BL and Cm are medians of triangle ABC right angled at A
Prove that:
4(BL square + CM square)=5BC square

Asked by Kartiki | 26th Aug, 2016, 07:46: PM

Expert Answer:

begin mathsize 16px style In space increment ABC comma
BC squared equals AB squared plus space AC squared... left parenthesis By space Pythagoras space theorem right parenthesis
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis straight i right parenthesis
In space increment ABL comma
BL squared equals AL squared plus space AB squared... left parenthesis By space Pythagoras space theorem right parenthesis
rightwards double arrow BL squared equals AC squared over 4 plus space AB squared... left parenthesis Since space BL space is space the space median space on space AC right parenthesis
rightwards double arrow 4 BL squared equals AC squared plus space 4 AB squared... left parenthesis ii right parenthesis
In space increment AMC comma
CM squared equals AC squared plus space AM squared... left parenthesis By space Pythagoras space theorem right parenthesis
rightwards double arrow CM squared equals AC squared plus AB squared over 4... left parenthesis Since space CM space is space the space median space on space AB right parenthesis
rightwards double arrow 4 CM squared equals 4 AC squared plus space AB squared... left parenthesis iii right parenthesis
Adding space equations space left parenthesis ii right parenthesis space and space left parenthesis iii right parenthesis comma space we space get
4 BL squared plus 4 CM squared equals AC squared plus space 4 AB squared plus 4 AC squared plus space AB squared
rightwards double arrow 4 BL squared plus 4 CM squared equals 5 AC squared plus space 5 AB squared
rightwards double arrow 4 BL squared plus 4 CM squared equals 5 left parenthesis AC squared plus space AB squared right parenthesis
rightwards double arrow 4 left parenthesis BL squared plus CM squared right parenthesis equals 5 BC squared....... left parenthesis from space left parenthesis straight i right parenthesis right parenthesis end style

Answered by Rebecca Fernandes | 28th Aug, 2016, 09:23: AM