If the sides of a triangle are 5 cm, 4 cm and `sqrt(41)` cm, then the area of the triangle is

Options

A. 20 `cm^(2)`
B. `(5 + 4 + sqrt(41)) cm^(2)`
C. `(5 + 4 + sqrt(41))/(2) cm^(2)`
D. 10 `cm^(2)`

Answer

To find the area of the triangle with sides 5 cm, 4 cm, and \(\sqrt{41}\) cm, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the type of triangle**:
We have the sides of the triangle as 5 cm, 4 cm, and \(\sqrt{41}\) cm. To determine if this is a right triangle, we can use the Pythagorean theorem.

2. **Apply the Pythagorean theorem**:
According to the Pythagorean theorem, for a triangle with sides \(a\), \(b\), and hypotenuse \(c\):
\[ a^2 + b^2 = c^2 \]
Here, we can take \(a = 4\) cm, \(b = 5\) cm, and \(c = \sqrt{41}\) cm.

3. **Calculate \(a^2 + b^2\)**:
\[ 4^2 + 5^2 = 16 + 25 = 41 \]

4. **Calculate \(c^2\)**:
\[ (\sqrt{41})^2 = 41 \]

5. **Verify the Pythagorean theorem**:
Since \(a^2 + b^2 = c^2\) (i.e., \(41 = 41\)), we confirm that this is a right triangle.

6. **Use the formula for the area of a right triangle**:
The area \(A\) of a right triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, we can take the base as 4 cm and the height as 5 cm.

7. **Substitute the values into the area formula**:
\[ A = \frac{1}{2} \times 4 \times 5 \]

8. **Calculate the area**:
\[ A = \frac{1}{2} \times 20 = 10 \text{ cm}^2 \]

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

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