A point P is given outside of a circle Gamma. A tangent from P touches Gamma at T with PT=45. A line from P cuts Gamma at the 2 points A, B. If PA=25, what is the length of PB?
We can use the tangent-secant theorem that says that
PT^2 = PA ( PA + BA)
45^2 = 25 (25 + BA)
2025 = 25^2 + 25BA
2025 = 625 + 25BA
2025 - 625 = 25BA
1600 = 25BA divide both sides by 25
64 = BA
So
PB = (PA + BA) = (25 + 64) = 89
