# Thread: Circle geometry, finding length of a tangent

1. ## Circle geometry, finding length of a tangent

Two circles of radii 5cm and 8cm touch each other externelly, calculate the length of the common tangent.

and

if the radii of two intersecting circles are 51cm and 74cm and the length of their common chord is 48cm
A) Calculate the length of the line joining their centres
B) Calculate the length of the common tangent

In both questions i get them all set up draw them out right, i just can't seem to figure out what the common tangent length is (mathematically and literally) maybe there is a rule i am forgetting?

thanks for the help

2. Originally Posted by JordanW
Two circles of radii 5cm and 8cm touch each other externelly, calculate the length of the common tangent.

...
B) Calculate the length of the common tangent

In both questions i get them all set up draw them out right, i just can't seem to figure out what the common tangent length is (mathematically and literally) maybe there is a rule i am forgetting?

thanks for the help
I've attached a sketch.

The triangle which contains the length of the common tangent, is a right triangle.

The hypotenuse is R + r

The short slanted leg is R - r

Thus the length of the common tangent is:

$l_t=\sqrt{(R+r)^2-(R-r)^2} = \sqrt{R^2+2rR +r^2-(R^2-2rR +r^2)} = 2\sqrt{rR}$

3. Hello, JordanW!

Two circles of radii 5cm and 8cm touch each other externally.
Calculate the length of the common tangent.
Code:
                                                  o Q
*   |
* * *               8     *         |
*           *             *             |
*               *       *                 |3
*                 *  *                     |
5   *                         |
*         P   *      *                      |
*         o - - - - - - - - - - - - - - - - * C
*         |         *   *                   |
|                                 |
*       5|        *                        |5
*       |       *          *              |
*     |     *                 *         |
* o * - - - - - - - - - - - - - - * o
A                                 B

The centers of the circles are $P$ and $Q.$
Their common tangent is $AB.$
$PA = 5,\;QB = 8$

Through $P$, draw $PC \parallel AB.$
Then: . $CB = 5,\;QC = 3$

In right triangle $QCP\!:\;\;PC^2 + QC^2 \:=\:PQ^2 \quad\Rightarrow\quad PC^2 + 3^2\:=\:13^2$

Hence: . $PC^2 \:=\:169 - 9 \:=\:160 \quad\Rightarrow\quad PC \:=\:4\sqrt{10}$

Since $AB = PC,\;\;AB \:=\:4\sqrt{10}$

4. Originally Posted by JordanW
...

if the radii of two intersecting circles are 51cm and 74cm and the length of their common chord is 48cm
A) Calculate the length of the line joining their centres
...
Draw a sketch.

Let c denote half of the common chord. c is simultaneously the leg of two right triangles where you know the length of the hypotenuses.

Therefore

$|\overline{M_1 M_2}| = \sqrt{R^2-c^2} + \sqrt{r^2-c^2}$

$|\overline{M_1 M_2}| = 70 + 45 = 115$