Multiple choice

Let $AP$ be a diameter of a circle of radius $r$ and $PT$ be the tange to the circle at the point $P$ such that the line $AT$ intersects the circle at $B$. If $PT=8units$ and $BT=4units$, then $r$ is equal to

  1. $4\sqrt { 3 } units$
  2. $4units$
  3. $\cfrac { 4 }{ \sqrt { 3 } } units$
  4. $2\sqrt { 3 } units$
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A Correct answer
Explanation

By the power of a point theorem, PT^2 = TB * TA. 8^2 = 4 * TA => 64 = 4 * TA => TA = 16. Since TB = 4, AB = 12. AB is a chord. In a circle, if AP is diameter, angle ABP = 90. In triangle ABP, AP^2 = AB^2 + BP^2. Using geometry properties, r = 4*sqrt(3).

AI explanation

Since PT is a tangent and AP is a diameter, triangle APT is a right triangle, so AP equals 2r. By the tangent secant theorem, PT squared equals BT multiplied by AT, which gives 8 squared equals 4 times AT, so AT is 16. Using the Pythagorean theorem in triangle APT, AP squared plus PT squared equals AT squared, meaning 4r squared plus 64 equals 256, so 4r squared is 192. Solving this gives r squared as 48, making r equal to 4 root 3 units.

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