Power of a point connects lengths formed when chords, secants, or tangents meet a circle. This cheat sheet helps students recognize which segment product rule applies in each diagram. It is useful for solving missing lengths, checking geometric relationships, and preparing for proofs involving circles.
The main idea is that certain products of segment lengths stay equal for the same point and circle.
For intersecting chords inside a circle, multiply the two pieces of one chord and set that equal to the product of the two pieces of the other chord. For two secants from an outside point, multiply each whole secant by its external part. For a tangent and a secant from the same outside point, the tangent length squared equals the external part times the whole secant.
These rules are all forms of the power of a point equation.
Key Facts
- If two chords intersect inside a circle, then , where is the intersection point.
- If two secants are drawn from an external point , then , where and are external parts and and are whole secants.
- If a tangent and a secant are drawn from an external point , then , where is the tangent length.
- A whole secant equals its external part plus its internal part, so .
- Power of a point can be positive, zero, or negative depending on whether the point is outside, on, or inside the circle.
- For an outside point, the power can be written as when a tangent from the point exists.
- For an inside intersection of chords, use the two segment pieces on each chord, not the whole chord lengths.
- Lengths in power of a point problems must be positive, so discard any negative solution that represents a segment length.
Vocabulary
- Power of a Point
- A value that relates a point to a circle through products of segment lengths drawn from that point.
- Chord
- A segment whose endpoints both lie on a circle.
- Secant
- A line or segment that intersects a circle at two points.
- Tangent
- A line or segment that touches a circle at exactly one point.
- External Segment
- The part of a secant from the outside point to the nearer point where it meets the circle.
- Whole Secant
- The full segment from the outside point to the farther point where the secant meets the circle.
Common Mistakes to Avoid
- Using only the inside pieces for secants from an external point is wrong because the formula requires external part times whole secant, such as .
- Forgetting to add the external and internal parts is wrong because the whole secant is , not just the portion inside the circle.
- Squaring the secant instead of the tangent is wrong because the tangent-secant theorem is .
- Mixing up chord pieces after an inside intersection is wrong because each product must use the two pieces on the same chord, as in .
- Keeping a negative answer for a length is wrong because geometric segment lengths are positive, even if an algebraic equation has a negative root.
Practice Questions
- 1 Two chords intersect at inside a circle. If , , and , find .
- 2 From point , two secants meet a circle. One has external part and whole length . The other has external part . Find whole length .
- 3 From point , a tangent has length and a secant has external part . Find whole secant length .
- 4 Explain how you can tell whether to use , , or from a diagram.