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In the diagram below, P is a point inside square ABCD. Find x.

 Jan 14, 2024
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We can place the diagram in the coordinate plane so that A = (0,s), B = (0,0), C = (s,0), and D = (s,s), where s is the side length of the square.  Let P = (x,y).  Then

 

x^2 + (y - s)^2 = 1

x^2 + y^2 = 4

(x - s)^2 + y^2 = 9

 

Solving, we get s = 2 + sqrt(3), x = sqrt(3)/4, y = 1/2 + sqrt(3)/8.  Then by the cosine law, angle APB = 120 degrees.

 Mar 8, 2024

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