`Delta`ABC is a right angle triangle and `angleABC = 90^@`. If the length of AB and BC is 12 cm and 16 cm respectively, then find the length of BD. (Where, BD`bot`AC)

Options

A. 6.4 cm
B. 7.2 cm
C. 8.4 cm
D. 9.6 cm

Answer

To find the length of \( BD \) in triangle \( ABC \), where \( \angle ABC = 90^\circ \), and the lengths of \( AB \) and \( BC \) are given as 12 cm and 16 cm respectively, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the triangle and its dimensions**:
- We have a right triangle \( ABC \) with \( AB = 12 \) cm and \( BC = 16 \) cm.
- Since \( ABC \) is a right triangle, we can use the Pythagorean theorem to find the length of \( AC \).

2. **Calculate the length of \( AC \)**:
\[ AC = \sqrt{AB^2 + BC^2} = \sqrt{12^2 + 16^2} = \sqrt{144 + 256} = \sqrt{400} = 20 \text{ cm} \]

3. **Set up the relationship using similar triangles**:
- Since \( BD \) is perpendicular to \( AC \), triangles \( ABC \) and \( CDB \) are similar (by AA criterion: \( \angle ABC = \angle BDC = 90^\circ \) and \( \angle ACB = \angle CDB \)).
- Therefore, we can set up the ratio:
\[ \frac{AC}{BC} = \frac{AB}{BD} \]

4. **Substitute the known values into the ratio**:
- We know \( AC = 20 \) cm, \( BC = 16 \) cm, and \( AB = 12 \) cm. Plugging these values into the ratio gives:
\[ \frac{20}{16} = \frac{12}{BD} \]

5. **Cross-multiply to solve for \( BD \)**:
\[ 20 \cdot BD = 12 \cdot 16 \]
\[ 20 \cdot BD = 192 \]
\[ BD = \frac{192}{20} = 9.6 \text{ cm} \]

### Final Answer:
The length of \( BD \) is \( 9.6 \) cm.

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