Point \( (2,1) \) and \( (6,7) \) are opposite vertices of a square which is nscribed in a circle find the centre and equation of a circle

To find the center and equation of the circle inscribed around the square with opposite vertices at points \( (2,1) \) and \( (6,7) \), we can follow these steps: 1. **Find the center of the square**: The center of the square is the midpoint of the line segment connecting the two opposite vertices. 2. **Calculate the midpoint**: The midpoint \( M \) of the points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] 3. **Calculate the radius of the circle**: The radius of the circle is half the length of the diagonal of the square. The length of the diagonal can be calculated using the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] 4. **Write the equation of the circle**: The equation of a circle with center \( (h, k) \) and radius \( r \) is given by: \[ (x - h)^2 + (y - k)^2 = r^2 \] Now, let's perform the calculations step by step. ### Step 1: Find the midpoint Let \( (x_1, y_1) = (2, 1) \) and \( (x_2, y_2) = (6, 7) \). Calculating the midpoint: \[ M = \left( \frac{2 + 6}{2}, \frac{1 + 7}{2} \right) = \left( \frac{8}{2}, \frac{8}{2} \right) = (4, 4) \] ### Step 2: Calculate the length of the diagonal Using the distance formula: \[ d = \sqrt{(6 - 2)^2 + (7 - 1)^2} = \sqrt{4^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52} = 2\sqrt{13} \] ### Step 3: Calculate the radius The radius \( r \) is half the length of the diagonal: \[ r = \frac{d}{2} = \frac{2\sqrt{13}}{2} = \sqrt{13} \] ### Step 4: Write the equation of the circle The center of the circle is \( (4, 4) \) and the radius is \( \sqrt{13} \). Thus, the equation of the circle is: \[ (x - 4)^2 + (y - 4)^2 = (\sqrt{13})^2 \] This simplifies to: \[ (x - 4)^2 + (y - 4)^2 = 13 \] ### Final Answer - The center of the circle is \( (4, 4) \). - The equation of the circle is \( (x - 4)^2 + (y - 4)^2 = 13 \).