What is the area of a circle whose area is equal to that of a triangle with sides 7cm, 24cm and 25cm?

Options

A. `80 cm^(2)`
B. `84 cm^(2)`
C. `88 cm^(2)`
D. `90 cm^(2)`

Answer

To find the area of a circle whose area is equal to that of a triangle with sides 7 cm, 24 cm, and 25 cm, we can follow these steps:

### Step 1: Determine if the triangle is a right triangle
First, we need to check if the triangle with sides 7 cm, 24 cm, and 25 cm 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, we have:
- \( a = 7 \, \text{cm} \)
- \( b = 24 \, \text{cm} \)
- \( c = 25 \, \text{cm} \) (hypotenuse)

Now, we check:
\[ c^2 = a^2 + b^2 \]
\[ 25^2 = 7^2 + 24^2 \]
\[ 625 = 49 + 576 \]
\[ 625 = 625 \]
Since the equation holds true, the triangle is indeed a right triangle.

### Step 2: Calculate the area of the triangle
For a right triangle, the area can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, we can take the base as 7 cm and the height as 24 cm:
\[ \text{Area} = \frac{1}{2} \times 7 \times 24 \]
\[ \text{Area} = \frac{1}{2} \times 168 \]
\[ \text{Area} = 84 \, \text{cm}^2 \]

### Step 3: Set the area of the circle equal to the area of the triangle
Now that we have the area of the triangle, we set it equal to the area of the circle. The area of a circle is given by the formula:
\[ \text{Area of Circle} = \pi r^2 \]
We want to find the radius \( r \) such that:
\[ \pi r^2 = 84 \]

### Step 4: Solve for the radius \( r \)
To find \( r \), we rearrange the equation:
\[ r^2 = \frac{84}{\pi} \]
Taking the square root of both sides gives:
\[ r = \sqrt{\frac{84}{\pi}} \]

### Step 5: Find the area of the circle
Using the value of \( r \) we found, we can now calculate the area of the circle:
\[ \text{Area of Circle} = \pi r^2 = \pi \left(\frac{84}{\pi}\right) = 84 \, \text{cm}^2 \]

### Final Answer
The area of the circle is \( 84 \, \text{cm}^2 \).

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