lesson
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Imagine an air traffic controller tracking an aircraft approaching an airport boundary.
To verify whether the plane is flying along the perimeter, cutting inside, or drifting outside, the radar calculates one essential measurement: distance.
A circle is defined as the set of all points in a plane that are an equal distance, called the radius (r), from a fixed central point (h,k).
In 1637, French philosopher RenΓ© Descartes connected algebra and geometry on a grid, showing that geometric shapes are simply collections of coordinate pairs.
To find the distance between the center (h,k) and any point (x,y), we draw a right triangle and apply the Pythagorean theorem: a2+b2=c2.
Taking the square root gives the distance formula: d=(xβh)2+(yβk)2β.
πA clean 2D coordinate geometry diagram showing the derivation of the circle distance formula. A circle centered at C(h, k) with a radius line extending to point P(x, y) on the circumference. Draw dashed right-triangle legs showing the horizontal change Ξx = (x - h) in blue and vertical change Ξy = (y - k) in orange, with hypotenuse r in green. Label the right angle. A clean info box at the bottom displays: r = β((x - h)Β² + (y - k)Β²). Palette: white background (#ffffff), dark slate text (#1e2945), soft grid lines (#e2e8f0), border radius 12px.
How do we use this distance formula to test whether specific coordinate pairs sit exactly on a circle?
Proving a Point Lies on a Circle
To prove a point lies on a circle, calculate its distance to the center. If that distance equals the circle's radius r, the point lies on the circle.
Let's test whether the point Q(β1,3) lies on a circle centered at C(2,β1) that is known to pass through P(5,3).
First, find the radius by calculating the distance from center C(2,β1) to the known point P(5,3):
r=(5β2)2+(3β(β1))2β=32+42β=9+16β=25β=5