Concept Check Match each equation in Column I with its graph in Column II. I II47. (x - 3)² + (y - 2)² = 25 A. 48. B. 49. C. 50. D.

Verified step by step guidance

1. Identify the type of equation given in Column I. The equation \((x - 3)^2 + (y - 2)^2 = 25\) is in the form of \((x - h)^2 + (y - k)^2 = r^2\), which represents a circle.
2. Determine the center of the circle from the equation. The center \((h, k)\) is \((3, 2)\).
3. Determine the radius of the circle. The equation \((x - 3)^2 + (y - 2)^2 = 25\) can be rewritten as \((x - 3)^2 + (y - 2)^2 = 5^2\), so the radius \(r\) is 5.
4. Match the equation with the correct graph in Column II. Look for a graph that shows a circle centered at \((3, 2)\) with a radius of 5.
5. Verify the match by checking if the graph in Column II corresponds to the circle's center and radius identified from the equation.

Key Concepts

Circle Equation

The equation (x - h)² + (y - k)² = r² represents a circle with center at the point (h, k) and radius r. In this case, the equation (x - 3)² + (y - 2)² = 25 indicates a circle centered at (3, 2) with a radius of 5, since the square root of 25 is 5.

Graphing Circles

To graph a circle, one must identify its center and radius from the standard form of the equation. The center is plotted first, followed by marking points that are a distance equal to the radius in all directions from the center, creating a circular shape. Understanding this process is crucial for matching the equation to its corresponding graph.

Coordinate Plane

The coordinate plane consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Each point on the plane is defined by an ordered pair (x, y). Familiarity with plotting points and interpreting their positions relative to the axes is essential for accurately matching equations to their graphical representations.