ABC is a right angled triangle of which A is the right angle, BD is drawn perpendicular to BC meets CA produced in D. If AB = 12, AC = 16, BC = 20, then BD =

Options

A. 15
B. 25
C. 10
D. 225

Explanation

To solve the problem, we will use the properties of right-angled triangles and the concept of similar triangles.

### Step-by-Step Solution:

1. **Identify the Triangle and Given Values:**
- We have a right-angled triangle ABC where angle A is the right angle.
- The lengths of the sides are given as:
- AB = 12
- AC = 16
- BC = 20

2. **Understanding the Geometry:**
- BD is drawn perpendicular to BC and meets CA produced at point D.
- We need to find the length of BD.

3. **Using Trigonometric Ratios:**
- Let angle \( \theta \) be the angle at B in triangle BDC.
- In triangle BDC, we can express the tangent of angle \( \theta \) as:
\[ \tan(\theta) = \frac{BD}{BC} = \frac{BD}{20} \]

4. **Finding the Angle \( \theta \) in Triangle ABC:**
- In triangle ABC, we can also express the tangent of angle \( \theta \) as:
\[ \tan(\theta) = \frac{AB}{AC} = \frac{12}{16} = \frac{3}{4} \]

5. **Setting the Equations Equal:**
- Since both expressions represent \( \tan(\theta) \), we can set them equal to each other:
\[ \frac{BD}{20} = \frac{3}{4} \]

6. **Solving for BD:**
- Cross-multiplying gives us:
\[ 4 \cdot BD = 3 \cdot 20 \]
\[ 4 \cdot BD = 60 \]
\[ BD = \frac{60}{4} = 15 \]

7. **Conclusion:**
- The length of BD is 15 units.

### Final Answer:
BD = 15 units.