What you’ll learn
Define circles geometrically.
Key idea
Circle boundary uses exact equality; the center has zero distance and is interior for a positive radius.
The circle is its exact-distance boundary; interior membership does not imply circumference membership.
Worked example
Classify examples and non-examples for the concept 'Define circles geometrically'.
In a plane, O=(-3,-2); radius r=3. Set S contains ALL points whose distance from O is at most 3. Is S only the circle boundary? Classify with its defining condition.
- The circumference uses distance equal to 3; at-most includes the interior and center distance zero.
- P=(0, -2) has distance 3, Q=(-3, 0) has distance 2, R=(-7, -2) has distance 4; P is on, Q inside and R outside the circle.
Answer: No — S is the filled disk, not only the circle of radius 3.
Quick Check
Classify examples and non-examples for the concept 'Define circles geometrically'.
In a plane, O=(-3,0); radius r=4. Set S contains ALL points whose distance from O is exactly 4. Is S only the circle boundary? Classify with its defining condition.