In `Delta ABC, /_B = 90^@` and `BD _|_ AC`.
If AC = 9 cm and AB = 7 cm, find AD. 

Answer

To solve the problem, we will use the properties of right triangles and the Pythagorean theorem. Here are the steps to find the length of AD in triangle ABC where ∠B = 90° and BD ⊥ AC.

### Step-by-Step Solution:

1. **Understand the Triangle Setup:**
- We have triangle ABC with ∠B = 90°.
- BD is perpendicular to AC, which means BD is the height from point B to side AC.
- We know AC = 9 cm and AB = 7 cm.

2. **Define the Segments:**
- Let AD = x cm.
- Therefore, CD = AC - AD = 9 - x cm.

3. **Apply the Pythagorean Theorem in Triangle ABD:**
- According to the Pythagorean theorem:
\[ AB^2 = AD^2 + BD^2 \]
- Substituting the known values:
\[ 7^2 = x^2 + BD^2 \]
- This simplifies to:
\[ 49 = x^2 + BD^2 \quad \text{(Equation 1)} \]

4. **Apply the Pythagorean Theorem in Triangle BCD:**
- Again using the Pythagorean theorem:
\[ BC^2 = BD^2 + CD^2 \]
- Substituting CD = 9 - x:
\[ BC^2 = BD^2 + (9 - x)^2 \]
- Expanding this:
\[ BC^2 = BD^2 + (81 - 18x + x^2) \quad \text{(Equation 2)} \]

5. **Use the Pythagorean Theorem in Triangle ABC:**
- Since ∠B = 90°, we can apply the Pythagorean theorem:
\[ AC^2 = AB^2 + BC^2 \]
- Substituting the known values:
\[ 9^2 = 7^2 + BC^2 \]
- This simplifies to:
\[ 81 = 49 + BC^2 \]
- Therefore:
\[ BC^2 = 81 - 49 = 32 \quad \text{(Equation 3)} \]

6. **Substitute Equation 3 into Equation 2:**
- From Equation 2, we have:
\[ 32 = BD^2 + (81 - 18x + x^2) \]
- Rearranging gives:
\[ BD^2 = 32 - 81 + 18x - x^2 \]
- Thus:
\[ BD^2 = 18x - x^2 - 49 \quad \text{(Equation 4)} \]

7. **Equate Equation 1 and Equation 4:**
- From Equation 1:
\[ BD^2 = 49 - x^2 \]
- Set this equal to Equation 4:
\[ 49 - x^2 = 18x - x^2 - 49 \]
- Simplifying gives:
\[ 49 + 49 = 18x \]
- Therefore:
\[ 98 = 18x \]
- Solving for x:
\[ x = \frac{98}{18} = \frac{49}{9} \approx 5.44 \text{ cm} \]

### Final Answer:
AD = \( \frac{49}{9} \) cm or approximately 5.44 cm.