Answer:
Given: Coordinates of ΔPQR are P( 0 , 0 ) , Q( 2a , 0 ) and R( 2b , 2c )
To Proof: Line containing the midpoints x & Y of two sides PQ & PR of a triangle is parallel to the third side QR .i.e XY║QR
First We find Mid Point of side PQ and PR say X & Y respectively and then we find slope of line XY and QR. As if there slope is equal then they are parallel.
Formula of Mid point is given by
[tex]Coordinates\,of\,Mid\,Point\,=\,(\frac{x_2+x_1}{2},\frac{y_2+y_1}{2})[/tex]
So, Coordinated of Mid Point of PQ , X = [tex](\frac{2a+0}{2},\frac{0+0}{2})[/tex]
= [tex](\frac{2a}{2},\frac{0}{2})[/tex]
= ( a , 0 )
Coordinates of Mid Point of PR, Y = [tex](\frac{2b+0}{2},\frac{2c+0}{2})[/tex]
= [tex](\frac{2b}{2},\frac{2c}{2})[/tex]
= ( b , c )
Slope of a line is given by,
[tex]Slope=\frac{y_2-y_1}{x_2-x_1}[/tex]
Slope of Line XY = [tex]\frac{c-0}{b-a}[/tex]
= [tex]\frac{c}{b-a}[/tex]
Slope of Line QR = [tex]\frac{2c-0}{2b-2a}[/tex]
= [tex]\frac{2c}{2(b-a)}[/tex]
= [tex]\frac{c}{b-a}[/tex]
Since, Slope of XY = Slope of QR
⇒ XY is parallel to QR
Hence proved
