Question

select the correct answer. given: △abc with median segments \\(\overline{ax}\\), \\(\overline{by}\\), and \\(\overline{cz}\\) prove: medians meet at point o. it is given that △abc has median segments \\(\overline{ax}\\), \\(\overline{by}\\), and \\(\overline{cz}\\). because ____, then \\(az = zb = \frac{1}{2}\\), \\(ay = cy = \frac{1}{2}\\), and \\(bx = cx = \frac{1}{2}\\). the ratios of az to zb is 1, of ay to cy is 1, and of bx to cx is 1 by substitution. therefore, △aoc, △boc, and △aob are similar to each other. then the medians meet at point o. what is the reasoning for the second step? a. medians intersect at one point b. medians divide each side of the triangle into two parts c. medians divide each side of the triangle in half d. medians intersect at multiple points

Explanation:

To determine the reasoning for the second step, we analyze the properties of medians in a triangle. A median of a triangle is a segment from a vertex to the midpoint of the opposite side. By definition, a median divides the opposite side into two equal (or half) parts.

• Option A states medians intersect at one point, which is true (the centroid), but this doesn't explain why \( AZ = ZB \), \( AY = CY \), or \( BX = CX \).
• Option B says medians divide each side into two parts, but it doesn't specify they are equal parts.
• Option C says medians divide each side of the triangle in half, which is the definition of a median (connecting a vertex to the midpoint of the opposite side, hence dividing the side into two equal - half - length segments). This explains why \( AZ = ZB \), \( AY = CY \), and \( BX = CX \) (since \( \overline{AX} \), \( \overline{BY} \), and \( \overline{CZ} \) are medians).
• Option D is incorrect because medians intersect at exactly one point (the centroid), not multiple points.

Answer:

C. medians divide each side of the triangle in half