Question

in triangle def, dg = 10 cm. what is cg?
5 cm
10 cm
15 cm
20 cm

Explanation:

Step1: Identify centroid property

Point $G$ is the centroid of $\triangle DEF$ (since $A, B, C$ are midpoints of the sides, so the lines are medians, and their intersection is the centroid). The centroid divides each median into a ratio of $2:1$, where the segment from vertex to centroid is twice the segment from centroid to the midpoint of the opposite side.

Step2: Apply ratio to $DG$ and $CG$

For median $DC$, $DG:CG = 2:1$. Given $DG = 10$ cm, let $CG = x$. Then $\frac{DG}{CG} = \frac{2}{1}$.
Substitute $DG = 10$:
$\frac{10}{x} = \frac{2}{1}$
Solve for $x$: $2x = 10 \implies x = \frac{10}{2} = 5$

Answer:

5 cm