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In triangle PQR, PQ = 13, QR = 14, and PR = 18. Let M be the midpoint line QR. Find PM.

 Jul 30, 2022
 #1
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Let P = (0,0)   Q = (13,0)

We can determine the coordiantes of R  by finding the intersection of two circles

 

x^2  + y^2 =  18^2     →   x^2 + y^2  =  324  (1)

(x -13)^2  + y^2 = 14^2    →  x^2 - 26x + 169  + y^2 = 196  → x^2 - 26x  + y^2 =  27     (2) 

 

Subtract  (2) from (1)

 

26x  =  297

x = 297/26

 

And    (297/26)^2  + y^2  =  324

 

y^2  = 324  - ( 297/26)^2

 

y^2  = 130815 / 676   take the positive root for y

 

y = sqrt (130815) / 26

 

R  =  ( 297/26 ,  sqrt (130815) /26)

 

M  =  ( (297 / 26 + 13)   / 2 , sqrt (130815 / 52 )    =   (635/52 , sqrt (130815) / 52 )

 

PM  =   (1/52)  sqrt  ( 635^2 + 130815)  =  (1/52) sqrt (534050)  ≈  14.05

 

 

cool cool cool

 Jul 30, 2022

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