ABC is a triangle right angled at Aand a perpendicular AD is drawn on the hypotenuse BC. What is BC.AD equal to ?

Options

A. AB .AC
B. AB . AD
C. CA .CD
D. AD . DB

Answer

To solve the problem, we need to establish the relationship between the lengths of the sides of triangle ABC, which is a right triangle, and the length of the perpendicular AD drawn from vertex A to the hypotenuse BC.

### Step-by-Step Solution:

1. **Identify the Triangle and the Perpendicular**:
- We have a right triangle ABC, where angle A is 90 degrees.
- A perpendicular line AD is drawn from point A to the hypotenuse BC.

2. **Understand the Similar Triangles**:
- The perpendicular AD divides triangle ABC into two smaller triangles: ABD and ADC.
- Triangles ABC, ABD, and ADC are all similar to each other because they share angles.

3. **Use the Property of Similar Triangles**:
- From the property of similar triangles, we know that the ratios of corresponding sides are equal.
- Therefore, we can write the following relationships:
- \( \frac{AB}{BC} = \frac{AD}{AB} \) (for triangle ABD)
- \( \frac{AC}{BC} = \frac{AD}{AC} \) (for triangle ADC)

4. **Expressing BC in Terms of AD**:
- From the first relationship:
\[ BC \cdot AD = AB^2 \]
- From the second relationship:
\[ BC \cdot AD = AC^2 \]

5. **Final Relationship**:
- Since both expressions equal \( BC \cdot AD \), we can conclude:
\[ BC \cdot AD = AB \cdot AC \]
- Thus, the final answer is:
\[ BC \cdot AD = AB \cdot AC \]

### Conclusion:
The value of \( BC \cdot AD \) is equal to the product of the lengths of the other two sides \( AB \) and \( AC \).