Which is precisely defined using the undefined terms point and plane? circle line segment perpendicular lines ray

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UpStudy AI · Accepted Answer

Consider the definition of a circle in Euclidean geometry. A circle is defined as

$$
\{\,P \in \text{plane} : \text{distance}(P, C) = r\,\}
$$

where:
- $$ C $$ is a given point in the plane (called the center),
- $$ r $$ is a fixed positive number (the radius).

This definition uses only the undefined terms “point” (for $$ P $$ and $$ C $$) and “plane” (as the setting in which the points lie).

Now, let us briefly compare with the other options:

- A **line segment** is typically described as the set of points on the line between two given endpoints. Although the endpoints are points, the concept of a “line” is also involved. In many axiomatic systems, the line is another undefined term.
- A **ray** is defined as the set of points on a line starting at one endpoint and extending indefinitely in one direction, again invoking the undefined term “line.”
- **Perpendicular lines** are described in terms of lines meeting at a right angle, hence relying on the undefined concept of a line (as well as the notion of an angle).

Since the circle is entirely defined by specifying a center point, a radius, and a plane, it is the figure that is precisely defined using the undefined terms “point” and “plane.”
A circle is precisely defined using the undefined terms "point" and "plane."

Which is precisely defined using the undefined terms point and plane? circle line segment perpendicular lines ray

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