Two circles, one with a radius of 6 and one with a radius of 2, have a common tangent with a length of 17. What is the distance between the circles' centers?

Answer 1

17 represents the longest leg of a right triangle because a tangent to a circle forms a perpendicular line to the radius. From the shorter radius, draw a perpendicular line from the radius to the longer radius which is parallel to the tangent and equal to it. That is equivalent to a perfect rectangle. That will form the leg of the needed triangle which you apply the Pythagorean formula. "C" is the hypotenuse you are looking for. The shorter leg is the difference between the two radii, say "b" = radius₁-radius₂

Use c² = a²+ b² Take the square root of both sides

There are other configurations. This is just one of them.

Answer 2

There are 2 possible answers depending on the way the tangent is situated.

If the tangent does not cross the line of centers, then the tangent is one side of a trapezoid and the distance

between centers is sqrt(289-14).

If the tangent does cross the line of centers there will be 2 similar triangles and the distance between centers

sqrt{(51/4)²+6²)] + sqrt{[17-(51/4)]²+2²}