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# Interior and exterior points of a circle

Points and circle: Look the figure above, point O is the center of the circle, d1 is the distance from the center of the circle to point A, d2 is the distance from the center of the circle to point B and d3 is the distance from the center of the circle to point C. The radius of the circle is r.
r is the radius of the circle.
If point A lies interior of the circle, then d1 < r
If point B lies on the circle, then d2 = r
If point C lies exterior of the circle, then d3 > r
Example:
In a xy-coordinate plane, the center of a circle lies on the origin. The radius of the circle is 5. Given the coordinates of points A, B, C, D are A = (2, 3), B = ( 4, 3), C = (3, 4), D = (4, 4) respectively. Find which points are interior or exterior or on the circle?
Solution
Find the distance from the center of the circle to the point.
For point A = (2, 3):
distance d1 = square root of (x2 + y2)
= square root of (22 + 32)
= square root of (4 + 9)
= square root of 13 = 3.6
For point B = (4, 3):
distance d2 = square root of (x2 + y2)
= square root of (42 + 32)
= square root of (16 + 9)
= square root of 25 = 5
For point C = (3, 4):
distance d3 = square root of (x2 + y2)
= square root of (32 + 42)
= square root of (9 + 16)
= square root of 25 = 5
For point D = (4, 4):
distance d4 = square root of (x2 + y2)
= square root of (42 + 42)
= square root of (16 + 16)
= square root of 32 = 5.66
Note: the radius of the circle is 5.
The distance from the center of the circle to the point A is d1 = 3.6 < 5 , so the point A lies interior of the circle.
The distance from the center of the circle to the point B is d2 = 5 , so the point B lies on the circle.
The distance from the center of the circle to the point C is d3 = 5 , so the point C lies on the circle.
The distance from the center of the circle to the point D is d4 = 5.66 > 5 , so the point D lies outside of the circle.