In square PQRS:
(i) if `PQ=3x-7 and QR=x+3`, find PS.
(ii) if PR=5x and QS=9x-8. find QS.

Answer

To solve the question step by step, we will break it down into two parts as given.

### Part (i): Finding PS

1. **Identify the sides of the square**:
- We know that in a square, all sides are equal. Therefore, \(PQ = QR = PS = SR\).

2. **Set up the equations**:
- Given \(PQ = 3x - 7\) and \(QR = x + 3\), we can set these two expressions equal to each other:
\[ 3x - 7 = x + 3 \]

3. **Rearrange the equation**:
- Move \(x\) to one side and constants to the other side:
\[ 3x - x = 3 + 7 \]
- This simplifies to:
\[ 2x = 10 \]

4. **Solve for \(x\)**:
- Divide both sides by 2:
\[ x = \frac{10}{2} = 5 \]

5. **Substitute \(x\) back to find PS**:
- Now substitute \(x = 5\) into either \(PQ\) or \(QR\) to find PS:
\[ PS = QR = x + 3 = 5 + 3 = 8 \]

### Part (ii): Finding QS

1. **Identify the diagonals of the square**:
- In a square, the diagonals are equal. Therefore, \(PR = QS\).

2. **Set up the equations**:
- Given \(PR = 5x\) and \(QS = 9x - 8\), we can set these two expressions equal to each other:
\[ 5x = 9x - 8 \]

3. **Rearrange the equation**:
- Move \(9x\) to one side and constants to the other side:
\[ 5x - 9x = -8 \]
- This simplifies to:
\[ -4x = -8 \]

4. **Solve for \(x\)**:
- Divide both sides by -4:
\[ x = \frac{-8}{-4} = 2 \]

5. **Substitute \(x\) back to find QS**:
- Now substitute \(x = 2\) into \(QS\):
\[ QS = 9x - 8 = 9(2) - 8 = 18 - 8 = 10 \]

### Final Answers:
- (i) PS = 8 cm
- (ii) QS = 10 cm