Chord
GeometryA chord is a line segment with both endpoints on a circle, with the diameter being the longest possible chord.
Formula
\text{chord length} = 2r\sin(\theta/2) \text{ (where } \theta \text{ is the central angle)}
Definition
A chord is a straight line connecting two points on a circle. Unlike a diameter, a chord does not have to go through the center. A diameter is the longest chord because it passes through the center.
Example
If you draw a line across a circle that does not go through the center, you have drawn a chord. A slice cut off a circle by a chord is called a segment. The diameter is a special chord that cuts the circle in half.
Key Insight
The word "chord" is also used in music for notes played together - but in geometry, a chord is the "string" stretched across a circle. Imagine a bow and arrow: the bowstring is a chord of the curved bow.
Definition
A chord is a line segment whose two endpoints lie on a circle. The diameter is the longest chord. Chord length formula: for a circle of radius $r$ and central angle $\theta$: chord $= 2r\sin(\theta/2)$. The perpendicular from the center to a chord bisects the chord (and the arc it subtends).
Example
Circle radius $10$, central angle $60^\circ$: chord $= 2\cdot10\sin(30^\circ) = 20\cdot0.5 = 10$ (equilateral triangle case, since the chord equals the radius when the central angle is $60^\circ$).
Key Insight
Two chords that intersect inside a circle satisfy the intersecting chords theorem: if chords $AB$ and $CD$ intersect at $P$ inside the circle, then $AP \times PB = CP \times PD$. This elegant result can be proved using similar triangles formed by the intersecting chords.
Definition
A chord of a circle of radius $r$ subtending central angle $\theta$ has length $2r\sin(\theta/2)$. The power of a point $P$ with respect to a circle (center $O$, radius $r$) is $d^2 - r^2$, where $d = |OP|$. The intersecting chords theorem (for $P$ inside the circle) and the secant-tangent theorem (for $P$ outside) are both instances of this power: $PA \times PB = |\text{power of } P|$ for any chord or secant through $P$.
Example
Point $P$ inside circle, two chords through $P$: $AP=3$, $PB=4$, $CP=2$. Then $PD = AP \times PB/CP = 12/2 = 6$. Power of $P = -(AP \times PB) = -12$ (negative since $P$ is inside). $|OP|^2 = r^2 - 12$.
Key Insight
The power of a point is an invariant: it depends only on the circle and the point, not on which chord or secant is drawn through the point. This invariance - the same product for all chords - is a beautiful projective property and underlies radical axes (the locus of equal power points with respect to two circles), used in classical construction problems.