I don't know how to do this question, could someone please explain?

- Apr 20th 2009, 02:28 AMiamanoobatmathIf the radius of the small circle is 8 cm, what is the circumference of the...
I don't know how to do this question, could someone please explain?

- Apr 20th 2009, 03:26 AMSoroban
Hello, iamanoobatmath!

I hope you can follow my labelling.

(I couldn't make a diagram of this one!)

The vertex of the angle is $\displaystyle A.$

Draw a line from $\displaystyle A$ through the centers of the circles.

It first intersects the small circle at $\displaystyle B$,

the center of the smaller circle is $\displaystyle C$,

$\displaystyle D$ is the point of tangency of the two circles,

the center of the larger circle is $\displaystyle E$.

From C, draw a radius down to the point of tangency, $\displaystyle F.$

From E, draw a radius down to the point of tangency, $\displaystyle G.$

We have: .$\displaystyle BC = CD = CF = 8$

Let $\displaystyle DE = EG = r$

In right triangle $\displaystyle CFA,\;\angle CAF = 30^o\,\text{ and }\,CF = 8.$

. . Hence, the hypotenuse $\displaystyle AC = 16.$

Since $\displaystyle \Delta EGA \sim \Delta CFA$, we have: .$\displaystyle \frac{EG}{EA} \:=\:\frac{CF}{CA}$

. . That is: .$\displaystyle \frac{r}{r+24} \:=\:\frac{8}{16} \quad\Rightarrow\quad r \:=\:24$

Therefore, the circumference of the larger circle is: .$\displaystyle 48\pi$ cm.