# Circles

# Perpendicular Tangent Theorem

How many different tangents can there be at one given point on a circle? That's not an easy question to answer, is it? We feel like there *ought *to be only one tangent line for each point on the circle, but at the same time we don't really know why that should be the case.

How about a postulate? That sounds good. We'll just *assume* that for any point on a circle, there is only one line tangent to the circle at that point. In math, assumptions are always true, unless they contradict statements already known to be true. So let's just make this assumption and hopefully nothing bad will happen.

Given ⊙*O* and point *P* on ⊙*O*, there is exactly one line tangent to ⊙*O* at *P*. Remember, "exactly one" means, "no less no more than one."

By the way, the phrase "going off on a tangent," comes from geometry. The idea is that when you're on-topic, you're following the curve of the circle, but at some "point," you veer off into space along some line and have trouble getting back. Sometimes, to the point of getting lost in space.

How's this for a tangent? It's a guy with a trenchcoat and a cowboy hat using a sling to swing a tennis ball over his head in slow motion. The only way it could be *more* tangential is if he releases it.

Oh wait. He does.

When he releases it, the ball travels along a straight line, tangent to the circular path. Here's an overhead diagram of this situation. The string is a radius of the circular path of the ball.

Notice that when the sling man wants to hit a target directly in front of him, he releases the ball when the string is pointing directly off to the side. If he wanted to hit a target over to his left, he would release the ball when the string was straight out in front of him.

In other words, when the ball is released, it travels along a path *perpendicular* to the string's position at that moment. In fact, a line perpendicular to the radius at a point on the circle is always a tangent line. We feel a theorem coming on.

Given a circle ⊙*O* and a point *P* on ⊙*O*, if a line *m* through *P* is perpendicular to the radius *OP*, then *m *is tangent to ⊙*O* at *P*. Let's start by drawing a picture of the situation, adding in a point *Q* on *m* somewhere other than *P*.

Wherever *Q* is on *m*, the distance *OQ* must be greater than *OP*, because the shortest path between a point and a line is a perpendicular segment. By definition of the exterior of a circle, *Q* (and therefore any point on *m* other than *P*) must be in the exterior of ⊙*O*. Therefore, we know for sure that *m* intersects ⊙*O* at no point other than *P* (where it does intersect ⊙*O*). Therefore, by definition of tangent line, *m* is tangent to ⊙*O*.

We can use this theorem to prove its converse. Sneaky, but totally legal.

If a line *m* is tangent to ⊙*O* at *P*, then *m *is perpendicular to *OP*. Let's start by remembering that for any line and a given point on that line, there is exactly *one* line through that point that is perpendicular to that line. Using that principle, let's draw the unique line *q* that is perpendicular to *OP* at point *P*. By the theorem we just proved, *q* is tangent to ⊙*O* at *P*.

But wait a second. Now we have two lines, *m* and *q*, that are both tangent to ⊙O at P. This violates the Postulate that says there can only be one line tangent to a given circle at a given point…unless *m* and *q* are the same line! In other words, *m* = *q*. Since *m *= *q* and *q *is, by our definition, perpendicular to *OP*, *m* must also be perpendicular to *OP*.

We can combine these two into one biconditional theorem. Let's call it the **Perpendicular Tangent Theorem**. Or, you know, Hubert. Your call.

Given a line *m* through a point *P* on ⊙*O*, *m *is perpendicular to *OP* if and only if *m *is tangent to ⊙*O* at *P*. In other words, a tangent line is always perpendicular to the circle's radius at the point of intersection.

Now you know all you need to know in order to be an expert sling-hunter, just like that trenchcoat-wearing cowboy. Shmoop is not responsible for any injuries you may sustain or inflict while practicing.

### Sample Problem

Lines *m* and *n* are tangent to ⊙*O* at points *P* and *Q*, respectively. Where must *P* and *Q* be located on ⊙*O* so that lines *m* and *n* are parallel to each other?

According to the Perpendicular Tangent Theorem, tangent lines are always perpendicular to a circle's radius at the point of intersection. In other words, *m* is perpendicular to *OP* and *n* is perpendicular to *OQ*. In order for *m* and *n* to be parallel (never intersect), this means ∠*POQ* has to be 180°. In other words, *P* and *Q* have to be at opposite ends of ⊙*O* (meaning that *PQ* is a diameter of ⊙*O*).