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Question
In ΔABC, E and F are mid-points of sides AB and AC respectively. If BF and CE intersect each other at point O,
prove that the ΔOBC and quadrilateral AEOF are equal in area.
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Solution
E and F are the midpoints of the sides AB and AC.
Consider the following figure.
Therefore, by midpoint theorem, we have, EF || BC
Triangles BEF and CEF lie on the common base EF and between the parallels, EF and BC
Therefore, Ar.( ΔBEF ) = Ar.( ΔCOF )
⇒ Ar.( ΔBOE ) + Ar.( ΔEOF ) = Ar.( ΔEOF ) + Ar.( ΔCOF )
⇒ Ar.(ΔBOE ) = Ar.( ΔCOF )
Now BF and CE are the medians of the triangle ABC
Medians of the triangle divide it into two equal areas of triangles.
Thus, we have, Ar. (ΔABF) = Ar. (ΔCBF)
Subtracting Ar. ΔBOE on both the sides, we have
Ar. (ΔABF) - Ar. (ΔBOE) = Ar. (ΔCBF) - Ar. (ΔBOE)
Since, Ar. ( ΔBOE ) = Ar. ( ΔCOF ),
Ar. (ΔABF) - Ar. (ΔBOE) = Ar. (ΔCBF) - Ar. (ΔCOF)
Ar. ( quad. AEOF ) = Ar. ( ΔOBC ) , hence proved
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