In a quadrilateral ABCD it is being given that M is the midpoint of AC.Prove that ar(ABMD)=ar(DMBC)

Asked by suprakash | 23rd Dec, 2015, 10:41: PM

Expert Answer:

 
begin mathsize 14px style In space increment ADC comma space since space straight M space is space the space midpoint space of space AC comma space DM space is space the space median. And comma space median space divides space the space triangle space into space 2 space triangles space of space equal space areas. rightwards double arrow straight A left parenthesis triangle ADM right parenthesis equals straight A left parenthesis triangle CDM right parenthesis space space space space space space.... left parenthesis 1 right parenthesis Also comma space in space triangle ABC comma space straight M thin space is space the space midpoint space of space AC comma space BM space is space the space median. rightwards double arrow straight A left parenthesis triangle ABM right parenthesis equals straight A left parenthesis triangle CBBM right parenthesis space space space space space.... left parenthesis 2 right parenthesis Adding space equations space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis comma space we space get straight A left parenthesis triangle ADM right parenthesis plus straight A left parenthesis triangle ABM right parenthesis equals straight A left parenthesis triangle CDM right parenthesis plus straight A left parenthesis triangle CBM right parenthesis rightwards double arrow straight A left parenthesis ABMD right parenthesis equals straight A left parenthesis DMBC right parenthesis space space space space.... left parenthesis proved right parenthesis end style

Answered by Rashmi Khot | 24th Dec, 2015, 10:16: AM

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