Explanation:

Let’s break it down:

- The tangent-secant theorem (or power of point theorem) says: For a point P outside the circle, the square of the tangent (PQ) equals product of the whole secant (PB) and its external part (PA).

- So, \(PQ^2 = PA \times PB\).

- We know \(PQ = 6\) cm, so \(6^2 = PA \times PB\rightarrow 36 = PA \times PB\).

- AB is given as 5 cm, meaning \(PB = PA + 5\).

- Plug that in: \(36 = PA \times (PA + 5)\). So, \(PA^2 + 5PA - 36 = 0\).

- Quadratic time: \(PA = \frac{-5 \pm \sqrt{25 + 144}}{2} = \frac{-5 \pm 13}{2}\).

- The positive root is \((13-5)/2 = 4\).

Option:3, 4 cm, is correct. ??

Here’s why the other options don’t fit:

- Option 1 (9 cm): Plug in 9, you get PB = 14, but \(9 \times 14 = 126\), not 36.

- Option 2 (6 cm): 6 × 11 = 66.

- Option 4 (3 cm): 3 × 8 = 24.

What this really means is:

If AB is 5 cm and the tangent is 6 cm, the only value that fits the tangent-secant relationship is 4 cm, which is what you picked.