Properties of Tangent Lines
A tangent line touches a circle at exactly one point. At that point, the tangent is always perpendicular to the radius.
Introduction
Tangent ⊥ radius at the point of tangency
Past Knowledge
Circle vocab (10.1.1). Perpendicular lines. Pythagorean Theorem (8.1.1).
Today's Goal
Prove tangent ⊥ radius and apply the two-tangent theorem.
Future Success
Secant-tangent angles (10.2.3), segment lengths (10.2.5).
Key Concepts
Tangent-Radius Theorem
A tangent line is perpendicular to the radius drawn to the point of tangency.
Two-Tangent Theorem
If two tangent segments are drawn to a circle from the same external point, then:
- The two tangent segments are congruent
- The line from the external point to the center bisects the angle between the tangents
Theorem & Proof
Two-Column Proof: Two Tangent Segments from One External Point Are Congruent
Given: and are tangent to circle at and
Prove:
| # | Statement | Reason |
|---|---|---|
| 1 | , | Tangent ⊥ radius |
| 2 | Both are radii | |
| 3 | Reflexive Property | |
| 4 | HL Congruence (right triangles, step 2, step 3) | |
| 5 | CPCTC |
∎ Two right triangles share a hypotenuse and have equal legs (radii), so HL gives congruence.
Worked Examples
Pythagorean with Tangent
Tangent segment = 8, radius = 6. Find the distance from the external point to the center.
Right triangle: →
d = 10
Two-Tangent Segments
From external point P, PA = 3x + 2 and PB = 5x − 6 are tangent to circle O. Find PA.
Two tangents from same point are congruent: →
PA = PB = 14
Circumscribed Polygon
A triangle is circumscribed about a circle. The sides (tangent lengths from vertices) split as: from A: 4 and 4, from B: 6 and 6, from C: 5 and 5. Find the triangle's perimeter.
Wait — tangent segments from each vertex are equal. Side AB = 4 + 6 = 10. Side BC = 6 + 5 = 11. Side CA = 5 + 4 = 9.
Perimeter = 10 + 11 + 9 = 30.
Perimeter = 30
Common Pitfalls
Forgetting the Right Angle
When you see a tangent, immediately mark a 90° angle at the point of tangency. This is essential for using the Pythagorean Theorem.
Distance vs. Tangent Length
The tangent segment length is NOT the distance from the external point to the center. Those form two legs and a hypotenuse of a right triangle.
Real-Life Applications
Roads & Roundabouts
A road tangent to a circular roundabout meets it at exactly 90° to the radius — the perpendicular tangent ensures smooth traffic flow without a sharp turn.
Belt Drives & Pulleys
A drive belt tangent to two pulleys touches each at a tangent point. The belt length depends on the tangent segment lengths between the circles.
Practice Quiz
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