1 = 6.

    The area of the triangle BHE is %281%2F2%29%2Aabs%28BH%29%2Aabs%28HE%29 = %281%2F2%29%2A9%2A8 = 36.


    Hence, the area of the quadrilateral ABHE is the sum of the areas of the triangles 6 + 36 = 42.


5.  Now the area of the shaded part under the question is 12*8 - 42 = 96-42 = 54.

Answer. The area of the shaded region ABGFEA is 54 square units.

Beautiful problem. Thanks.