Power of a Point · Explore together
A product that stays put
Rotate a secant, record two distances, and look for a relationship that survives.
One person controls the experiment. The other predicts and questions. Swap roles after a trial.
Scroll across the diagram to see both intersections.
Both distances start at P. PB includes the part inside the circle.
Predict with a partner. As the line turns, do you expect PA, PB, or their product to stay fixed? Give a reason before moving it.
Angles are limited to lines that meet the circle twice.
- PA
- 2.234
- PB
- 7.163
- PA × PB
- 16.000
Record three different angles. Does your evidence support your prediction?
| Angle | PA | PB | Product |
|---|---|---|---|
| No angles recorded yet. | |||
Turn the picture into an equation. A point t units along the ray has coordinates (−5 + t cos θ, t sin θ). Substitute into x² + y² = 9 to get t² − 10 cos θ · t + 16 = 0.
The two intersection distances are its roots. Their product is the constant term, 16 = PO² − r². This is the power of P. The fixed circle and fixed external point matter.
Now explain to your partner why rotating the line changes the middle coefficient, but not the product. What happens at a tangent?
Compare your evidence
What can you explain now?
Ask your partner to restate your reasoning. An experiment suggests a pattern; your explanation should say why it holds and when it applies.
- Where must both distances begin?
- Why is the far distance the whole secant?
- What would change if the point moved or the circle grew?