A circle touches all the sides of a quadrilateral PQRS, whose sides. PQ=2cm, QR =3cm and RS= 4cm. What is the length of PS?

Options

A. 3 cm
B. 2 cm
C. 1 cm
D. 4 cm

Answer

To find the length of side PS in the quadrilateral PQRS, we can use the property of tangential quadrilaterals. In such quadrilaterals, the sum of the lengths of opposite sides is equal.

### Step-by-step Solution:

1. **Identify the sides of the quadrilateral:**
- Given:
- PQ = 2 cm
- QR = 3 cm
- RS = 4 cm
- PS = ? (This is what we need to find)

2. **Apply the property of tangential quadrilaterals:**
- The property states that for a quadrilateral with an inscribed circle (tangential quadrilateral), the sum of the lengths of opposite sides is equal:
\[ PQ + RS = QR + PS \]

3. **Substitute the known values into the equation:**
- Substitute PQ, QR, and RS into the equation:
\[ 2 + 4 = 3 + PS \]

4. **Simplify the equation:**
- Calculate the left side:
\[ 6 = 3 + PS \]

5. **Isolate PS:**
- To find PS, subtract 3 from both sides:
\[ PS = 6 - 3 \]
\[ PS = 3 \text{ cm} \]

### Final Answer:
The length of PS is **3 cm**.