The standard form of a circle's equation is $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ represents the center of the circle and $r$ represents the radius. This equation allows us to easily identify the center and radius of a circle given its equation, and conversely, to write the equation of a circle given its center and radius. Let's practice!
Match the terms with their definitions:
| Term | Definition |
|---|---|
| 1. Radius | A. The set of all points equidistant from a center point. |
| 2. Diameter | B. The point from which all points on a circle are equidistant. |
| 3. Center | C. A line segment passing through the center of a circle, connecting two points on the circle. |
| 4. Circle | D. The distance from the center of a circle to any point on the circle. |
| 5. Chord | E. A line segment connecting two points on a circle. |
Match the correct letters (A-E) to the numbers (1-5).
The standard form equation of a circle is $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ represents the ______ of the circle and $r$ represents the ______. If the center of a circle is at the origin, $(0, 0)$, the equation simplifies to ______. Therefore, to write the equation of a circle, you need to know the coordinates of the ______ and the length of the ______.
Explain how you can determine if an equation represents a circle, and what steps you would take to rewrite it in standard form if it's not already in that form. Provide an example.
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