Question

if g is the centroid of \\( \triangle a c e, a g = 26, b c = 44 \\), and \\( d g = 12 \\), find each measure.
answer attempt 1 out of 2
\\( g f = \\)
\\( a f = \\)
\\( f c = \\)
\\( g b = \\)
\\( d b = \\)

Explanation:

Step1: Recall centroid property

The centroid divides each median in a ratio of \(2:1\). Also, the median bisects the side of the triangle.

Step2: Find \(GF\)

Since \(G\) is the centroid and \(AG = 26\), for median \(AF\), \(AG:GF=2:1\). Let \(GF=x\), then \(AG = 2x\). Given \(AG = 26\), so \(2x=26\), \(x = 13\). So \(GF=13\)

Step3: Find \(AF\)

\(AF=AG + GF\). Substitute \(AG = 26\) and \(GF = 13\). Then \(AF=26 + 13=39\)

Step4: Find \(FC\)

Since \(BC = 44\) and \(F\) is the mid - point of \(BC\) (because \(AF\) is a median), \(FC=\frac{BC}{2}\). So \(FC=\frac{44}{2}=22\)

Step5: Find \(GB\)

Since \(DG = 12\) and for median \(BD\), \(BG:GD = 2:1\). Let \(BG = 2y\), \(GD=y\). Given \(y = 12\), then \(BG=2\times12 = 24\)

Step6: Find \(DB\)

\(DB=BG + GD\). Substitute \(BG = 24\) and \(GD = 12\). Then \(DB=24+12=36\)

Answer:

\(GF = 13\)
\(AF = 39\)
\(FC = 22\)
\(GB = 24\)
\(DB = 36\)