Question

Prove the centroid of a triangle cuts the median 2:1.

A) Midpoint Theorem
B) Angle Bisector Theorem
C) Centroid Theorem
D) Pythagorean Theorem

Answer 1

To prove that the centroid divides the median in a 2:1 ratio, we apply the Centroid Theorem. This theorem specifically states the proportion in which the centroid cuts the median, resulting in segments with a 2:1 length ratio.

Step-by-step explanation:

The question is asking to prove that the centroid of a triangle divides the median in a 2:1 ratio. The correct theorem to use for this proof is the Centroid Theorem.

In any triangle, the centroid is the point of intersection of the triangle's three medians (the segments from each vertex to the midpoint of the opposite side). The Centroid Theorem states that the centroid divides each median into two segments, with the segment nearest the vertex being twice as long as the segment that connects to the midpoint of the side. This means that if the length of the entire median is 3x, the longer segment adjacent to the vertex is 2x and the shorter section from the centroid to the midpoint is x, thus establishing the 2:1 ratio.