Example 3:  Line RS is tangent to circles P and Q at points R and S.  Find the length of segment RS.

Solution:  Draw radii from P to R and from A to S.  These radii are perpendicular to the tangent line, and the total length of segment PQ is 30 units: 

 

Now construct TS parallel to PQ to form parallelogram TSQP:

Then triangle TRS is a right triangle, so we can use the Pythagorean Theorem to find RS:

RS² + 10² = 30²

RS² + 100 = 900

RS² = 800

So RS is the square root of 800, which simplifies:

or approximately 28.28 units. 

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