A tangent to two circles

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Question from erson, a student:

this is the last.....
please help me
i humbly request.....if possible....may i get the answer tomorrow...
because i'll be needing this the day after tomorrow...

find the length of the tangent segment AB to two circles whose radii are a and b respectively, when the circles touch each other.

the illustration looks like this...hope you'll understand...
there are 2 circles - one is big one is small. they touch each other. and there is this irregular 4 sided polygon that connects them...there is a line that connects them from their center point and another from the tip of the circles...and that's it...i cannot explain very well
please bear with me

I think you are looking at this diagram:

circles1

Remember that radii meet points of tangency at right angles, so that means the two radii drawn are parallel to each other. Now if I draw a blue line parallel to the tangent line here:

circles1

Then I have created a rectangle. Therefore the length of the blue line is the same as the length x you are trying to find. As well, you know the lengths of two of the sides of the triangle you see there and can show that it is a right triangle:

circles1

Solve for x.

Cheers,
Stephen La Rocque.

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