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Help pls :)

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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

2+0 Answers

#1
+8341
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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
+1326
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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

Jul 7, 2020