All sides of a quadrilateral ABCD touch a circle. If `AB = 6 cm, BC = 7.5 cm, CD = 3 cm`, then `DA` is

Options

A. 3.5 cm
B. 4.5 cm
C. 2.5 cm
D. 1.5 cm

Answer

To find the length of side DA in the quadrilateral ABCD, where all sides touch a circle, we can use the property that states the sum of the lengths of opposite sides are equal.

### Step-by-Step Solution:

1. **Identify the known lengths**:
- AB = 6 cm
- BC = 7.5 cm
- CD = 3 cm
- DA = ?

2. **Use the property of tangential quadrilaterals**:
For a quadrilateral that is tangential (i.e., it has an inscribed circle), the following relationship holds:
\[ AB + CD = BC + DA \]

3. **Substitute the known values into the equation**:
\[ 6 + 3 = 7.5 + DA \]

4. **Calculate the left side**:
\[ 9 = 7.5 + DA \]

5. **Isolate DA**:
To find DA, subtract 7.5 from both sides:
\[ DA = 9 - 7.5 \]

6. **Perform the subtraction**:
\[ DA = 1.5 \text{ cm} \]

### Final Answer:
Thus, the length of side DA is **1.5 cm**.

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