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In the diagram, the circle is inscribed in the square.  This means that the circle and the square share points $S$, $T$, $U$, and $V$, and the width of the square is exactly equal to the diameter of the circle.  Rounded to the nearest tenth, what is $XY/XV$, to the nearest tenth?

 

 Sep 10, 2023

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Let the width = W

Let X = (W, W)

Let V = ( (1/2)W , 0)

 

VX =   sqrt  ( (W - (1/2)W)^2  + W^2  =  sqrt [ W/4 + W^2 ]  = Wsqrt (5) / 2

 

XY =  W sqrt (2)

 

XY / XV =  W sqrt (2)  / ( (W sqrt (5) / 2)  =  2sqrt (2) / sqrt (5)  =  sqrt (8) / sqrt (5)  = sqrt (8/5) ≈ 1.3

 

 

cool cool cool

 Sep 10, 2023

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