ABC is a right-angled triangle with the right angle at vertex B. BD is the altitude through B. Given BD = 12 cm and AD = 9 cm.
Find AC.

Answer

To find the length of AC in the right-angled triangle ABC, we can use the properties of right triangles and the altitude drawn from the right angle.

### Step-by-Step Solution:

1. **Identify the given values**:
- BD (altitude from B to AC) = 12 cm
- AD = 9 cm

2. **Use the property of right triangles**:
In a right triangle, the square of the length of the altitude (BD) from the right angle to the hypotenuse (AC) is equal to the product of the segments into which it divides the hypotenuse (AD and DC). This can be expressed as:
\[ BD^2 = AD \cdot DC \]

3. **Substitute the known values into the equation**:
\[ 12^2 = 9 \cdot DC \]
\[ 144 = 9 \cdot DC \]

4. **Solve for DC**:
\[ DC = \frac{144}{9} = 16 \text{ cm} \]

5. **Find AC**:
The length of AC can be found by adding the lengths of AD and DC:
\[ AC = AD + DC \]
\[ AC = 9 + 16 = 25 \text{ cm} \]

### Final Answer:
The length of AC is **25 cm**.