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.
Understanding Power of a Point Intersecting Chords and Secants
The deeper idea behind every version of this theorem is that a point has one fixed numerical relationship with a particular circle. It does not depend on which line through the point is chosen. In terms of the circle’s center, the power equals the square of the distance from the point to the center minus the square of the radius.
This explains the sign of the result. A point beyond the circle has a positive power because its distance from the center is greater than the radius. A point on the circle has zero power.
A point within the circle has negative power. Segment problems usually use positive lengths, but the signed version helps explain the full geometry.
The segment relationships are not separate tricks to memorize. They can be proved using similar triangles. When two chords cross, the angles formed at the crossing point create vertical angle pairs.
Other angle pairs match because they intercept the same arcs of the circle. The resulting triangles have equal angle measures, so their corresponding sides have matching ratios. Multiplying those ratios produces the length relationship.
The tangent case comes from the same structure. A tangent touches the circle at one point, while a secant cuts through it at two points.
As the two intersection points on one secant move together, that secant becomes a tangent. This is why a tangent length appears squared.
Careful labeling matters more than fast algebra. For a secant beginning outside a circle, read its points in order from the outside point toward the far side of the circle. The first section is the external part.
The entire distance to the far intersection is the whole secant. Many wrong answers come from using only the part inside the circle where the whole length is needed. Make a small note beside each line before substituting numbers.
If a diagram is crowded, redraw only the circle, the outside point, and the two lines. A clean sketch often reveals whether a given length is a piece or a total.
Students meet this topic most often in geometry proofs, circle constructions, and coordinate geometry. In coordinate work, the center and radius can be used to find a point’s power without drawing chords at all. The same idea leads to the radical axis of two circles.
This is the set of points with equal power to both circles, and it forms a straight line when the circles are different. For problem solving, check that every length uses the same unit, that a whole secant is longer than its external part, and that the final value makes sense in the picture. A negative algebra result cannot represent an ordinary segment length, so it signals an impossible measurement or an incorrect setup.