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.