AB and CD are two chords of a circle intersecting at a point P inside the circle. If
AP=3cm, PB= 2.5cm and CD= 6.5 cm, determine CP.

Answer

To find the length of CP in the given problem, we can use the property of intersecting chords in a circle. According to this property, if two chords AB and CD intersect at point P, then the products of the lengths of the segments of each chord are equal.

### Step-by-step solution:

1. **Identify the given lengths:**
- AP = 3 cm
- PB = 2.5 cm
- CD = 6.5 cm

2. **Calculate the total length of chord AB:**
- The total length of chord AB can be calculated as:
\[ AB = AP + PB = 3 \text{ cm} + 2.5 \text{ cm} = 5.5 \text{ cm} \]

3. **Let CP = x cm.**
- Since CD = 6.5 cm, we can express DP (the other segment of chord CD) as:
\[ DP = CD - CP = 6.5 \text{ cm} - x \]

4. **Apply the intersecting chords theorem:**
- According to the theorem, we have:
\[ AP \times PB = CP \times DP \]
- Substituting the known values:
\[ 3 \text{ cm} \times 2.5 \text{ cm} = x \times (6.5 \text{ cm} - x) \]
- This simplifies to:
\[ 7.5 = x(6.5 - x) \]

5. **Rearranging the equation:**
- Expanding the right side:
\[ 7.5 = 6.5x - x^2 \]
- Rearranging gives us:
\[ x^2 - 6.5x + 7.5 = 0 \]

6. **Solve the quadratic equation:**
- We can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1, b = -6.5, c = 7.5 \):
\[ x = \frac{6.5 \pm \sqrt{(-6.5)^2 - 4 \cdot 1 \cdot 7.5}}{2 \cdot 1} \]
- Calculate the discriminant:
\[ (-6.5)^2 - 4 \cdot 1 \cdot 7.5 = 42.25 - 30 = 12.25 \]
- Now substituting back into the formula:
\[ x = \frac{6.5 \pm \sqrt{12.25}}{2} \]
- Since \( \sqrt{12.25} = 3.5 \):
\[ x = \frac{6.5 \pm 3.5}{2} \]

7. **Calculate the two possible values for x:**
- First value:
\[ x = \frac{6.5 + 3.5}{2} = \frac{10}{2} = 5 \]
- Second value:
\[ x = \frac{6.5 - 3.5}{2} = \frac{3}{2} = 1.5 \]

8. **Conclusion:**
- Therefore, the possible lengths for CP are:
\[ CP = 5 \text{ cm or } 1.5 \text{ cm} \]