Question

in \\( \triangle cde \\), \\( i \\) is the centroid. if \\( g e = 26 \\), find \\( d g \\).

Explanation:

Step1: Recall the centroid property

The centroid of a triangle divides each median into a ratio of \(2:1\). Also, the length of the whole median \(DG + GE\) (since \(G\) is on the median from \(D\) to \(E\) and \(I\) is the centroid). And the centroid property implies that \(DG=\frac{2}{3}\) of the median \(DE\) and \(GE = \frac{1}{3}\) of the median \(DE\). So, \(DE=DG + GE\) and \(DG = 2GE\) (another way to look at the ratio: if we let \(DG = 2x\) and \(GE=x\), then \(DE=3x\)).

Step2: Substitute the value of \(GE\)

Given \(GE = 26\). Since \(DG = 2GE\) (from the centroid - median ratio property). Then \(DG=2\times26\).

Answer:

\(52\)