Question

Right triangle ABC is located at A (-1,-2), B (-1,1), and C (3,1) on a coordinate plane. What is the equation of a circle A with radius segment AC?

A. (x+2)² +(y+1)² =10
B. (x+1)² +(y−2)² =10
C. (x+1)² +(y+2)² =10
D. (x−2)² +(y+1)² =10

Answer 1

The equation of a circle A with radius segment AC is (x+1)² +(y+2)² =10

Therefore, correct answer is C. (x+1)² +(y+2)² =10

Step-by-step explanation:

To find the equation of the circle with radius segment AC, we need the center and the radius. The center of the circle is the midpoint of segment AC, which can be calculated as the average of the coordinates of A and C. Midpoint: ((-1 + 3)/2, (-2 + 1)/2) = (1, -0.5). The radius is the distance from A to C, which is the square root of the sum of the squared differences in x and y coordinates: √((3 - (-1))² + (1 - (-2))²) = √(16 + 9) = √25 = 5. Therefore, the equation of the circle is (x - 1)² + (y + 0.5)² = 5², which simplifies to (x + 1)² + (y + 2)² = 10.

Understanding how to derive the equation of a circle from its center and radius is fundamental in coordinate geometry. This knowledge is crucial for solving problems related to circles on a coordinate plane.

Therefore, correct answer is C. (x+1)² +(y+2)² =10