ABC is a triangle which is right angled at A and a perpendicular AD is drawn on the hypotenuse BC. If BC = 8 and AD = 3, then what is the value of `ABxxAC`?

Options

A. 12
B. 24
C. 32
D. 36

Answer

To solve the problem, we need to find the value of \( AB \times AC \) in the right triangle \( ABC \) where \( A \) is the right angle, \( BC \) is the hypotenuse, and \( AD \) is the perpendicular drawn from \( A \) to \( BC \).

### Step-by-Step Solution:

1. **Identify Given Values**:
- The length of the hypotenuse \( BC = 8 \).
- The length of the perpendicular \( AD = 3 \).

2. **Use the Area of Triangle Formula**:
The area of triangle \( ABC \) can be calculated in two ways:
- Using the base \( BC \) and height \( AD \):
\[ \text{Area} = \frac{1}{2} \times BC \times AD \]
- Using the sides \( AB \) and \( AC \) as base and height:
\[ \text{Area} = \frac{1}{2} \times AB \times AC \]

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:
\[ \frac{1}{2} \times AB \times AC = \frac{1}{2} \times BC \times AD \]

4. **Cancel the Common Factor**:
We can cancel \( \frac{1}{2} \) from both sides:
\[ AB \times AC = BC \times AD \]

5. **Substitute the Known Values**:
Substitute the values of \( BC \) and \( AD \):
\[ AB \times AC = 8 \times 3 \]

6. **Calculate the Result**:
\[ AB \times AC = 24 \]

Thus, the value of \( AB \times AC \) is \( 24 \).