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Circle basics

C = 2\pi r, \qquad A = \pi r^2, \qquad \text{chord length} = 2\sqrt{r^2-d^2}

where $d$ is the distance from the center to the chord. Also worth remembering: a tangent line is always perpendicular to the radius at the point of tangency, and two tangent segments drawn from the same external point to a circle are always equal in length.

Power of a Point

Power of a Point captures three configurations with the same underlying idea — the product of distances from a fixed point to a circle along any line through that point is constant.

Two chords intersecting inside the circle (at point $P$ ):

PA \cdot PB = PC \cdot PD

Two secants from an external point $P$ :

PA \cdot PB = PC \cdot PD

A tangent and a secant from an external point $P$ (tangent length $PA$ ):

PA^2 = PB \cdot PC

All three are really the same theorem in disguise — a tangent is just the limiting case of a secant whose two intersection points have merged into one.

Worked example

Problem: From external point $P$ , a tangent to a circle has length 12, and a secant from $P$ passes through the circle, hitting it first at $B$ and then at $C$ , with $PB = 8$ . Find $BC$ .

Solution: By the tangent-secant relationship, $PA^2 = PB \cdot PC$ :

12^2 = 8 \cdot PC \;\Longrightarrow\; PC = \dfrac{144}{8} = 18

So $BC = PC - PB = 18 - 8 = 10$ .

Practice problems

1.Two chords intersect inside a circle. One is split into pieces of length 4 and 9; the other is split into pieces of length 6 and $x$ . Find $x$ .

2.From an external point, a tangent has length 6. Find the power of that point with respect to the circle.

3.From external point $P$ , two secants are drawn. The first hits the circle at distances 5 and 12 from $P$ ; the second hits it at distance 6 and $x$ from $P$ . Find $x$ .

4.A chord of a circle with radius 10 is 12 units long. Find its distance from the center.

Circles & Power of a Point

Circle basics

Power of a Point

Practice problems