# Chord of a Circle

### Chord of a Circle in Coordinate Geometry:

#### 1. **Circle Equation:**

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

#### 2. **Chord Equation:**

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

#### 3. **Chord Length:**

- The distance between two points $({x}_{1},{y}_{1})$ and $({x}_{2},{y}_{2})$ can be found using the distance formula: $d=\sqrt{({x}_{2}-{x}_{1}{)}^{2}+({y}_{2}-{y}_{1}{)}^{2}}$
- . 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 $({x}_{\text{mid}},{y}_{\text{mid}})$ of a chord with endpoints $({x}_{1},{y}_{1})$ and $({x}_{2},{y}_{2})$, 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: $m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$.