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geometry problem

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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.

Jun 15, 2022

#1
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Here :  https://web2.0calc.com/questions/geometry_95436

Jun 15, 2022
#2
+9458
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By Menelaus' theorem,

$$\dfrac{QO}{ON} \cdot \dfrac{NR}{RP}\cdot \dfrac{PM}{MQ} = 1\\ \dfrac{QO}{ON} \cdot \dfrac12 \cdot \dfrac11 = 1\\ \dfrac{QO}{ON} = 2\\ QO = 2 ON$$

Since QN = 12, QO = 8 and ON = 4.

Since PN = NR and PR = 18, NR = 9.

Then by Pythagorean theorem,

$$OR = \sqrt{NR^2 + ON^2} = \sqrt{4^2 + 9^2} = \sqrt{97}$$

Jun 15, 2022
edited by MaxWong  Jun 15, 2022