## Length of a Tangent Between Concentric Circles

The diagram shows two concentric circles, radii
$r_1, \: r_2$
.

Between the circles is a tangent to the inner circle of length
$l$
which just touches the outer circle. To find the length of this line in terms of
$r_1, \: r_2$
complete the triangle using the line and two radii of the large circle as shown. Draw a line from the centre of the circle bisecting the tangent and label the angles between the radii as
$\theta$
.

The
$cos \theta = \frac{r_1}{r_2}$
.
The angle subtended at the centre of the circle by the tangent is
$2 \theta$
.
$cos 2\theta =2 cos^2 \theta -1 =2 ( \frac{r_1}{r_2})^2 -1$
.
Now use the Cosine Rule to find the length of the tangent.
\begin{aligned} l^2 &= r_2^2+r_2^2-2 r_2^2 cos 2 \theta \\ & =2r_2^2 -2r_2^2 (2( \frac{r_1}{r_2})^2 -1) \\ &= 4r_2^2-4r_1^2 \end{aligned}
.
Then
$l= \sqrt{4r_2^2-4r_1^2}= 2 \sqrt{r_2^2-r_1^2}$