A right angled isosceles triangle is inscribed in a circle of radius r. What is the area of the remaining portion of the circle ?

Options

A. `(pi r^2)/(2)`
B. `(pi - 1/2) r^2`
C. `(pi - 1) r^2`
D. `(pi - 2) r^2`

Answer

To find the area of the remaining portion of a circle when a right-angled isosceles triangle is inscribed in it, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Circle's Area**:
The area of a circle is given by the formula:
\[ \text{Area of Circle} = \pi r^2 \]
where \( r \) is the radius of the circle.

2. **Determine the Triangle's Side Length**:
Since the triangle is a right-angled isosceles triangle inscribed in the circle, we can denote the lengths of the equal sides of the triangle as \( A \). The hypotenuse of the triangle will be the diameter of the circle, which is \( 2r \).

3. **Apply the Pythagorean Theorem**:
According to the Pythagorean theorem:
\[ (2r)^2 = A^2 + A^2 \]
Simplifying this gives:
\[ 4r^2 = 2A^2 \]
Dividing both sides by 2:
\[ 2r^2 = A^2 \]
Taking the square root of both sides:
\[ A = r\sqrt{2} \]

4. **Calculate the Area of the Triangle**:
The area of a triangle is given by:
\[ \text{Area of Triangle} = \frac{1}{2} \times \text{base} \times \text{height} \]
For our isosceles triangle, both the base and height are equal to \( A \):
\[ \text{Area of Triangle} = \frac{1}{2} \times A \times A = \frac{1}{2} \times (r\sqrt{2}) \times (r\sqrt{2}) = \frac{1}{2} \times 2r^2 = r^2 \]

5. **Find the Remaining Area**:
Now, we can find the area of the remaining portion of the circle by subtracting the area of the triangle from the area of the circle:
\[ \text{Remaining Area} = \text{Area of Circle} - \text{Area of Triangle} \]
Substituting the values we calculated:
\[ \text{Remaining Area} = \pi r^2 - r^2 = (\pi - 1) r^2 \]

### Final Answer:
The area of the remaining portion of the circle is:
\[ \text{Remaining Area} = (\pi - 1) r^2 \]