# If in a parallelogram ABDC, the coordinates of A,B and C are respectively (1,2), (3,4) and (2,5) , then the equation of the diagonal AD is:Option 1)$5x-3y+1=0$Option 2)$3x-5y+1=0$Option 3)$5x+3y-11=0$Option 4)$3x+5y-13=0$

P Plabita

Mid-point formula -

$x= \frac{x_{1}+x_{2}}{2}$

$y= \frac{y_{1}+y_{2}}{2}$

- wherein

If the point P(x,y) is the mid point of line joining A(x1,y1) and B(x2,y2) .

Two – point form of a straight line -

$y-y_{1}=\left ( \frac{y_{2}-y_{1}}{x_{2}-x_{1}} \right )(x-x_{1})$

- wherein

The lines passes through  $(x_{1}y_{1})$  and  $(x_{2}\, y_{2})$

As BD and AC are parallel

$\frac{n-4}{m-3}=\frac{5-2}{2-1}$

${n-4}=3({m-3})$..............................(1)

As AB and CD are parallel

$\frac{n-5}{m-2}=\frac{4-2}{3-1}=\frac{2}{2}=1$

${n-5}=({m-2})$..............................(2)

Solving (1) and (2)

m=4 and n=7

$\vec{DA} \: \: is\: \: (y-2)=(\frac{7-2}{4-1})(x-1)$

$=>5x-3y+1=0$

Option 1)

$5x-3y+1=0$

Option 2)

$3x-5y+1=0$

Option 3)

$5x+3y-11=0$

Option 4)

$3x+5y-13=0$

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