In `Delta ABC,AD` and CE are perpendiculars from A and C on the opposite sides. If `AD = 6 cm. BC = 14 cm, AB = 10cm, `find CE.

Answer

To solve the problem, we need to find the length of CE in triangle ABC, given the following information:

- AD = 6 cm (perpendicular from A)
- BC = 14 cm (base)
- AB = 10 cm (another side)

### Step-by-Step Solution:

1. **Understand the Triangle and Given Information**:
- We have triangle ABC with points A, B, and C.
- AD and CE are the perpendicular heights from points A and C to the opposite sides (BC and AB respectively).

2. **Use the Area Formula for Triangle**:
- The area of triangle ABC can be calculated in two different ways using the heights AD and CE.
- The area using height AD is:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times BC \times AD \]
- The area using height CE is:
\[ \text{Area} = \frac{1}{2} \times AB \times CE \]

3. **Set the Two Area Expressions Equal**:
- Since both expressions represent the area of the same triangle, we can set them equal to each other:
\[ \frac{1}{2} \times BC \times AD = \frac{1}{2} \times AB \times CE \]

4. **Substitute the Known Values**:
- Substitute BC = 14 cm, AD = 6 cm, and AB = 10 cm into the equation:
\[ \frac{1}{2} \times 14 \times 6 = \frac{1}{2} \times 10 \times CE \]

5. **Simplify the Equation**:
- The \(\frac{1}{2}\) cancels out from both sides:
\[ 14 \times 6 = 10 \times CE \]

6. **Calculate the Left Side**:
- Calculate \(14 \times 6\):
\[ 14 \times 6 = 84 \]
- So, we have:
\[ 84 = 10 \times CE \]

7. **Solve for CE**:
- Divide both sides by 10 to find CE:
\[ CE = \frac{84}{10} = 8.4 \text{ cm} \]

### Conclusion:
Thus, the length of CE is **8.4 cm**.