In the right angled triangle, the hypotenuse is 65 cm and one of its sides is equal to 16 cm. What is its area (in `cm^(2)`)?

Options

A. 450
B. 324
C. 360
D. 504

Answer

To find the area of the right-angled triangle with a hypotenuse of 65 cm and one side of 16 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the sides of the triangle**:
- Let the hypotenuse (the longest side) be \( c = 65 \) cm.
- Let one of the other sides (let's say \( a \)) be \( 16 \) cm.
- We need to find the length of the other side (let's call it \( b \)).

2. **Apply the Pythagorean theorem**:
- According to the Pythagorean theorem, for a right-angled triangle, the relationship between the sides is given by:
\[ c^2 = a^2 + b^2 \]
- Plugging in the known values:
\[ 65^2 = 16^2 + b^2 \]

3. **Calculate the squares**:
- Calculate \( 65^2 \) and \( 16^2 \):
\[ 65^2 = 4225 \]
\[ 16^2 = 256 \]

4. **Substitute and solve for \( b^2 \)**:
- Substitute the values into the equation:
\[ 4225 = 256 + b^2 \]
- Rearranging gives:
\[ b^2 = 4225 - 256 \]
\[ b^2 = 3969 \]

5. **Find \( b \)**:
- Take the square root of both sides:
\[ b = \sqrt{3969} \]
- Calculate \( b \):
\[ b = 63 \text{ cm} \]

6. **Calculate the area of the triangle**:
- The area \( A \) of a right-angled triangle is given by:
\[ A = \frac{1}{2} \times a \times b \]
- Substitute the values of \( a \) and \( b \):
\[ A = \frac{1}{2} \times 16 \times 63 \]
- Calculate the area:
\[ A = \frac{1}{2} \times 1008 = 504 \text{ cm}^2 \]

### Final Answer:
The area of the triangle is \( 504 \text{ cm}^2 \).