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Question
If AD is a median of a triangle ABC, then prove that triangles ADB and ADC are equal in
area. If G is the mid-point of median AD, prove that ar (Δ BGC) = 2 ar (Δ AGC).
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Solution
Draw AM ⊥ BC
Since, AD is the median of ΔABC
∴ BD = DC
⇒ BD = AM = DC × AM
⇒ ` 1/2 (BD xx AM ) = 1/2 ( DC xx AM)`
⇒ ar (Δ ABC) = ar (Δ ACD) ........ (1)
In ΔBGC , GDis the median
∴ ar (BGD) = area (OGD) ......... (2)
In ΔACD , CG is the median
∴ area (AGC) = area (Δ CGD) ......... (3)
From (1) and (2) , we have
Area (ΔBGD) = ar (Δ AGC)
But, ar (ΔBGC) = 2ar (BGD)
∴ ar (BGC ) = 2ar (Δ AGC)
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