Ohio Assessments for Educators (OAE) Mathematics Practice Exam

What are the coordinates of the center of a sphere given by the formula (x-a)² + (y-b)² + (z-c)² = r²?

1. (a, b, c)

2. (b, c, a)

3. (c, a, b)

4. (0, 0, 0)

The correct answer is: (a, b, c)

The correct coordinates of the center of a sphere represented by the equation \((x-a)² + (y-b)² + (z-c)² = r²\) are \((a, b, c)\). This formula is in standard form for the equation of a sphere in three-dimensional space, where the terms \((x-a)\), \((y-b)\), and \((z-c)\) indicate the horizontal, vertical, and depth distances from the center of the sphere to any point on its surface, respectively. In this formula, \(a\), \(b\), and \(c\) directly correspond to the coordinates of the center of the sphere along the X, Y, and Z axes. Thus, the center can be clearly identified as the point located at these specific coordinates: \((a, b, c)\). The other sets of coordinates presented do not correspond to the center of the sphere defined by this specific equation, as they rearrange or substitute the terms \(a\), \(b\), and \(c\) in ways that do not reflect their roles in defining the sphere's center.