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.

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.

Answer 1

For point P to be the centroid of triangle JKL, it must satisfy certain properties. The centroid of a triangle is the point where the three medians intersect. The median is a line segment that joins a vertex of the triangle to the midpoint of the opposite side.

Given the options you provided, the following statement must be true for point P to be the centroid:

JM = ML, LO = OK, and KN = NJ.

This statement indicates that the segments created by the intersection of the medians (from points J, K, and L to the midpoints of the opposite sides) are equal. This equality is a characteristic of the centroid, which divides each median into two segments such that one segment (from the vertex to the centroid) is twice the length of the other segment (from the centroid to the midpoint of the opposite side).

The other options, such as perpendicular bisector conditions or equal side lengths, do not specifically pertain to the definition or properties of a centroid in triangles.