Two circles of radii 3 in. and 6 in. have their centers 15 in. part. find the length in inches of the common internal tangent

See illustration attached.

We could find the length of the common internal tangent using pythagorean theorem (ΔAOB). The common internal tangent is represented by MN, which has the same length as AO in ΔAOB

In triangle AOB, AB = 15 in and OB = (3 + 6) = 9 in
Pythagorean theorem states that the sum of the squares of the perpendicular sides is equal to the square of hypotenuse. That means
AO² + OB² = AB²
AO² + 9² = 15²
AO² = 15² - 9²
AO² = 225 - 81
AO² = 144
AO = √144
AO = 12

The length of AO is 12 in, thus the length of common internal tangent MN is also 12 in.