Question

The point $Q$ is the image of the point $P(1,5)$ about the line $y=x$ and $R$ is the image of the point $Q$ about the line $y=-x$. The circumcenter of the $△PQR$ is

• A. $(5,1)$
• B. $(-5,1)$
• C. $(1,-5)$
• D. $(0,0)$

Answer 1

The correct option is D $(0,0)$

Since $Q$ is the image of $P$ about the line $y=x,PA=QA$ and $PQ$ is perpendicular to the line $y=x$.

So, the line $y=x$ bisects side $PQ$ and is perpendicular to it.

So. the line $y=x$ is the perpendicular bisector of side $PQ$.

Similarly, the line $y=-x$ is the perpendicular bisector of side $QR$.

Circumcentre of a triangle is the point of intersection of perpendicular bisectors of its sides.

So, the circumcentre of triangle $\Delta PQR$ is the point of intersection of perpendicular bisectors of its sides,i.e, the lines $y=x$ and $y=-x$.

Origin $(0,0)$ is the point of intersection of lines $y=x$ and $y=-x$.

Hence, $(0,0)$ is the circumcentre of $\Delta PQR$

So, the answer is option (D).