If the sides of a quadrilateral ABCD touch a circle and AB = 6 cm, CD = 5 cm, BC = 7 cm, then the length of AD in cm is :

Options

A. a) 8
B. b) 9
C. c) 4
D. d) 6

Answer

To solve the problem, we need to find the length of side AD in the quadrilateral ABCD, given that it touches a circle and we know the lengths of the other sides.

### Step-by-Step Solution:

1. **Understand the Properties of the Quadrilateral**:
When a quadrilateral touches a circle, the sum of the lengths of its opposite sides is equal. This means:
\[ AB + CD = AD + BC \]

2. **Write Down the Given Values**:
- \( AB = 6 \, \text{cm} \)
- \( CD = 5 \, \text{cm} \)
- \( BC = 7 \, \text{cm} \)
- \( AD = ? \)

3. **Substitute the Known Values into the Equation**:
Using the property from step 1, we can substitute the known values:
\[ 6 + 5 = AD + 7 \]

4. **Simplify the Equation**:
Calculate the left side:
\[ 11 = AD + 7 \]

5. **Isolate AD**:
To find \( AD \), subtract 7 from both sides:
\[ AD = 11 - 7 \]

6. **Calculate AD**:
\[ AD = 4 \, \text{cm} \]

Thus, the length of side AD is **4 cm**.