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Question
The medians BE and CF of a triangle ABC intersect at G. Prove that the area of ∆GBC = area of the quadrilateral AFGE.
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Solution
Given: The medians BE and CF of a triangle ABC intersect at G
To prove: ar (ΔGBC) = ar (AFCE)
Proof: As median CF divides a triangle into triangle of equal area. So, ar (ΔBCF) = ar (ΔACF)
ar (ΔGBF) + ar (ΔGBC) = ar (AFGE) + ar (ΔGCE) ...(I)
Now, median BE divides a triangle into two triangle of equal area.
So, ar (ΔGBF) + ar (AFGE) = ar (ΔGCE) + ar (ΔGBC) ...(II)
Now, subtracting (II) from (I), we get
ar (ΔGBC) – ar (AFGE) = ar (ΔAFGE) – ar (ΔGBC)
ar (ΔGBC) + ar (ΔGBC) = ar (ΔAFGE) + ar (ΔAFGE)
2ar (ΔGBC) = 2ar (AFGE)
Hence, ar (ΔGBC) = ar (AFGE).
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