ABCD is a rectangle with side length AB = 3 and BC =11, and AECF is a rectangle with side AF = 7 and FC = 9. The two rectangles overlap, and the area that is common to both rectangles = m/n, where m and n are relatively positive integers. Find m + n.

The value of m + n is 28.

Expert Answers

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Refer to the attached image. Denote the point of intersection of AD and CE as G and the length of GE as `x , ` then `GC = 7 - x ` and `GD = sqrt ( ( 7 - x )^2 - 9 ) .`

The angles EAG and DCG are equal as complementary angles to the vertical angles EGA and DGC. Because of this, `tan ( EAG ) = x / 9 = sqrt ( ( 7 - x )^2 - 9 ) / 3 , ` or `x = 3 sqrt ( x^2 - 14x +40 ) .`

Square this equation: `x^2 = 9x^2 - 126x + 360 , ` or `8x^2 - 126x + 360 = 0 , ` or `4x^2 - 61x + 180 = 0 .`

The discriminant is `D = 61^2 - 16 * 180 = 841 = 29^2 , ` so the roots are

`x_( 1 , 2 ) = ( 61 +- 29 ) / 8 , ` `x_1 = 4 , ` `x_2 = 90 / 8 .`

The second root is greater than 7, so we ignore it. Now, `x = 4 , ` and the area of the overlapping part is the area of AECF minus twice the area of AEG; in other words,

`7 * 9 - 9x = 9 ( 7 - x ) = 27 = 27 / 1 .`

This way, m = 27, n = 1, and m + n = 28.

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