Power of a point is a geometry idea that connects a point to a circle through products of segment lengths. It explains why certain chord, secant, and tangent measurements are equal even when the lines look different. This matters because it turns circle diagrams into equations that can be solved with algebra.
It is especially useful for finding missing lengths in problems involving intersecting lines and circles.
The basic mechanism is that a line through a point cuts the circle in related segments, and the product of those segments stays constant for that same point. If the point is inside the circle, two chords create equal products of chord pieces. If the point is outside the circle, secants and tangents create equal products using outside and whole lengths.
These relationships are all versions of one idea: the power of the point relative to the circle.
Key Facts
- Intersecting chords inside a circle: PA · PB = PC · PD
- Two secants from an external point: PA · PB = PC · PD, where PA and PC are outside segments and PB and PD are whole secant lengths
- Tangent and secant from an external point: PT^2 = PA · PB
- Two tangents from the same external point are equal: PT = PU
- Power of a point with circle center O and radius r: Pow(P) = OP^2 - r^2
- A point inside the circle has negative power, a point on the circle has zero power, and a point outside the circle has positive power
Vocabulary
- Power of a Point
- The power of a point is a value that describes the product relationship between segments drawn from that point to a circle.
- Chord
- A chord is a line segment whose endpoints both lie on a circle.
- Secant
- A secant is a line that intersects a circle at two points.
- Tangent
- A tangent is a line that touches a circle at exactly one point.
- External Segment
- An external segment is the part of a secant from an outside point to the nearer point where the secant meets the circle.
Common Mistakes to Avoid
- Using only the inside part of a secant in the formula, which is wrong because the secant product uses outside segment times whole secant length.
- Mixing up chord and secant formulas, which is wrong because intersecting chords use two internal pieces while secants from outside use external and whole lengths.
- Taking the square root too early in tangent problems, which is wrong because PT^2 equals a product and the product must be calculated first.
- Assuming all segments from the same outside point are equal, which is wrong because only tangent segments from the same external point are guaranteed equal.
Practice Questions
- 1 Two chords intersect inside a circle at P. One chord has segments 6 and 10, and the other has segments x and 15. Find x.
- 2 From an external point P, a tangent PT has length 12. A secant from P has outside segment 8 and whole length x. Find x.
- 3 A point P is outside a circle. One line from P is a tangent, and another line from P is a secant. Explain why the tangent length squared can equal a product involving the secant lengths even though the two lines touch the circle in different ways.