In a right-angled triangle ABC, right angled at B, an altitude BD is dropped on AC. If AB=8 and BC=6, what is the length of AD?

Options

A. 2.4
B. 3.6
C. 4.8
D. 6.4

Answer

To find the length of \( AD \) in the right-angled triangle \( ABC \) where \( AB = 8 \) and \( BC = 6 \), we can follow these steps:

### Step 1: Calculate the length of \( AC \) using the Pythagorean theorem.
In triangle \( ABC \), since it is a right triangle with \( AB \) and \( BC \) as the two legs, we can use the Pythagorean theorem:

\[ AC^2 = AB^2 + BC^2 \]

Substituting the known values:

\[ AC^2 = 8^2 + 6^2 \]

Calculating the squares:

\[ AC^2 = 64 + 36 = 100 \]

Taking the square root:

\[ AC = \sqrt{100} = 10 \]

### Step 2: Set up the relationship using cosine.
Let \( \theta \) be the angle \( \angle ABC \). We can express \( \cos \theta \) in two different ways.

First, in triangle \( ABC \):

\[ \cos \theta = \frac{AB}{AC} = \frac{8}{10} = 0.8 \]

Next, in triangle \( ABD \):

\[ \cos \theta = \frac{AD}{AB} = \frac{AD}{8} \]

### Step 3: Equate the two expressions for \( \cos \theta \).
Since both expressions equal \( \cos \theta \), we can set them equal to each other:

\[ \frac{AD}{8} = \frac{8}{10} \]

### Step 4: Solve for \( AD \).
Cross-multiplying gives:

\[ AD \cdot 10 = 8 \cdot 8 \]

Calculating the right side:

\[ 10AD = 64 \]

Dividing both sides by 10:

\[ AD = \frac{64}{10} = 6.4 \]

### Conclusion
The length of \( AD \) is \( 6.4 \).

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