Given the problem, let's break it down:

Step 1: Understand the Given Information

• Chord AB is 24 cm long.
• The distance from the center $O$ to the chord AB is 5 cm.
• Tangents at points $A$ and $B$ meet at point $P$.

We need to find the length of the line segment $PA$.

Step 2: Analyze the Geometry of the Problem

1. Draw the Circle: Let $O$ be the center of the circle. Since $AB$ is a chord, and the perpendicular distance from $O$ to $AB$ is 5 cm, drop a perpendicular $OM$ from $O$ to $AB$. This point $M$ is the midpoint of the chord $AB$.

2. Calculate Radius $r$ of the Circle:

   • The length of $AB = 24$ cm, so the half-length $AM = MB = 12$ cm.
   • We can use the Pythagorean theorem in the right triangle $OMA$ to find the radius $r$.

   $OA = r, \quad OM = 5 \text{ cm}, \quad AM = 12 \text{ cm}$ $OA^2 = OM^2 + AM^2$ $r^2 = 5^2 + 12^2$ $r^2 = 25 + 144 = 169$ $r = \sqrt{169} = 13 \text{ cm}$

   So, the radius $r = 13$ cm.

Step 3: Use Power of a Point Theorem

The power of point $P$ with respect to the circle is equal to the square of the length of the tangent from $P$ to the circle. Since $PA = PB$ (tangents from a point to a circle are equal):

$PA^2 = \text{Power of point } P = \overline{OP}^2 - r^2$

But we also know from the Power of a Point theorem that: $PA^2 = OA^2 - OM^2$ $PA^2 = 13^2 - 5^2 = 169 - 25 = 144$ $PA = \sqrt{144} = 12 \text{ cm}$

Final Answer

The length of the line segment $PA$ is $\boxed{12}$ cm.

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Would you like more details or have any questions about this solution? Here are five related questions for further exploration:

1. What is the power of a point theorem and how is it applied in this context?
2. If the chord length was different, how would the length of $PA$ change?
3. What are other methods to find the radius of the circle when given a chord?
4. Can you determine the length of the tangents from any external point to a circle?
5. How does the location of point $P$ affect the lengths of $PA$ and $PB$?

Tip: Always check if you can apply the Pythagorean theorem when dealing with distances in circles involving chords and radii.