In a circle chords PO and TS are produced to meet at R. If RO 14.4 cm, PO = 11.2 cm and SR = 12.8 cm then the length of chord TS is :

Options

A. 18cm
B. 14.2 cm
C. 12.4 cm
D. 16 cm

Answer

To solve the problem, we will use the intersecting secants theorem, which states that if two secants intersect outside a circle, then the product of the lengths of one secant segment is equal to the product of the lengths of the other secant segment.

### Step-by-step Solution:

1. **Identify the Given Values:**
- RO = 14.4 cm
- PO = 11.2 cm
- SR = 12.8 cm
- We need to find the length of chord TS.

2. **Calculate RP:**
- RP is the total length from R to P, which is the sum of RO and PO.
\[ RP = RO + PO = 14.4 \, \text{cm} + 11.2 \, \text{cm} = 25.6 \, \text{cm} \]

3. **Use the Intersecting Secants Theorem:**
- According to the theorem:
\[ RO \times RP = RS \times RT \]
- We know RO and RP, and we can express RT in terms of TS and SR.
- Let TS = x, then RT = SR + TS = 12.8 cm + x.

4. **Set Up the Equation:**
- Substitute the known values into the equation:
\[ 14.4 \times 25.6 = 12.8 \times (12.8 + x) \]

5. **Calculate the Left Side:**
\[ 14.4 \times 25.6 = 369.84 \, \text{cm}^2 \]

6. **Expand the Right Side:**
\[ 12.8 \times (12.8 + x) = 12.8 \times 12.8 + 12.8x = 163.84 + 12.8x \]

7. **Set Up the Equation:**
\[ 369.84 = 163.84 + 12.8x \]

8. **Solve for x:**
- Subtract 163.84 from both sides:
\[ 369.84 - 163.84 = 12.8x \]
\[ 206 = 12.8x \]
- Divide both sides by 12.8:
\[ x = \frac{206}{12.8} = 16.09375 \, \text{cm} \approx 16 \, \text{cm} \]

9. **Conclusion:**
- The length of chord TS is approximately 16 cm.