Find the area of `DeltaABC` in which `angleB = 90^(@)`, BC = 8 cm and AC=10 cm. If `BDbotAC`, then find BD.

Options

A. 4. 8 cm
B. 8. 4 cm
C. 3.7 cm
D. none of these

Answer

To solve the problem, we need to find the area of triangle ABC and then determine the length of BD, where BD is perpendicular to AC.

### Step-by-Step Solution:

1. **Identify the given values:**
- Angle B = 90°
- BC = 8 cm
- AC = 10 cm

2. **Use the Pythagorean theorem to find AB:**
Since triangle ABC is a right triangle at B, we can use the Pythagorean theorem:
\[ AC^2 = AB^2 + BC^2 \]
Substituting the known values:
\[ 10^2 = AB^2 + 8^2 \]
\[ 100 = AB^2 + 64 \]
\[ AB^2 = 100 - 64 \]
\[ AB^2 = 36 \]
\[ AB = \sqrt{36} = 6 \text{ cm} \]

3. **Calculate the area of triangle ABC:**
The area of a right triangle can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, we can take BC as the base and AB as the height:
\[ \text{Area} = \frac{1}{2} \times BC \times AB = \frac{1}{2} \times 8 \times 6 \]
\[ \text{Area} = \frac{1}{2} \times 48 = 24 \text{ cm}^2 \]

4. **Relate the area to BD:**
The area can also be expressed using AC as the base and BD as the height:
\[ \text{Area} = \frac{1}{2} \times AC \times BD \]
Setting the two area expressions equal:
\[ 24 = \frac{1}{2} \times 10 \times BD \]
\[ 24 = 5 \times BD \]
\[ BD = \frac{24}{5} = 4.8 \text{ cm} \]

### Final Answers:
- The area of triangle ABC is \( 24 \text{ cm}^2 \).
- The length of BD is \( 4.8 \text{ cm} \).