Chords AB and CD of a circle when extended meet at point X. Given AB =4 cm, BX=6 cm and XD=6cm, calculate the length of CD.

Answer

To solve the problem step by step, we will use the intersecting chords theorem, which states that if two chords intersect each other, the products of the lengths of the segments of each chord are equal.

### Step-by-Step Solution:

1. **Identify the Given Lengths**:
- Chord AB = 4 cm
- Segment BX = 6 cm
- Segment XD = 6 cm
- Let the length of chord CD be represented as \( x \).

2. **Determine the Length of AX**:
- Since AB = 4 cm and BX = 6 cm, we can find AX:
\[ AX = AB + BX = 4 \, \text{cm} + 6 \, \text{cm} = 10 \, \text{cm} \]

3. **Set Up the Equation Using the Intersecting Chords Theorem**:
- According to the theorem:
\[ AX \cdot BX = CX \cdot DX \]
- We know:
- \( AX = 10 \, \text{cm} \)
- \( BX = 6 \, \text{cm} \)
- \( CX = x \) (the part of CD we want to find)
- \( DX = 6 \, \text{cm} \)
- Therefore, we can write:
\[ 10 \cdot 6 = x \cdot 6 \]

4. **Simplify the Equation**:
- This simplifies to:
\[ 60 = 6x \]

5. **Solve for x**:
- Divide both sides by 6:
\[ x = \frac{60}{6} = 10 \, \text{cm} \]

6. **Calculate the Length of CD**:
- Since CD consists of CX and DX:
\[ CD = CX + DX = x + 6 = 10 + 6 = 16 \, \text{cm} \]

### Final Answer:
The length of CD is **16 cm**.