In a circle, chord AB which is at a distance of 5 cm from the centre of the circle is 24 cm long. The tangents at A and B meet at point P. What is the length of the line segment PA?

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.

Learn how to calculate the length of line segment PA in a circle where chord AB is given. Detailed step-by-step solution using circle geometry, tangents, Pythagorean theorem, and Power of a Point theorem. Suitable for students in Grades 9-12.

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