Question

If $(2,1),(-1,-2),(3,3)$ are midpoints of sides $BC,CA,AB$ of $\Delta ABC$, then equation of $AB$ is

• A. $x-y=\frac{1}{2}$
• B. $x+y=1$
• C. $x-y=9$
• D. $x=y$

Answer 1

Answer: (D)

$x=y$

In $\Delta ABC$, $E$ and $D$ are mid points of $AC$ and $BC$

$\therefore AB\parallel DE$

$\therefore$ Slope of AB$\left(m_{AB}\right)$= Slope of $DE\left(m_{DE}\right)$ ............(i)

Now $ED$ passes through $E=(x_1,y_1)=(-1,-2)$ and $D=(x_2,y_2)=(2,1)$

Therefore, by (i)

$\left(m_{AB}\right)=\left(m_{DE}\right)=\frac{y_2-y_1}{x_2-x_1}=\frac{2-(-1)}{1-(-2)}=1$ ..........(ii)

We take the equation of $AB$ as $y=mx+c$ ..........(iii)

Also $AB$ passes through $(3,3)$

$\Rightarrow (x,y)=(3,3)$ ......(iv)

$\therefore$ substituting from (ii) and (iv), equation (iii) becomes $3=1\times 3+c$

$\Rightarrow c=0$

$\therefore$ equation of $AB$ is $y=x$

$\Rightarrow x=y$