How is it possible for two circles to have only two common tangents?
Hint
Circles can "protect" each other from tangent lines... the "closer" two circles are, the safer they are from tangent lines.
Answer
The circles must intersect at two points.
Example 2
How is it possible for two circles to have only one common tangent?
Hint
Circles can "protect" each other from tangent lines... the "closer" two circles are, the safer they are from tangent lines.
Answer
One circle must be inside the other, and they've gotta intersect at exactly point.
Example 3
A diameter is a secant. True or false?
Hint
Read the definitions of each of those terms carefully.
Answer
False! It's true that secants and diameters both intersect a circle twice, but technically, a diameter is a chord, which is a line segment. A secant is an infinite line, not just a segment.
Example 4
If a line m is perpendicular to the radius of a circle, then m is not a secant of the circle. True or false?
Hint
Tangents must be perpendicular to the radius of a circle. What about secants?
Answer
False
Example 5
In the figure below, segments PB and PA are tangent to ⊙O at B and A, respectively. Also, ∠AOB is a right angle. Prove that ∠APB is a right angle.
Hint
PBOA is a quadrilateral and therefore the measures of its angles must add up to 360°.
Answer
If we recognize that a tangent segment is part of a tangent line, by the Perpendicular Tangent Theorem, we know that PB is perpendicular to OB. Likewise, PA is perpendicular to OA. Therefore, ∠PAO and ∠PBO are both right angles. PBOA is a quadrilateral, so the measures of all its angles must add up to 360°. Since each angle in PBOA other than ∠APB is a right angle, ∠APB must also be a right angle.
