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

Printable View

- Nov 8th 2009, 07:38 PMukorovquestion on properties of circle 9
Referring to the attached pic, find the radii of both circles.

- Nov 9th 2009, 05:40 AMGrandad
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$$\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.

Grandad - Nov 9th 2009, 09:48 AMukorov