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Printjmc
geometry senior
Problem
A circle with center has radius units and circle has radius units. The circles are externally tangent to each other at point . Segment is the common external tangent to circle and circle at points and , respectively. What is the length of segment ? Express your answer in simplest radical form.
Solution
We create a diagram with the given information from the problem:
We draw in radii and and connect and . Then we drop a perpendicular from to that intersects at :
and are right angles as tangents create right angles with radii at points of tangency. is a rectangle, and is right. We use Pythagorean theorem on : we have and , so . Then as well.
Finally, is the hypotenuse of right triangle with . Hence .
We draw in radii and and connect and . Then we drop a perpendicular from to that intersects at :
and are right angles as tangents create right angles with radii at points of tangency. is a rectangle, and is right. We use Pythagorean theorem on : we have and , so . Then as well.
Finally, is the hypotenuse of right triangle with . Hence .
Final answer
8\sqrt{2}