In a triangle ABC, AB = 12, BC = 18 and AC = 15. The medians AX and BY intersect sides BC and AC at X and Y respectively. If AX and BY intersect each other at O, then what is the value of OX?
(√23)/(√2)
O (intersection of medians AX and BY) is the centroid, so \(OX=\tfrac{1}{3}AX\).
Step 1 — Median length (Apollonius):
\[AX=\tfrac{1}{2}\sqrt{2AB^2+2AC^2-BC^2}=\tfrac{1}{2}\sqrt{2(144)+2(225)-324}=\tfrac{1}{2}\sqrt{414}=\tfrac{3\sqrt{46}}{2}\]
Step 2: \(OX=\tfrac{1}{3}\cdot\tfrac{3\sqrt{46}}{2}=\tfrac{\sqrt{46}}{2}=\dfrac{\sqrt{23}}{\sqrt 2}\).
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