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ABCD is a square with AB = 25. P is a point within ABCD such that PA = 24 and PB = 7. What is the value of PD^2?

 Jan 5, 2021
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Let A =(0,0)  B =(25,0)  C =(25,25) D =(0,25)

APB  is a 7-24-25 right triangle 

Its area is   (1/2) ( 7) (24) = 84

 

Also...its  area  =   (1/2)  AB   *  height

 

84 =  (1/2) 25  * height

 

168/25  = height =  y coordinate of P  = 6.72

 

And using the  equation  of a circle  x^2  + y^2 = 24^2...we can find the x coordinate as

 

x^2 +  (6.72)^2  =  24^2

x=  sqrt  [  24^2 - ( 6.72)^2  ]  =  23.04

 

So DP^2   =

 

( 23.04)^2  +  ( 25 - 6.72)^2   =   865

 

Also.....note something interesting.... 

 

AP^2  + CP^2  =  BP^2  + DP^2

24^2  + CP^2   = 7^2  + 865

576  + CP^2   = 914

CP^2 = 914 - 576   =   338

 

 

cool cool cool

 Jan 5, 2021

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