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From a point $P$ on the circle $x^{2} + y^{2} = 4r^{2}$, tangents are drawn to $x^{2} + y^{2} = r^{2}$ at $Q$ and $R$. Then which of the following statement(s) are true regarding the centroid of $ΔPQR$

$A.$ is at a distance $r$ from chord $QR$.;

$B.$ lies on $x^{2} + y^{2} = r^{2}$

$C.$ is at a distance $1.5r$ from $P$

$D.$ doesn't lie on the line joining the origin and $P$

My take: $B$

Please help.

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up vote 1 down vote accepted

Let $O$ be the centre of the circles. By symmetry the centroid lies on the line $OP$.

Where on $OP$? Join $Q$ and $R$, and let $M$ be the point where $OP$ meets $QR$. Then by properties of the centroid, the centroid lies on $PM$, one-third of the way up from $M$ towards $P$.

Note that $PO=2r$ and $OQ=r$. So $\angle QPO$ is $30^\circ$. Computation (special angles or Pythagorean Theorem) shows that $PM=\frac{3}{2} r$.

The point $X$ where the little circle meets $OP$ is therefore $1/3$ of the way up from $M$, since $XP=r$. It follows that $X$ is the centroid.

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enter image description here

Can you see Oklid's relation in the picture? (Focus on $\triangle PQO$)

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aye aye captain. – TheApe Jul 22 '12 at 18:52

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