Question : In triangle XYZ, G is the centroid. If XY = 11 cm, YZ = 14 cm, and XZ = 7 cm, then what is the value of GM?
Option 1: 6 cm
Option 2: 4 cm
Option 3: 2 cm
Option 4: 3 cm
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Correct Answer: 2 cm
Solution : Given: XY = 11 cm, YZ = 14 cm, and XZ = 7 cm In $\triangle XYZ$, where XM is the median $(XY)^2+ (XZ)^2= 2(YM^2+ XM^2)$ Putting the values, we get: ⇒ $11^2+7^2=2(7^2+XM^2)$ ⇒ $121 + 49 = 98 +2 XM^2$ ⇒ $XM^2 = \frac{72}{2} = 36$ ⇒ $XM = 6\ \text{cm}$ We know that the centroid divides the median in ratio 2 : 1. Thus, $GM = \frac{1}{3} × 6 = 2\ \text{cm}$ Hence, the correct answer is 2 cm.
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