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Points $X$, $Y$, and $Z$ are on the sides $\overline{QR}$, $\overline{PR}$, and $\overline{PQ}$, respectively, of right triangle $PQR$ such that $PZXY$ is a square.  If $PQ = 6$ and $PR = 6$, then what is the side length of the square?

 Dec 16, 2023
 #1
avatar+129449 
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           Q

                   

6         Z                X

                         

      

          P                Y               R

                            6

 

Let  the side of the square  = S

 

QZ / ZX  = XY / YR

QZ / S  = S / (6 - S)

QZ  =  S^2 / ( 6 - S)

 

6 = QZ  + S

6 = S^2 / ( 6 - S) + S

6 = [ S^2 + S ( 6 - S) ] / (6 - S)

6 (6 - S) = S^2 + 6S - S^2

36 - 6S  = 6S

36  = 12S

S  = 3

 

cool cool cool

 Dec 16, 2023

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