# Interior and exterior points of a circle

- Points and circle:
- Look the figure above, point O is the center of the circle, d
_{1} is the distance from the center of the
circle to point A, d_{2} is the distance from the center of the circle to point B and d_{3} 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 d
_{1} < r
- If point B lies on the circle, then d
_{2} = r
- If point C lies exterior of the circle, then d
_{3} > 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 d
_{1} = square root of (x^{2} + y^{2})
- = square root of (2
^{2} + 3^{2})
- = square root of (4 + 9)
- = square root of 13 = 3.6

- For point B = (4, 3):
- distance d
_{2} = square root of (x^{2} + y^{2})
- = square root of (4
^{2} + 3^{2})
- = square root of (16 + 9)
- = square root of 25 = 5

- For point C = (3, 4):
- distance d
_{3} = square root of (x^{2} + y^{2})
- = square root of (3
^{2} + 4^{2})
- = square root of (9 + 16)
- = square root of 25 = 5

- For point D = (4, 4):
- distance d
_{4} = square root of (x^{2} + y^{2})
- = square root of (4
^{2} + 4^{2})
- = 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 d
_{1} = 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 d
_{2} = 5 , so the point B lies on the circle.

- The distance from the center of the circle to the point C is d
_{3} = 5 , so the point C lies on the circle.

- The distance from the center of the circle to the point D is d
_{4} = 5.66 > 5 , so the point D lies outside of the circle.