Tangent Lines

TipLearning Objectives
  • Identify tangent lines to circles.
  • Use the radius–tangent perpendicularity rule.
  • Apply tangent-tangent and tangent-secant relationships.

Key Ideas

Tangent Line

A line that touches the circle at exactly one point.

Crucial Fact

The radius to the point of tangency is perpendicular to the tangent:

\[ \text{radius} \perp \text{tangent} \]

Tangent–Tangent Length Theorem

If two tangents come from the same point:

\[ \text{segments are equal}. \]

A tangent line touching a circle with the radius perpendicular at the point of tangency.

Common Problem Types

Using Perpendicularity

Right triangles appear automatically.

Solving for Radius or Distance

Use Pythagorean theorem.

Two Tangents From One Point

Set tangent segments equal.

Tangent–Secant Problems (Basic)

Often simplified SAT versions.

Strategies

  • Immediately mark a right angle at point of tangency.
  • Look for right triangles and use Pythagorean theorem.
  • When two tangents meet at external point → segments equal.

Worked Examples

Example 1

A tangent touches circle at point T, radius = 10, distance from center to external point = 26.
Find tangent length.

Right triangle:
\[ 10^2 + x^2 = 26^2 \Rightarrow x=24. \]

Example 2

If two tangents from same point are 11 and \(x\), then \(x=11\).

WarningCommon Mistakes
  • Forgetting tangents form right angles.
  • Treating secants like tangents.
  • Using diameter instead of radius in Pythagorean setups.

Practice Problems

  1. Radius = 5, center-to-external-point = 13 → tangent length?
  2. Tangents from same point: 9 and \(x\). Find \(x\).
  3. Radius = 7, tangent length = 24. Find center-to-external distance.
  1. \(x = 12.\)
  2. \(x=9.\)
  3. Use \(7^2 + 24^2 = d^2\) → \(d=25.\)

Summary

  • Radius ⟂ tangent.
  • Two tangents from same external point → equal.
  • See a tangent? Draw a right angle immediately.