In `Delta ABC /_BAC = 90^@` and `AD_I_BC`. If `BD = 3 cm` and `CD = 4 cm`, then the length of AD is

Options

A. `3.5` cm
B. `5 cm `
C. `2sqrt(3)` cm
D. `6 cm `

Answer

To solve the problem, we need to find the length of \( AD \) in triangle \( ABC \) where \( \angle BAC = 90^\circ \), and \( AD \) is perpendicular to \( BC \). We are given that \( BD = 3 \, \text{cm} \) and \( CD = 4 \, \text{cm} \).

### Step-by-Step Solution:

1. **Understand the Geometry**:
- We have a right triangle \( ABC \) with \( \angle BAC = 90^\circ \).
- Point \( D \) lies on line segment \( BC \) such that \( AD \) is perpendicular to \( BC \).

2. **Identify the Given Lengths**:
- \( BD = 3 \, \text{cm} \)
- \( CD = 4 \, \text{cm} \)

3. **Use the Relationship for Right Triangles**:
- In a right triangle where a perpendicular is drawn from the right angle to the hypotenuse, there is a relationship that states:
\[ AD^2 = BD \times CD \]
- This means that the square of the length of the perpendicular \( AD \) is equal to the product of the lengths \( BD \) and \( CD \).

4. **Calculate \( AD^2 \)**:
- Substitute the values of \( BD \) and \( CD \):
\[ AD^2 = 3 \, \text{cm} \times 4 \, \text{cm} = 12 \, \text{cm}^2 \]

5. **Find \( AD \)**:
- To find \( AD \), take the square root of \( AD^2 \):
\[ AD = \sqrt{12 \, \text{cm}^2} = \sqrt{4 \times 3} = 2\sqrt{3} \, \text{cm} \]

Thus, the length of \( AD \) is \( 2\sqrt{3} \, \text{cm} \).

### Final Answer:
The length of \( AD \) is \( 2\sqrt{3} \, \text{cm} \).

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