The hypotenuse of a right-angled triangle is `14sqrt(2)` cm long. What is the area of the circumcircle of the said triangle?
`["Use" pi = (22)/(7)]`

Options

A. `264 cm^(2)`
B. `308 cm^(2)`
C. `286 cm^(2)`
D. `312 cm^(2)`

Answer

To find the area of the circumcircle of a right-angled triangle with a hypotenuse of \( 14\sqrt{2} \) cm, we can follow these steps:

### Step 1: Identify the diameter of the circumcircle
In a right-angled triangle, the hypotenuse serves as the diameter of the circumcircle. Therefore, the diameter \( D \) is:
\[ D = 14\sqrt{2} \text{ cm} \]

### Step 2: Calculate the radius of the circumcircle
The radius \( r \) of the circumcircle is half of the diameter. Thus, we calculate:
\[ r = \frac{D}{2} = \frac{14\sqrt{2}}{2} = 7\sqrt{2} \text{ cm} \]

### Step 3: Use the formula for the area of a circle
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
Substituting the value of \( r \):
\[ A = \pi (7\sqrt{2})^2 \]

### Step 4: Simplify \( (7\sqrt{2})^2 \)
Calculating \( (7\sqrt{2})^2 \):
\[ (7\sqrt{2})^2 = 7^2 \cdot (\sqrt{2})^2 = 49 \cdot 2 = 98 \]
Thus, we have:
\[ A = \pi \cdot 98 \]

### Step 5: Substitute the value of \( \pi \)
We are given \( \pi = \frac{22}{7} \). Therefore:
\[ A = \frac{22}{7} \cdot 98 \]

### Step 6: Calculate the area
To simplify:
\[ A = \frac{22 \cdot 98}{7} \]
Calculating \( \frac{98}{7} \):
\[ \frac{98}{7} = 14 \]
Now substituting back:
\[ A = 22 \cdot 14 = 308 \text{ cm}^2 \]

### Final Result
The area of the circumcircle of the triangle is:
\[ \boxed{308 \text{ cm}^2} \]