# question on properties of circle 9

• Nov 8th 2009, 07:38 PM
ukorov
question on properties of circle 9
Referring to the attached pic, find the radii of both circles.
• Nov 9th 2009, 05:40 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 \$\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.

• Nov 9th 2009, 09:48 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 \$\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.