Suppose we have a small circle inside a larger circle, the circles touching tangentially at the bottom, A horizontal line is dr awn tangentially to the small circle to meet the large circle. What is the length of this line?
![](/igcse-maths/circle-inside-circle-1.png)
Label the centre of the large circle as O. From the centre of the large circle draw a vertical line to meet the horizontal line at C. The triangle OCB is a right angled triangle, and OC=x=12-10=2cm.
![](/igcse-maths/circle-inside-circle-2.png)
Then
\[CB=\sqrt{10^2-2^2}=\sqrt{96}=\sqrt{16} \sqrt{6}=4 \sqrt{6}cm\]
.
Then
\[AB=2CB=8 \sqrt{6}cm\]