Final answer:
The points (7,1) and (-5,5) both have an equal distance of √50 from the origin (0,0), indicating they lie on the same circle centered at the origin.
Step-by-step explanation:
Examining the definition of a circle can help determine whether the points (7,1) and (-5,5) lie on the same circle with its center at (0,0). The set of all points in a plane that are at a specific distance (referred to as the radius) from a given point (referred to as the center) is known as a circle in mathematics. We compute the distances between the two provided points and compare if they are equal to the origin (0,0) to see if they lie on the same circle.
To do this, we use the distance formula: Distance = √(x² + y²), where (x, y) are the coordinates of the point. For (7,1), the distance is √(7² + 1²) = √(49 + 1) = √50. For (-5,5), the distance is √((-5)² + 5²) = √(25 + 25) = √50. Since both distances are equal, the points (7,1) and (-5,5) do indeed lie on the same circle centered at (0,0).