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Chapter 2: Problem 77

What is a circle? Without using variables, describe how the definition of a circle can be used to obtain a form of its equation.

Short Answer

Expert verified
A circle is defined as a set of all points in a plane that are at a constant distance (known as the radius) from a fixed point (known as the center). It's equation can be formed from this definition by considering all points that satisfy this condition. In a simplified scenario where the circle's center is at the origin of a coordinate plane, the circle's equation can be described as, 'Every point on the circle is a distance 'r' away from the origin.'

Step by step solution

01

Define a Circle

The first thing to know is that a circle is a two-dimensional shape. All its points are equidistant from a fixed point., which we call the 'center' of the circle. That constant distance from the center to any point on the circle is known as the 'radius'.
02

Describe Circle Properties

The properties of a circle play a key role in deriving its equation. Some fundamental properties to remember are: All radii in a circle are equal. Any line segment joining the center to a point on the circle (the radius) will always have a constant length regardless of its position in the circle.
03

General Form of Circle's Equation

The general form of a circle's equation, based on these definitions and properties, can be described as all points (P) in a two-dimensional space that have a constant distance (the radius) from a fixed point (the center, C). This condition must be met for all possible points on the circle, and results in the formation of a circular shape.
04

Simplified Form of Circle's Equation

It's also helpful to consider a simplified form of the circle's equation in which the center of the circle is the origin point of the coordinate plane (0,0). Here the circle's equation is simplified to: 'Every point (P) on the circle is a distance 'r' away from the origin.' This displays the definition of a circle as a set of points equidistant from a single point, when the single point is the origin.

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