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 $CP$?
+23254 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.
