10.4 Tangents
Notes
A tangent line meets a circle at exactly one point.
- A tangent is perpendicular to the radius at the point of tangency.
- Two tangent segments from the same outside point are congruent.
- Use the Pythagorean Theorem with the right angle at the point of tangency.
- Example: radius 5, distance from center to outside point 13. Tangent length $\displaystyle = \sqrt{169 - 25} = 12$.
Practice
$\overline{PA}$ is tangent to circle $O$ at $A$. $OA = 8$ and $PA = 15$. Find $OP$.
$\angle OAP = 90^\circ$, so $\displaystyle OP = \sqrt{64 + 225} = 17$.
$\overline{PA}$ and $\overline{PB}$ are tangent to a circle from point $P$. $PA = 4x - 1$ and $PB = 2x + 9$. Find $PA$.
Tangent segments from the same point are congruent: $4x - 1 = 2x + 9$, so $x = 5$.
Line $\ell$ touches circle $O$ at $T$. What is the angle between $\ell$ and radius $\overline{OT}$?
A tangent is perpendicular to the radius at the point of tangency.
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