Question

In a triangle $PQR,$the coordinates of the points $P$ and $Q$ are $(-2,4)$ and $(4,2)$ respectively. If the equation of the perpendicular bisector of $PR$is $2x-y+2=0,$ then what is the circumcenter of the $∆PQR$

• A. $(-2,-2)$
• B. $(0,2)$
• C. $(-1,0)$
• D. $(1,4)$

Answer 1

The correct option is A

$(-2,-2)$

Explanation for correct option:

Step -1 : Finding the perpendicular bisector:

Let's illustrate the diagram using the given information.

Given that, the co-ordinates of $P$ is $(-2,4)$

the co-ordinates of $Q$ is $(4,2)$

Equation of the perpendicular bisector of $PR$is $2x-y+2=0\dots \dots \left(i\right)\left(\text{given}\right)$.

According to the mid point theorem,

$\left(x,y\right)=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$

Using midpoint theorem to find midpoint of $PQ$.

The midpoint of $PQ$ is $\left(\frac{4-2}{2},\frac{4-2}{2}\right)=(1,1)$ [given $x_1=-2,x_2=4,y_1=4,y_2=2$]

As we know, Slope of a line $=\frac{y_2-y_1}{x_2-x_1}$

Slope of $PQ$ is $=\frac{y_2-y_1}{x_2-x_1}=-\frac{6}{6}=-1$

When two lines are perpendicular, their slopes are negative reciprocal of each other.

Therefore, the slope of the line perpendicular to $PQ$ will be $=-1\times \frac{1}{-1}=1$

The equation of the line which is the perpendicular bisector to $PQ$ is:

$$\begin{gathered} y-y_1=m(x-x_1) \\ \Rightarrow y-1=1(x-1) \\ \Rightarrow y=x\dots \dots \left(ii\right) \end{gathered}$$

Step-2 : Finding the circumcenter of $∆PQR:$

The circumcenter is the point at which the perpendicular bisector of all the sides intersects.

Hence, the perpendicular bisector of $PQ$ and $PR$ will intersect at the circumcenter.

Equating equation $\left(ii\right)$ and $\left(i\right)$ to get the point of circumcenter.

$$\begin{gathered} 2x-y+2=0 \\ \Rightarrow 2y-y+2=0\left(\text{from equation}\right(2\left)\right) \\ \Rightarrow y=-2 \end{gathered}$$

After putting the value of $y$ in equation $\left(2\right)$, we get,

$x=-2$

Therefore, the circumcenter $∆PQR:$ $(-2,-2)$.

Hence, the correct answer is Option (A).