# Thread: question on properties of circle 9

1. ## question on properties of circle 9

Referring to the attached pic, find the radii of both circles.

2. Hello ukorov
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.

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.

PQ = 6 = w + z
w = 6 - z .....(1)
TA = TP + w = 10 + w
TB = TQ + z = 8 + z
hence 10 + w = 8 + z .....(2)
(1) to (2):
10 + 6 - z - 8 = z
2z = 8
z = 4
hence larger circle has radius 4cm.

Thanks