ShmoopTube

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Here’s your shmoop du jour, brought to you by circle diagrams.

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Leave your dirty minds at the door, please.

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In this figure, O1 and O2 are circles with equal radii of 4 centimeters.

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Find the length of the tangent O2P.

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And here are the potential answers:

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O2P or not to pee, that is the question.

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So no matter how much this problem tries to fool us into thinking it’s about circles…

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…we know it’s really a triangle problem.

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Okay, so it’s missing a leg, so let’s go ahead and draw in that missing leg right now…

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but be careful, we’re not drawing it straight up to the end of the line… only to the letter “P”…

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Now, by property of a tangent, the radius O1 P has to be perpendicular to O2 P…

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…making O1 P O2 a right triangle.

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And whenever we have a right triangle… we can always call in our good friend Pythagoras

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to help us out of our jam.

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So (O1P)2 + (O2P)2 = (O1O2)2

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We’re looking for O2 P, so we don’t know that one yet… but we can plug in the other

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values because we’re told that the radius of both circles is 4.

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So… O2 P squared plus 4 squared = 8 squared…or O2 P squared plus 16 = 64.

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Subtract 16 from both sides, and O2 P squared = 48…

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…which, after we take the square root of both sides,

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reduces to the square root of 3 times 16, and we can take the root of 16 and pull it out of there….

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…leaves us with 4 square root of 3 – answer C.

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