The vertices of a right angled triangle are A(2,-2), B(-2,1) and C (5,2). The equation of circumcircle is

Options

A. `(x-2) (x+2) +(y+2) (y+1)=0`
B. `(x+2) (x-5) +(y-1) (y-2)=0`
C. `(x-5) (x-2)+(y-2) (y+2)=0`
D. none

Answer

To find the equation of the circumcircle of the right-angled triangle with vertices A(2, -2), B(-2, 1), and C(5, 2), we can follow these steps:

### Step 1: Identify the vertices
The vertices of the triangle are:
- A(2, -2)
- B(-2, 1)
- C(5, 2)

### Step 2: Determine the lengths of the sides
We need to calculate the lengths of the sides AB, BC, and CA to identify the hypotenuse.

1. **Length of AB**:
\[ AB = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} = \sqrt{(-2 - 2)^2 + (1 - (-2))^2} = \sqrt{(-4)^2 + (3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

2. **Length of BC**:
\[ BC = \sqrt{(x_C - x_B)^2 + (y_C - y_B)^2} = \sqrt{(5 - (-2))^2 + (2 - 1)^2} = \sqrt{(7)^2 + (1)^2} = \sqrt{49 + 1} = \sqrt{50} \]

3. **Length of CA**:
\[ CA = \sqrt{(x_A - x_C)^2 + (y_A - y_C)^2} = \sqrt{(2 - 5)^2 + (-2 - 2)^2} = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 3: Verify the right angle
To confirm that the triangle is right-angled, we check if the Pythagorean theorem holds:
\[ BC^2 = AB^2 + CA^2 \]
Calculating:
\[ (\sqrt{50})^2 = 5^2 + 5^2 \implies 50 = 25 + 25 \implies 50 = 50 \]
Thus, triangle ABC is indeed a right-angled triangle with BC as the hypotenuse.

### Step 4: Find the circumcircle
The circumcircle of a right-angled triangle has its center at the midpoint of the hypotenuse.

1. **Midpoint of BC**:
\[ \text{Midpoint M} = \left( \frac{x_B + x_C}{2}, \frac{y_B + y_C}{2} \right) = \left( \frac{-2 + 5}{2}, \frac{1 + 2}{2} \right) = \left( \frac{3}{2}, \frac{3}{2} \right) \]

2. **Radius of the circumcircle**:
The radius is half the length of the hypotenuse BC:
\[ r = \frac{BC}{2} = \frac{\sqrt{50}}{2} = \frac{5\sqrt{2}}{2} \]

### Step 5: Write the equation of the circumcircle
The equation of a circle with center (h, k) and radius r is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Substituting the values:
\[ \left( x - \frac{3}{2} \right)^2 + \left( y - \frac{3}{2} \right)^2 = \left( \frac{5\sqrt{2}}{2} \right)^2 \]
Calculating the right side:
\[ \left( \frac{5\sqrt{2}}{2} \right)^2 = \frac{25 \cdot 2}{4} = \frac{50}{4} = 12.5 \]
Thus, the equation becomes:
\[ \left( x - \frac{3}{2} \right)^2 + \left( y - \frac{3}{2} \right)^2 = 12.5 \]

### Step 6: Rearranging the equation
Expanding and rearranging:
\[ \left( x - \frac{3}{2} \right)^2 + \left( y - \frac{3}{2} \right)^2 - 12.5 = 0 \]

### Final Answer
The equation of the circumcircle is:
\[ \left( x - \frac{3}{2} \right)^2 + \left( y - \frac{3}{2} \right)^2 = 12.5 \]