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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=14, then find OR.

 

Thank you in advance

 Jul 7, 2020
 #1
avatar+8336 
+1

Obviously. O is the centroid of \(\triangle PQR\).

 

Because PN = NR, and QN is perpendicular to PR, PQ = QR.

 

And, 

\(PQ^2 = PN^2 + NQ^2 = \left(\dfrac12 \cdot 14\right)^2 + 12^2 = 193\\ PQ = \sqrt{193} = QR\)

 

Using Apollonius's theorem, 

 

\(MR = \dfrac12 \sqrt{2\cdot 14^2 + 2\cdot \sqrt{193}^2 - \sqrt{193}^2} = \dfrac32 \sqrt{65}\)

 

Because centroid divides each medians into 2 : 1,

 

\(OR = \dfrac23 MR = \boxed{\sqrt{65}}\)

 Jul 7, 2020
 #2
avatar+1055 
+2

Side PR is divided by the median into 2 equal line segments.

 

Median is divided by the centroid into 2 line segments whose ratio is 1:2

 

ON = QN / 3 = 4

 

NR = PR / 2 = 7

 

OR = sqrt ( 42 + 72 ) = 8.062257748    smiley

 Jul 7, 2020

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