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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