Question

in the following diagram, point \\(g\\) represents the centroid of the triangle. which of the following statements is true?

\\(\bigcirc \overline{ag} = \frac{1}{2}\overline{gf}\\)
\\(\odot \overline{ag} = \frac{2}{3}\overline{af}\\)
\\(\bigcirc \overline{gf} = \frac{2}{3}\overline{af}\\)
\\(\bigcirc \overline{ag} = \overline{cg}\\)

Explanation:

Identify the properties of a centroid

The problem states that point \(G\) is the centroid of \(\triangle ABC\). By definition, the centroid of a triangle is the intersection of its medians. A median is a line segment connecting a vertex to the midpoint of the opposite side. Here, \(AF\) is a median from vertex \(A\) to side \(BC\).

Apply the Centroid Theorem

The Triangle Centers concept, specifically the Centroid Theorem, states that the centroid of a triangle divides each median into two segments in a \(2:1\) ratio. The segment from the vertex to the centroid is twice as long as the segment from the centroid to the opposite side.
For median \(AF\) with centroid \(G\):

$$ AG = 2 \cdot GF $$

This means the total length of the median is:

$$ AF = AG + GF = 2 \cdot GF + GF = 3 \cdot GF $$

Express segment ratios

Using the relationships from the previous step, we can express the lengths of the segments relative to the entire median \(AF\):

$$ AG = \frac{2}{3} AF $$

$$ GF = \frac{1}{3} AF $$

Evaluate the given statements

Let's check each of the multiple-choice options:

• \(\overline{AG} = \frac{1}{2} \overline{GF}\): Incorrect, because \(AG = 2 \cdot GF\).
• \(\overline{AG} = \frac{2}{3} \overline{AF}\): Correct, as derived in the previous step.
• \(\overline{GF} = \frac{2}{3} \overline{AF}\): Incorrect, because \(GF = \frac{1}{3} AF\).
• \(\overline{AG} = \overline{CG}\): Incorrect, the distances from the centroid to different vertices are not necessarily equal unless the triangle is equilateral.

Answer:

• (A) \(\overline{AG} = \frac{1}{2} \overline{GF}\)
• (B) \(\overline{AG} = \frac{2}{3} \overline{AF}\) (Correct answer)
• (C) \(\overline{GF} = \frac{2}{3} \overline{AF}\)
• (D) \(\overline{AG} = \overline{CG}\)