What is the area of the triangle ABC with sides a = 10 cm, c=4 cm and angle B = `30^(@)`?

Options

A. `16cm^(2)`
B. `12cm^(2)`
C. `10 cm^(2)`
D. `8 cm^(2)`

Answer

To find the area of triangle ABC with sides \( a = 10 \, \text{cm} \), \( c = 4 \, \text{cm} \), and angle \( B = 30^\circ \), we can use the formula for the area of a triangle given two sides and the included angle:

\[ \text{Area} = \frac{1}{2} \times a \times c \times \sin(B) \]

### Step-by-Step Solution:

1. **Identify the values**:
- Side \( a = 10 \, \text{cm} \)
- Side \( c = 4 \, \text{cm} \)
- Angle \( B = 30^\circ \)

2. **Substitute the values into the area formula**:
\[ \text{Area} = \frac{1}{2} \times 10 \times 4 \times \sin(30^\circ) \]

3. **Calculate \( \sin(30^\circ) \)**:
- We know that \( \sin(30^\circ) = \frac{1}{2} \).

4. **Substitute \( \sin(30^\circ) \) into the formula**:
\[ \text{Area} = \frac{1}{2} \times 10 \times 4 \times \frac{1}{2} \]

5. **Perform the multiplication**:
\[ \text{Area} = \frac{1}{2} \times 10 \times 4 \times \frac{1}{2} = \frac{1}{2} \times 10 \times 2 = 10 \, \text{cm}^2 \]

### Final Answer:
The area of triangle ABC is \( 10 \, \text{cm}^2 \).