PQRS is a cyclic quadrilaeral in which PQ = 14.4 cm QR = 12.8 cm and SR = 9.6 cm If PR bisects QS, what is the length of PS?

Options

A. 16.4 cm
B. 13.6 cm
C. 19,2 cm
D. 15.8 cm

Answer

To solve the problem, we will use the properties of cyclic quadrilaterals and the concept of the angle bisector theorem.

### Step-by-Step Solution:

1. **Identify the Given Information**:
- We have a cyclic quadrilateral PQRS.
- Lengths of the sides are given as:
- \( PQ = 14.4 \, \text{cm} \)
- \( QR = 12.8 \, \text{cm} \)
- \( SR = 9.6 \, \text{cm} \)
- The diagonal \( PR \) bisects \( QS \).

2. **Use the Angle Bisector Theorem**:
- According to the angle bisector theorem, if a point \( P \) on side \( QS \) is such that \( PR \) bisects \( QS \), then:
\[ \frac{PQ}{QR} = \frac{PS}{SR} \]
- Let \( PS = x \). Then, we can substitute the known values:
\[ \frac{14.4}{12.8} = \frac{x}{9.6} \]

3. **Cross-Multiply to Solve for \( x \)**:
- Cross-multiplying gives us:
\[ 14.4 \cdot 9.6 = 12.8 \cdot x \]
- Calculate \( 14.4 \cdot 9.6 \):
\[ 14.4 \cdot 9.6 = 138.24 \]
- Now, we have:
\[ 138.24 = 12.8 \cdot x \]

4. **Solve for \( x \)**:
- Divide both sides by \( 12.8 \):
\[ x = \frac{138.24}{12.8} \]
- Calculate \( x \):
\[ x = 10.8 \, \text{cm} \]

5. **Conclusion**:
- Therefore, the length of \( PS \) is \( 10.8 \, \text{cm} \).

### Final Answer:
The length of \( PS \) is \( 10.8 \, \text{cm} \).