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.
Calculate AB .

Answer

To solve the problem, we need to find the length of side AB in triangle ABC, where BD is the altitude from vertex B to side AC. We are given that BD = 12 cm and AD = 9 cm.

### Step-by-Step Solution:

1. **Identify the Right Triangle**:
We have triangle ABC with a right angle at B. The altitude BD divides the triangle into two smaller right triangles: ABD and BDC.

2. **Apply the Pythagorean Theorem**:
In triangle ABD, we can apply the Pythagorean theorem. According to the theorem:
\[ AB^2 = AD^2 + BD^2 \]

3. **Substitute the Given Values**:
We know that:
- \(AD = 9 \, \text{cm}\)
- \(BD = 12 \, \text{cm}\)

Now substituting these values into the equation:
\[ AB^2 = 9^2 + 12^2 \]

4. **Calculate the Squares**:
Calculate \(9^2\) and \(12^2\):
\[ 9^2 = 81 \]
\[ 12^2 = 144 \]

5. **Add the Squares**:
Now add the two results:
\[ AB^2 = 81 + 144 = 225 \]

6. **Find the Length of AB**:
To find AB, take the square root of both sides:
\[ AB = \sqrt{225} = 15 \, \text{cm} \]

### Final Answer:
Thus, the length of AB is \(15 \, \text{cm}\).

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