Overview

Tangents create right angles with radii and equal-length segments from the same external point.

Core Skills

Use Radius-Perpendicular Fact

The radius to the tangent point is perpendicular to the tangent line. This creates right triangles for length and angle chasing.

Use Equal Tangents

From an external point $P$, tangents to the circle have equal lengths, so $PA=PB$.

Combine with Power of a Point

If a secant and a tangent meet at $P$, then $PT^2=PA\cdot PB$.

Key Ideas

• Radius to a tangent point is perpendicular to the tangent.
• Tangents from the same external point are equal in length.

Worked Example

Point $P$ is outside a circle, and tangents touch at $A$ and $B$. Show $PA=PB$.

Both are tangents from the same external point, so the lengths are equal.

More Examples

Example 1: Right Angle

If $O$ is the center and $T$ is a tangent point, what is $\angle OTB$ where $TB$ is the tangent line?

$90^\circ$.

Example 2: Tangent Length

If $OP=13$ and $r=5$, find the tangent length from $P$.

$PT^2=OP^2-r^2=169-25=144$, so $PT=12$.

Example 3: Secant-Tangent

If $PA=4$ and $PB=9$ on a secant from $P$, find the tangent length.

$PT=\sqrt{36}=6$.

Strategy Checklist

• Use the radius-perpendicular fact to build right triangles.
• Apply equal tangents from the same point.
• Use $PT^2=PA\cdot PB$ when a secant is present.

Common Pitfalls

• Confusing tangent segments with secant segments.
• Forgetting that the tangent-radius angle is $90^\circ$.
• Treating tangent lengths as diameters.

Practice Problems