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 $\displaystyle P$ and $\displaystyle Q.$

Their common tangent is $\displaystyle AB.$

$\displaystyle PA = 5,\;QB = 8$

Through $\displaystyle P$, draw $\displaystyle PC \parallel AB.$

Then: .$\displaystyle CB = 5,\;QC = 3$

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

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

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