In a triangle ABC, `angle B=90^(@), AB=8 cm, BC=6cm`. BD is perpendicular to AC, then find the length of CD, AD and BD.

Answer

To solve the problem step by step, we will follow the approach outlined in the video transcript:

### Step 1: Identify the triangle and given values
We have a right triangle ABC where:
- Angle B = 90°
- AB = 8 cm
- BC = 6 cm

### Step 2: Find the length of AC using the Pythagorean theorem
According to the Pythagorean theorem:
\[ AC^2 = AB^2 + BC^2 \]

Substituting the known values:
\[ AC^2 = 8^2 + 6^2 \]
\[ AC^2 = 64 + 36 \]
\[ AC^2 = 100 \]

Now, taking the square root:
\[ AC = \sqrt{100} = 10 \text{ cm} \]

### Step 3: Find the length of CD using the similarity of triangles
Using the property of similar triangles, we can find CD:
\[ CD = \frac{BC^2}{AC} \]

Substituting the known values:
\[ CD = \frac{6^2}{10} \]
\[ CD = \frac{36}{10} = 3.6 \text{ cm} \]

### Step 4: Find the length of AD using the similarity of triangles
Similarly, we can find AD:
\[ AD = \frac{AB^2}{AC} \]

Substituting the known values:
\[ AD = \frac{8^2}{10} \]
\[ AD = \frac{64}{10} = 6.4 \text{ cm} \]

### Step 5: Find the length of BD using the formula for the area of the triangle
The length of BD can be found using the formula:
\[ BD = \frac{AB \times BC}{AC} \]

Substituting the known values:
\[ BD = \frac{8 \times 6}{10} \]
\[ BD = \frac{48}{10} = 4.8 \text{ cm} \]

### Summary of Results
- CD = 3.6 cm
- AD = 6.4 cm
- BD = 4.8 cm