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. 4
B. 6
C. 8
D. 9

Answer

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

### Step-by-Step Solution:

1. **Identify the given lengths:**
- AB = 6 cm
- CD = 5 cm
- BC = 7 cm
- Let AD = x (we need to find this).

2. **Use the property of tangential quadrilaterals:**
- According to the property, for a tangential quadrilateral (one whose sides touch a circle), the sum of the lengths of opposite sides is equal:
\[ AB + CD = BC + AD \]

3. **Substitute the known values into the equation:**
- Substitute AB, CD, and BC into the equation:
\[ 6 + 5 = 7 + x \]

4. **Simplify the equation:**
- Calculate the left side:
\[ 11 = 7 + x \]

5. **Solve for x (AD):**
- Rearranging the equation to isolate x:
\[ x = 11 - 7 \]
- Thus,
\[ x = 4 \]

6. **Conclusion:**
- The length of AD is 4 cm.

### Final Answer:
AD = 4 cm