Tangent-Secant Data and a Radius Assumption
A circular fountain has a tangent walkway PT and a straight sight line from point P that crosses the fountain edge at A and again at B. A groundskeeper measures PT = 12 m, PA = 8 m, and AB = 10 m, with P, A, and B collinear in that order. A designer uses the tangent-secant theorem to claim that the fountain’s radius must be 5 m because AB would then be a diameter. Another worker points out that the numbers are consistent with a diameter but do not prove one. The project needs to know whether the measured lengths are mutually consistent, whether the radius is uniquely determined, and what single additional measurement would settle the radius. No center is marked, and the fountain is not drawn to scale. The team must avoid assuming that any chord through an external point is a diameter.
1. Check the consistency of PT = 12 m, PA = 8 m, and AB = 10 m using the tangent-secant theorem. Compute PB and verify the relationship.
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Since P, A, and B are collinear in that order, PB = PA + AB = 8 + 10 = 18 m. The tangent-secant theorem states that PT² = PA · PB. Here PT² = 12² = 144, and PA · PB = 8 · 18 = 144. The two values agree exactly, so the three measurements are mutually consistent with a tangent from P and a secant through A and B. This consistency check does not, by itself, determine the circle’s radius or prove that AB is a diameter.
2. Does the given data force AB to be a diameter? Determine whether the radius is uniquely determined. Support your answer with algebra or a counterexample.
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Let O be the center, r the radius, and d = PO. The power of point P is d² − r² = PT² = 144. Since AB is a chord of length 10, its midpoint M is 13 m from P because PM = PA + AM = 8 + 5 = 13. The perpendicular from O to chord AB meets at M, so if OM = x, then r² = x² + 5² and d² = 13² + x². Substituting gives d² − r² = (169 + x²) − (25 + x²) = 144, which is always true regardless of x. Thus x is not determined. If x = 0, then r = 5 and AB is a diameter. If x = 12, then r = √(144 + 25) = 13, and the same measurements still satisfy the tangent-secant relationship. Therefore AB need not be a diameter, and the radius is not uniquely determined by the three lengths alone.
3. What additional measurement would determine the radius? Give a specific example and show how it would be used.
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One sufficient measurement is the distance from the center O to chord AB. If that distance is x, then the radius is r = √(x² + 5²), because the perpendicular from the center bisects the 10 m chord into two 5 m halves. For example, if the center is 12 m from the chord, then r = √(12² + 5²) = √169 = 13 m. Another sufficient measurement is PO = d; then r = √(d² − 144). A third option is the angle between tangent PT and secant PB. In right triangle PTO, with OT perpendicular to PT, that angle θ satisfies sin θ = r/d, and combining this with d² − r² = 144 gives r = 12 tan θ. Any of these measurements removes the ambiguity and determines the radius.