# 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 $\displaystyle x$
$\displaystyle O_1D=O_1F=DQ=FQ = x$
and if $\displaystyle PE = y$, then $\displaystyle PD = y$ (tangents from a point to a circle are equal)

$\displaystyle PQ = x+y=6$

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

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

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

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

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

Express the lengths of $\displaystyle TA$ and $\displaystyle TB$ in terms of $\displaystyle w$ and $\displaystyle z$. Then use the fact that $\displaystyle PQ = 6$ to form another equation. Solve these equations to find $\displaystyle z$, 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 $\displaystyle x$[/size]
$\displaystyle O_1D=O_1F=DQ=FQ = x$
and if $\displaystyle PE = y$, then $\displaystyle PD = y$ (tangents from a point to a circle are equal)

$\displaystyle PQ = x+y=6$

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

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

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

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

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

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