Overview

Power of a point unifies secant and tangent length relations. It is essential for circle problems with a point outside the circle.

Key Ideas

• For point $P$, $PA\cdot PB = PC\cdot PD$ for secants.
• Tangent: $PT^2 = PA\cdot PB$.
• Power formula: $\text{pow}(P) = OP^2 - r^2$.

Core Skills

Identify the Configuration

Decide if you have two secants, a secant and tangent, or a chord through an interior point. The same power equation applies.

Use Consistent Segment Order

Compute lengths from the external point to each intersection along the line. This prevents sign errors.

Convert to Radius Form

If center $O$ and radius $r$ are known, use $OP^2-r^2$ to compute power quickly.

Worked Example

A circle has radius $5$. Point $P$ is $8$ units from the center. If a secant through $P$ meets the circle at $A$ and $B$ with $PA=2$, find $PB$.

Power is $8^2-5^2=39$. So $2\cdot PB = 39$, giving $PB=39/2$.

More Examples

Example 1: Tangent Length

Point $P$ is $13$ units from the center of a circle of radius $5$. Find the tangent length.

$PT^2=13^2-5^2=144$, so $PT=12$.

Example 2: Two Secants

From $P$, one secant has $PA=3$ and $PB=15$. Another secant has $PC=5$. Find $PD$.

$3\cdot 15 = 5\cdot PD$, so $PD=9$.

Strategy Checklist

• Identify the external point and segment order.
• Use $PA\cdot PB = PC\cdot PD$ consistently.
• Switch to $OP^2-r^2$ if center and radius are known.

Common Pitfalls

• Mixing up the order of points along the secant.
• Using power-of-point for an interior point without absolute values.
• Squaring a secant length instead of using the product.

Practice Problems