Question

In the diagram, which must be true for point P to be the centroid of the triangle?

LN ⊥ JK, JO ⊥ LK, and JL ⊥ MK.
JL = LK = KJ
JM = ML, LO = OK, and KN = NJ.
LN is a perpendicular bisector of JK, JO is a perpendicular bisector of LK, and MK is a perpendicular bisector of JL.

Triangle J K L is shown. Lines are drawn from each point to the opposite side and intersect at point P. Line segments J O, K M, and L N are created.
I

Answer 1

Option 3: JM = ML, LO = OK, and KN = NJ.

Reason: the centroid is the intersection of the medians. That requires each cevian (JO, KM, LN) to connect a vertex to the midpoint of the opposite side, which is exactly the equalities in option 3. Options 1 and 4 describe altitudes or perpendicular bisectors (not medians), and 2 (equilateral) is not required.