In basic geometry, which statement best compares a line and a point?

Introduction / Context:
This conceptual question belongs to the foundations of Euclidean geometry. Understanding the basic undefined terms point, line, and plane, and knowing their dimensional properties, is essential for reasoning about shapes, figures, and geometric relationships. The task is to choose the statement that correctly describes and compares a point and a line, using standard geometric definitions.

Given Data / Assumptions:
- A point is considered an exact position or location in space with no length, width, or thickness.
- A line is a straight one-dimensional figure extending without end in both directions, with length but no thickness.
- Standard school-level Euclidean geometry terminology applies.
- We are selecting the statement that best captures the dimensional difference between a point and a line, without introducing incorrect additional claims.

Concept / Approach:
The key concept is geometric dimension. A point is zero-dimensional: it only indicates position, not size. A line is one-dimensional: it has length along one direction but no width or height. We examine each option to see whether it correctly states these facts and avoids incorrect claims, such as saying that a point cannot lie on a plane or that a line has planes on it.

Step-by-Step Solution:
Step 1: Evaluate option A: It says a point has no dimension and a line has one dimension. This aligns exactly with standard geometric definitions (0-dimensional point, 1-dimensional line). Step 2: Evaluate option B: It claims that a point cannot be on any line segment. This is false because any line segment is made up of infinitely many points and any endpoint is itself a point on the segment. Step 3: Evaluate option C: It incorrectly states that a line has several planes located on it. In Euclidean geometry, infinitely many planes can pass through a given line, but lines do not contain planes. Step 4: Evaluate option D: It claims that a point cannot lie on a plane. This is incorrect; a plane is also made up of infinitely many points, so a point can certainly lie on a plane. Step 5: From these evaluations, option A is the only statement that is both accurate and concise in comparing a line and a point.

Verification / Alternative check:
Referencing standard geometry: a point is described as that which has no part, and a line as breadthless length. This again reinforces that a point is 0-dimensional and a line is 1-dimensional. None of the other options match textbook definitions, confirming that option A is correct.

Why Other Options Are Wrong:
Option B: Misrepresents the nature of line segments and points by denying that points belong to segments.
Option C: Confuses the relationship between lines and planes. Lines lie in planes, but do not contain planes.
Option D: Incorrectly excludes a point from lying on a plane, although planes are sets of points.
Option None of these: Not correct because option A is fully accurate and sufficient.

Common Pitfalls:
Students sometimes mix up the hierarchy of geometric objects, thinking lines consist of segments only and forgetting both lines and planes are sets of points. Another confusion is to think that dimension refers to physical thickness, whereas in geometry it refers to independent directions needed to specify a position. Remembering that points (0D), lines (1D), planes (2D), and solids (3D) form a natural progression helps avoid these mistakes.

Final Answer:
The correct comparison is that a point has no dimension and a line has one dimension.