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If the medians of a triangle ABC intersect at G,prove that

  area of triangle BCG=1/3 area of triangle ABC

Asked by laxmi.agarwal055 21st February 2018, 7:04 PM
Answered by Expert
Answer:
begin mathsize 16px style Median space of space straight a space triangle space divides space it space into space two space triangles space of space equal space area.
rightwards double arrow Area space left parenthesis triangle ABD right parenthesis equals Area space left parenthesis triangle ACD right parenthesis space space space space space space left parenthesis AD space is space the space median right parenthesis space space space... left parenthesis straight i right parenthesis
Area space left parenthesis triangle GBD right parenthesis equals Area space left parenthesis triangle GCD right parenthesis space space space space space space left parenthesis GD space is space the space median space space space space space... left parenthesis ii right parenthesis
Subtracting space left parenthesis ii right parenthesis space from space left parenthesis straight i right parenthesis comma
rightwards double arrow Area space left parenthesis triangle AGB right parenthesis equals Area space left parenthesis triangle AGC right parenthesis space space space space space space.... left parenthesis iii right parenthesis
Similarly. space Area space left parenthesis AGB right parenthesis equals Area space left parenthesis triangle BGC right parenthesis space space space space space space space.... left parenthesis iv right parenthesis
From space left parenthesis iii right parenthesis space and space left parenthesis iv right parenthesis comma space
Area space left parenthesis triangle AGB right parenthesis equals Area space left parenthesis BGC right parenthesis equals Area space left parenthesis AGC right parenthesis
But space left parenthesis triangle AGB right parenthesis plus Area space left parenthesis BGC right parenthesis plus Area space left parenthesis AGC right parenthesis equals Area space left parenthesis triangle ABC right parenthesis
rightwards double arrow 3 cross times Area space left parenthesis triangle BGC right parenthesis equals Area space left parenthesis triangle ABC right parenthesis
rightwards double arrow Area space left parenthesis triangle BGC right parenthesis equals 1 third Area space left parenthesis triangle ABC right parenthesis end style
Answered by Expert 22nd February 2018, 12:05 PM
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