A circle diagram can contain several lengths on one line. Power-of-a-point relationships use particular products, so labelling external and whole lengths prevents a common substitution error.
Need help choosing a next step? Scan the contact QR code below or contact Hanlin on WeChat. Send AMC12 SECANT 1025 and tell us your school year and question.
Specify the configuration
From a point P outside a circle, a secant meets the circle first at A and then B. Suppose PA = 9 and PB = 16. A tangent from P touches the circle at T.
The tangent-secant theorem gives PT2 = PA × PB = 144, so the tangent length is PT = 12.
Identify the internal segment
The inside part AB is PB - PA = 7. It is not the second factor in this theorem. Using 9 × 7 would confuse the whole secant with its internal part.
All lengths must refer to the same external point P and the actual circle configuration.
Check applicability before multiplying
If the point were inside the circle, the diagram and relevant segment relationship would need reinterpreting. A theorem name alone does not establish its conditions.
Mark P-A-B in order before writing the product, and choose the positive root because a segment length is positive. This original example supports AMC 12 geometry reasoning and is not a released MAA problem.
Continue reading: AMC 12 Trigonometry: Solve Equations Without Losing Solutions.
Contact Hanlin
Planning a first attempt or reviewing a topic you find difficult? Scan the QR code and send AMC12 SECANT 1025. Include your school year, country or competition region, experience and the question you want to discuss.
On your phone, use the button or save the QR image and scan it within WeChat.
Official information and further reading: Source 1
