Tangent Lines and Their Properties
Prove that a tangent to a circle is perpendicular to the radius at the point of tangency. Apply the theorem that two tangent segments from an external point are congruent.
A lesson of Geometry, in Circles, about 10 minutes.
By the end of this lesson, you will be able to apply the tangent-radius perpendicularity theorem, use the two-tangent theorem to find equal tangent lengths from an external point, and solve problems involving tangent segments and circumscribed polygons.
The Big Idea
A tangent line touches a circle at exactly one point — and at that point, it is perfectly perpendicular to the radius. This rigidity generates all tangent theorems. The two-tangent theorem, in particular, is elegant and widely applicable: two tangent segments from the same external point are always equal.
Tangent-Radius Theorem
If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.
$$\ell \text{ tangent at } P \implies \overline{OP} \perp \ell$$
Converse: If a line is perpendicular to a radius at the circle's endpoint, it is tangent to the circle.
Two-Tangent Theorem
If two tangent segments are drawn from an external point to a circle, the tangent segments are congruent.
$$PA = PB$$
where $A$ and $B$ are the points of tangency and $P$ is the external point.
Proof: Draw $\overline{OA}$, $\overline{OB}$, $\overline{OP}$.
$\angle OAP = \angle OBP = 90°$ (tangent ⊥ radius)
$OP = OP$ (reflexive)
$OA = OB$ (radii)
$\triangle OAP \cong \triangle OBP$ by HL
$PA = PB$ by CPCTC ✓
Example 1 — Tangent-Radius
Tangent $\overline{PA}$ touches circle $O$ at $A$. $OA = 5$ and $OP = 13$. Find $PA$.
$\angle OAP = 90°$ (tangent ⊥ radius)
$$PA = \sqrt{OP^2 - OA^2} = \sqrt{169 - 25} = \sqrt{144} = 12$$
Example 2 — Two-Tangent Theorem
From external point $P$, two tangents touch circle $O$ at $A$ and $B$. $PA = 3x + 5$ and $PB = 5x - 7$. Find $PA$.
$$3x + 5 = 5x - 7 \implies x = 6$$
$PA = 3(6) + 5 = 23$
Example 3 — Circumscribed Polygon
A triangle is circumscribed about a circle (all three sides are tangent). The tangent lengths from vertices $A$, $B$, $C$ are $x$, $y$, $z$ respectively. Express the perimeter.
By the Two-Tangent Theorem, each vertex contributes two equal tangent lengths.
$$\text{Perimeter} = 2x + 2y + 2z = 2(x + y + z)$$
Specific example: If the sides of the triangle are $AB = 10$, $BC = 12$, $CA = 8$, find each tangent length.
Let $x$ = tangent from $A$, $y$ from $B$, $z$ from $C$.
$x + y = 10$, $y + z = 12$, $x + z = 8$