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Suppose line m intersects ⊙X at point Z and m is perpendicular to XZ. What is the maximum number of other points on ⊙X that m can intersect?

The Perpendicular Tangent Theorem tells us that in the situation described above, line m must be tangent to ⊙X at Z. By definition of tangent line, line m must intersect ⊙X in exactly one point. That means that the maximum number of points other than P on ⊙X that m can intersect is zero.

Example 2

Is there a maximum number of secant lines two circles can have in common? If so, what is it? (In other words, given two circles, how many lines m can you draw that m is a secant of both circles?)

In this case, it's best to draw a picture. Start with any two circles you like. Spend a few seconds drawing common secants and you will find that there is no maximum number of secant lines two circles can have in common. You could keep on drawing them for the rest of your life if you wanted to.

Example 3

In the figure below, segments CA and CB are tangent to ⊙O at points A and B, respectively. If OA = 8 cm and AC = 18 cm, what is the length of BC?

In the given situation, the Endpoint-Tangent Theorem applies. It tells us that AC is congruent to BC. Therefore BC = AC = 18 cm. Note that we didn't even need to know OA.