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D,E and F are midpoints of the sides BC,CA and AB respectively of a triangle ABC.prove that area of parallelogram FDCE=1/2 area of triangle ABC

Asked by laxmi.agarwal055 21st February 2018, 7:12 PM
Answered by Expert
Answer:
 
begin mathsize 16px style DF space is space straight a space diagonal space of space parallelogram space BDEF comma
rightwards double arrow Area left parenthesis triangle BDF right parenthesis equals Area left parenthesis triangle DEF right parenthesis space space space space space space.... left parenthesis straight i right parenthesis
Similarly comma space we space can space show space that
Area space left parenthesis triangle DCE right parenthesis equals Area left parenthesis triangle DEF right parenthesis space space space space space space space space.... left parenthesis ii right parenthesis
Area space left parenthesis triangle AFE right parenthesis equals Area left parenthesis triangle DEF right parenthesis space space space space space space space space space.... left parenthesis iii right parenthesis
From space left parenthesis straight i right parenthesis comma space left parenthesis ii right parenthesis space and space left parenthesis iii right parenthesis comma space we space have
Area space left parenthesis triangle BDF right parenthesis equals Area space left parenthesis triangle DCE right parenthesis equals Area space left parenthesis triangle AFE right parenthesis equals Area space left parenthesis triangle DEF right parenthesis
But comma space Area space left parenthesis triangle BDF right parenthesis plus Area space left parenthesis triangle DCE right parenthesis plus Area space left parenthesis triangle AFE right parenthesis plus Area space left parenthesis triangle DEF right parenthesis equals Area space left parenthesis triangle ABC right parenthesis
rightwards double arrow 4 space Area space left parenthesis triangle DEF right parenthesis equals Area space left parenthesis triangle ABC right parenthesis
rightwards double arrow Area space left parenthesis triangle DEF right parenthesis equals 1 fourth Area space left parenthesis triangle ABC right parenthesis
Now comma space Area space of space parallelogram space FDCE equals 2 cross times Area space left parenthesis triangle DEF right parenthesis equals 2 cross times 1 fourth Area space left parenthesis triangle ABC right parenthesis equals 1 half space Area space left parenthesis triangle ABC right parenthesis end style
Answered by Expert 22nd February 2018, 11:34 AM
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