A point is considered to be on the circumference of a circle if the distance from that point to the center of the circle is exactly equal to the radius of the circle. This is a fundamental concept in geometry that defines the relationship between a point and a circle.
To determine this, you can follow these steps:
- Identify the Center and Radius: First, you need to know the coordinates of the center of the circle and the radius. The center is often denoted as (h, k) in a Cartesian coordinate system, and the radius is denoted as r.
- Use the Distance Formula: The distance (d) between the center of the circle (h, k) and the point (x, y) can be calculated using the Distance Formula:
d = √((x - h)² + (y - k)²) - Compare Distances: Once you have calculated the distance d, compare this value to the radius r of the circle. If
d = r, then the point lies on the circle. Ifd < r, the point is inside the circle; and ifd > r, the point is outside the circle.
For example, let’s say you have a circle with its center at (2, 3) and a radius of 5. To determine if the point (5, 7) lies on the circle:
- Calculate the distance:
d = √((5 - 2)² + (7 - 3)²) = √(3² + 4²) = √(9 + 16) = √25 = 5 - Since
d = r(5 = 5), the point (5, 7) lies exactly on the circle.
In conclusion, remembering that a point is on a circle if the distance to the center equals the radius allows for efficient identification of point positions relative to circular shapes.