D, E, F are the mid-points of the sides BC,CA and AB respectively of triangle ABC. Prove that BDEF is a parallelogram whose area is half that of ABC. Also show that ar(tringle DEF) = 1/4 ar(triangle ABC) and ar(BDEF) = 1/2 ar(ABC).
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Since line segment joining the mid-points of two sides of a triangle is half of the third side. Therefore, D and E are mid-points of BC and AC respectively.
⇒ DE = 1 / 2 AB --- (i)
E and F are the mid - points of AC and AB respectively .
∴ EF = 1 / 2 BC --- (ii)
F and D are the mid - points of AB and BC respectively .
∴ FD = 1 / 2 AC --- (iii)
DE = EF = FD [using (i) , (ii) , (iii) ]
Hence, DEF is an equilateral triangle
⇒ DE = 1 / 2 AB --- (i)
E and F are the mid - points of AC and AB respectively .
∴ EF = 1 / 2 BC --- (ii)
F and D are the mid - points of AB and BC respectively .
∴ FD = 1 / 2 AC --- (iii)
DE = EF = FD [using (i) , (ii) , (iii) ]
Hence, DEF is an equilateral triangle
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