# question on properties of circle 9

• Nov 8th 2009, 08:38 PM
ukorov
question on properties of circle 9
Referring to the attached pic, find the radii of both circles.
• Nov 9th 2009, 06:40 AM
Hello ukorov
Quote:

Originally Posted by ukorov
Referring to the attached pic, find the radii of both circles.

In the attached diagram, I have marked the points of contact with the smaller circle as E and F.

Then, if the radius of the smaller circle is $x$
$O_1D=O_1F=DQ=FQ = x$
and if $PE = y$, then $PD = y$ (tangents from a point to a circle are equal)

$PQ = x+y=6$

In $\triangle TPQ, TP= 10$ cm (Pythagoras)

$TE = TF \Rightarrow 10-y = 8-x \Rightarrow y-x=2$

Solving these simultaneous equations: $y = 4, x = 2$.

So the radius of the smaller circle is $2$ cm.

Similarly let $PA = PC = w$ and $QC = QB = z$

Express the lengths of $TA$ and $TB$ in terms of $w$ and $z$. Then use the fact that $PQ = 6$ to form another equation. Solve these equations to find $z$, the radius of the larger circle.

• Nov 9th 2009, 10:48 AM
ukorov
Quote:

Hello ukorovIn the attached diagram, I have marked the points of contact with the smaller circle as E and F.

Then, if the radius of the smaller circle is $x$[/size]
$O_1D=O_1F=DQ=FQ = x$
and if $PE = y$, then $PD = y$ (tangents from a point to a circle are equal)

$PQ = x+y=6$

In $\triangle TPQ, TP= 10$ cm (Pythagoras)

$TE = TF \Rightarrow 10-y = 8-x \Rightarrow y-x=2$

Solving these simultaneous equations: $y = 4, x = 2$.

So the radius of the smaller circle is $2$ cm.

Similarly let $PA = PC = w$ and $QC = QB = z$

Express the lengths of $TA$ and $TB$ in terms of $w$ and $z$. Then use the fact that $PQ = 6$ to form another equation. Solve these equations to find $z$, the radius of the larger circle.