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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 $ΔPQ$R is-

Answer given is $(0,0)$

My attempt:

Replection of $(1,5)$ in $y=x$ is $(5,1)$ and in $y=-x$ is $(-5,-1)$. Now I don't see why origin is the circumcenter.

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Let $O = (0,0)$ then after reflection across $y=x$ $O$ goes to it self and the same is true after a reflection across $y=-x$. So $OP = OQ$ and $OP = OR$. So $O$ is on the same distance from $P,Q$ and $R$ so it must be a circumcenter of triangle $PQR$.

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  • $\begingroup$ Ok.. thanks. But why was my method wrong? $\endgroup$ – tatan Apr 16 '19 at 14:19
  • $\begingroup$ Why should it be? The line joining $(-1,-5)$ and $(1,5)$ is clearly not perpendicular to $y=-x$ . $\endgroup$ – tatan Apr 16 '19 at 14:24
  • $\begingroup$ Oh yes. I did some calc errors. But why do you say (-1,-5) is the reflection? $\endgroup$ – tatan Apr 16 '19 at 14:28
  • $\begingroup$ My bad, I thought you are reflecting P across y=-x $\endgroup$ – Aqua Apr 16 '19 at 14:34
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Let $Q$ be the image of the first reflection and $R$ the image of the second. Since the two lines of reflection are perpendicular, $\angle{QPR}$ is a right angle, so $QR$ must be a diameter of the circle. The midpoint of this segment is the origin.

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