Question

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.

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

Answer: C

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

Explanation:

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Answer 2

For point P to be the centroid of triangle JKL, the following must be true:

- P must be the intersection point of the three medians of the triangle, which are the line segments connecting each vertex to the midpoint of the opposite side.

- Each median must pass through P, dividing the median into two equal parts.

- The centroid is the center of mass of the triangle, so the three medians must intersect at a point that divides each median into two parts in the ratio of 2:1.

Option 3 satisfies all these conditions. If LN is a perpendicular bisector of JK, then it passes through the midpoint of JK, dividing it into two equal parts. Similarly, JO is a perpendicular bisector of LK and MK is a perpendicular bisector of JL, so they each pass through the midpoint of the opposite side, dividing it into two equal parts. Therefore, all three medians pass through P and divide each median into two parts in the ratio of 2:1, making P the centroid of triangle JKL.