Two circles have radii 14 cm and 6 cm, and their centres are 17 cm apart. What is the length of a direct common tangent to the two circles?

A direct (external) common tangent has length √$(d^2-(r_1-r_2{)}^2)=$ √$(17^2-8^2)=$ √(289 − 64) = √225 = 15 cm. Note the transverse formula √$(d^2-(r_1+r_2{)}^2)$ is useless here: 289 − 400 is negative, and that is the algebra telling us the circles overlap, since 8 < 17 < 20 means they cut each other at two points and no tangent can cross between them. The trap 8 cm reports the difference of the radii itself, 20 cm reports their sum, and √353 = √(289 + 64) adds where the formula subtracts.