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# ​Geometry- triangle medians and altitudes.

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What I've done so far is extend a lie from point Q to R to form an isoceles triangle. Then using the side lengths given in the problem, I found that line QR was \sqrt{193}. I don't really know what to do from there. Should I form a ratio for triangle QNR and ORN since they appear to be similar?

Jul 29, 2020

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QN = 12                PR = 14       PN = 7

F is a foot of the altitude from M to PN

Find angle QPN using QN and PN

Find QP and MP using the Pythagorean theorem

Find MF and PF using angle QPN and MP

Find angle MRF using MF and FR

And finally, find  OR = 5*sqrt(2).

Jul 29, 2020
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"And finally, find  OR = 5*sqrt(2)"         5*sqrt(2) = 7.07106781   INCORRECT !!!

The correct answer is:        OR = 7 / cos(29.7448813º) = sqrt(65)

Guest Jul 29, 2020
edited by Guest  Jul 29, 2020
#2
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There's a shortcut:::

Important:   The ratio of  QO  to  NO is  2 : 1        ( The Median Theorem )

NR = 1/2 * 14 = 7            ( QN ⊥ PR )

NO = 1/3 * 12 = 4

OR = sqrt (72 + 42) = √65

Jul 29, 2020
edited by Dragan  Jul 29, 2020
#3
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Ohh so I just needed to apply this theorem?

gwenspooner85  Jul 29, 2020