# Chord of a Circle

### Chord of a Circle in Coordinate Geometry:

#### 1. Circle Equation:

• A circle with center $\left(h,k\right)$ and radius $r$ is represented by the equation $\left(x-h{\right)}^{2}+\left(y-k{\right)}^{2}={r}^{2}$.

#### 2. Chord Equation:

• A chord in a circle can be defined by the coordinates of its endpoints $\left({x}_{1},{y}_{1}\right)$ and $\left({x}_{2},{y}_{2}\right)$ lying on the circle.

#### 3. Chord Length:

• The distance between two points $\left({x}_{1},{y}_{1}\right)$ and $\left({x}_{2},{y}_{2}\right)$ can be found using the distance formula: $d=\sqrt{\left({x}_{2}-{x}_{1}{\right)}^{2}+\left({y}_{2}-{y}_{1}{\right)}^{2}}$
• . This represents the length of the chord.

#### 4. Midpoint of a Chord:

• The midpoint of the chord is the average of the x-coordinates and y-coordinates of the endpoints:
• To find the midpoint $\left({x}_{\text{mid}},{y}_{\text{mid}}\right)$ of a chord with endpoints $\left({x}_{1},{y}_{1}\right)$ and $\left({x}_{2},{y}_{2}\right)$, use the midpoint formula:
${x}_{\text{mid}}=\frac{{x}_{1}+{x}_{2}}{2}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{y}_{\text{mid}}=\frac{{y}_{1}+{y}_{2}}{2}$

#### 5. Perpendicular Bisector of a Chord:

• The line passing through the midpoint of a chord and perpendicular to the chord is the perpendicular bisector of the chord.
• Its equation can be found using the negative reciprocal of the slope of the chord.

#### 6. Intersecting Chords Theorem in Coordinates:

• If two chords intersect within a circle, the product of the segments of one chord is equal to the product of the segments of the other. (AC × CB = EC × CD)

#### 7. Slope of Chords:

• The slope of the chord is calculated using the formula: $m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$.