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M is the midpoint of PQ and N is the midpoint of PR, and O is the intersection of QN and RM, as shown. If QN is perpendicular to PR, QN = 12, and PR = 18, then find OR.

 

 Feb 11, 2021
 #1
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Draw  MJ  perpendicular  to  PR  to meet PR  at J

Draw  MK  parallel  to  PR to meet QN  at  K

 

MK  divides  QN  into   equal  parts  so   KN  =  6 =   MJ  

 

And

 

PJ  / MJ  = PN /NQ

PJ/ 6  = 9/12 =  3/4

PJ  =  18/4  =  9/2  = 4.5

So  RJ   =  18 - 4.5      = 13.5

 

And

ON / NR   = MJ / RJ

ON / 9  = 6 / 13.5

ON  =  54/13.5   = 4

  

 

So using the Pythagorean Theorem:

 

OR  = sqrt  ( NR^2  +   ON^2)  =  sqrt ( 4^2  + 9^2)    = sqrt (97)

 

 

cool cool cool

 Feb 11, 2021

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