Properties of Diameters and Chords of a Circle
📐 Key Definitions
Circle: The set of all points in a plane that are equidistant from a fixed point called the center. The distance from the center to any point on the circle is the radius ($r$).
Chord: A line segment whose both endpoints lie on the circle. A chord connects two points on the circle.
Diameter: A chord that passes through the center of the circle. It is the longest possible chord. The diameter $d = 2r$.
Interactive visualization — available in the full lesson
Interactive diagram: Circle with center O, radius r, diameter AB, and chord CD labeled.
⭐ Property 1: The Diameter Is the Longest Chord
The diameter of a circle is always the longest chord. Every other chord is strictly shorter than the diameter.
If \( d \) is the diameter and \( c \) is the length of any other chord, then:
$$c \leq d = 2r$$
Why? Any chord that does not pass through the center has its endpoints connected by a path that "misses" the center. By the triangle inequality applied to the triangle formed by the center and the chord's endpoints, such a chord must be shorter than the diameter.
Equality holds if and only if the chord passes through the center — that is, the chord itself is a diameter.
Interactive visualization — available in the full lesson
Drag point P around the circle to see how the chord length compares to the diameter.