Multiple choice

The points A, B, and C are on a circle O. The tangent line at A and the secant BC intersect at P, B lying between C and P. If $\overline{BC}$ = 20 and $\overline{PA}$ = $10\sqrt{3}$ ,then $\overline{PB}$ equals:

  1. $5$
  2. $10$
  3. $10\sqrt{3}$
  4. $20$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

By the Tangent-Secant Theorem, PA^2 = PB * PC. Let PB = x. Then PC = x + 20. (10*sqrt(3))^2 = x(x+20). 300 = x^2 + 20x. x^2 + 20x - 300 = 0. (x+30)(x-10) = 0. Since length must be positive, x = 10.

AI explanation

By the Tangent-Secant Theorem, the square of the tangent length equals the product of the secant segment lengths, giving the equation $PA^2 = PB \times PC$. Substituting the values gives $(10\sqrt{3})^2 = PB \times (PB + 20)$, which simplifies to $PB^2 + 20PB - 300 = 0$. Factoring this quadratic equation yields $(PB + 30)(PB - 10) = 0$, so the valid positive length for PB is 10.

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