06/08/2020 (2011: National Sprint Round, Problem 27)

Q: Circle O has radius 8 units and circle P has radius 2 units. The circles are externally tangent to each other at point Q. Segment TS is tangent to both circle O and P at points T and S, respectively. What is the length of segment OS? Express your answer in simplest radical form.

A: 8√(2)

We first draw a picture:

From that we get:

Which means:

That means that ST = 8 units. We also know that OT = 8 units, and we use the Pythagorean Theorem (because STO is 90 degrees) and we get OS √(8² + 8²) = 8√(2).