What is the area (in `"cm"^2`) of a circle circumscribing a triangle whose sides are 28 cm, 45 cm and 53 cm?

Options

A. a) `506 (1)/(4) pi`
B. b) `508 (1)/(2) pi`
C. c) `718 (1)/(2) pi`
D. d) `702 (1)/(4) pi`

Answer

To find the area of a circle that circumscribes a triangle with sides 28 cm, 45 cm, and 53 cm, we can follow these steps:

### Step 1: Identify the sides of the triangle
We have a triangle with sides:
- \( a = 28 \) cm
- \( b = 45 \) cm
- \( c = 53 \) cm

### Step 2: Check if the triangle is a right triangle
To check if the triangle is a right triangle, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.

Here, the longest side is \( c = 53 \) cm. We check:
\[ c^2 = a^2 + b^2 \]
Calculating the squares:
\[ 53^2 = 28^2 + 45^2 \]
Calculating each term:
\[ 2809 = 784 + 2025 \]
Now, adding the two smaller squares:
\[ 784 + 2025 = 2809 \]
Since both sides are equal, the triangle is indeed a right triangle.

### Step 3: Calculate the radius of the circumcircle
For a right triangle, the radius \( R \) of the circumcircle can be calculated using the formula:
\[ R = \frac{c}{2} \]
where \( c \) is the length of the hypotenuse. Thus:
\[ R = \frac{53}{2} = 26.5 \text{ cm} \]

### Step 4: Calculate the area of the circumcircle
The area \( A \) of a circle is given by the formula:
\[ A = \pi R^2 \]
Substituting the value of \( R \):
\[ A = \pi \left( \frac{53}{2} \right)^2 \]
Calculating \( R^2 \):
\[ R^2 = \left( \frac{53}{2} \right)^2 = \frac{2809}{4} \]
Now substituting back into the area formula:
\[ A = \pi \cdot \frac{2809}{4} \]
Thus, the area of the circle is:
\[ A = \frac{2809\pi}{4} \text{ cm}^2 \]

### Final Answer
The area of the circle circumscribing the triangle is:
\[ \frac{2809\pi}{4} \text{ cm}^2 \]