1. **Problem Statement:** Prove that the diameter is the longest chord of a circle. 2. **Definitions and Formula:** - A chord is a line segment with both endpoints on the circle. - The diameter is a chord that passes through the center of the circle. - Let the radius of the circle be $r$. - The length of any chord can be found using the formula: $$\text{Chord length} = 2\sqrt{r^2 - d^2}$$ where $d$ is the perpendicular distance from the center of the circle to the chord. 3. **Explanation:** - The diameter passes through the center, so for the diameter, $d=0$. - Substitute $d=0$ into the chord length formula: $$\text{Diameter length} = 2\sqrt{r^2 - 0^2} = 2r$$ - For any other chord, $d > 0$, so: $$\text{Chord length} = 2\sqrt{r^2 - d^2} < 2r$$ 4. **Conclusion:** - Since $2r$ is the maximum possible chord length, the diameter is the longest chord of the circle.