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## 'Points in Pairs' printed from http://nrich.maths.org/

Two points, one inside a circle and the other outside, are related
in the following way :

A line starting at the centre of the circle and passing through the
first point ($P$) goes on to pass through the second point ($P'$)

Positions along the line are such that the ratio of $OP$ to the
radius of the circle matches the ratio of the radius of the circle
to $OP'$

For example if $OP$ happened to be $2/3$ of the radius then
$OP'$ would be $3/2$ of the radius.

You can use the interactivity below to help you explore how the
positions of a pair of points relate to each other.

Full Screen
Version
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Once you have a feel for that, leave the interactivity and go on
to the main problem which is best done by drawing sketches on
paper.

In the diagram above, $P$ and $P'$ are a connected pair, and $Q$
and $Q'$ the same, both pairs behaving like the points in the
interactivity.

Now for the problem : The radius of the circle is 10 units, $OP$
is 8 units and $OQ$ is 6 units.

If the distance $PQ$ is 5 units what is the distance $P'Q'$?