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We have a right triangle $\triangle ABC$ where the legs $AB$ and $BC$ have lengths $6$ and $3\sqrt{3},$ respectively. Medians $AM$ and $CN$ meet at point $P.$ What is the length of $CP$?

May 16, 2020

#1
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Medians AM and CN intersect at the centroid, so CP = 2*sqrt(3).

May 16, 2020
#2
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We have right triangle(ABC) with angle(B) the right angle.

AB = 6     N is the midpoint of AB, so BN = 3.

Triangle(NBC) is also a right triangle, with sides BN = 3 and BC = 3sqt(3).

By the Pythagorean Theorem, CN2  =  BN2 + BC2

CN2  =  (3)2 + ( 3sqrt(3) )2

CN2  =  9 + 27  =  36

CN  =  6

The point P is the centroid of the triangle, so CP = 2/3rds of CN.

This makes CP = 4.

May 17, 2020