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Basic Geometrical Ideas - Points, Lines, Line Segments, Rays

Grade 6CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Point: A point determines a location in space and has no dimensions (no length, width, or height). Visually, it is represented by a tiny dot and is named using a capital letter like AA, PP, or XX.

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Line Segment: A line segment is the shortest path between two fixed points. Visually, it is represented as a straight line with two dots at the ends, indicating that it has a definite length. It is denoted as AB‾\overline{AB}, where AA and BB are the endpoints.

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Line: When a line segment is extended indefinitely in both directions, it becomes a line. Visually, it is drawn with arrowheads at both ends to indicate it never ends. A line through points AA and BB is denoted as AB↔\overleftrightarrow{AB}. It has no fixed length.

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Ray: A ray is a portion of a line that starts at a fixed point (called the initial point) and goes on endlessly in one direction. Visually, it is represented with one endpoint and one arrowhead, such as OA→\overrightarrow{OA}, where OO is the starting point and the ray passes through AA.

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Intersecting Lines: If two distinct lines have a common point, they are called intersecting lines. Visually, they look like the letter XX. The common point where they meet is known as the point of intersection.

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Parallel Lines: Lines in the same plane that do not meet or intersect, no matter how far they are extended, are called parallel lines. Visually, they look like the opposite edges of a ruler or railway tracks. The distance between them remains constant everywhere. They are denoted as l∥ml \parallel m.

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Collinear Points: If three or more points lie on the same straight line, they are called collinear points. Visually, you can draw one single straight line that passes through all of them.

📐Formulae

Number of lines passing through 11 given point = ∞\infty (Infinite)

Number of lines passing through 22 distinct points = 11 (Unique)

Maximum number of intersection points for nn lines = n(n−1)2\frac{n(n-1)}{2}

Length of segment AB‾\overline{AB} = Distance between coordinates ∣x2−x1∣|x_2 - x_1|

💡Examples

Problem 1:

Given a line with points PP, QQ, and RR marked in order, name all the possible line segments.

Solution:

  1. Identify all pairs of points: (P,Q)(P, Q), (Q,R)(Q, R), and (P,R)(P, R).
  2. Write them using the line segment notation: PQ‾\overline{PQ}, QR‾\overline{QR}, and PR‾\overline{PR}.
  3. Note: PQ‾\overline{PQ} is the same as QP‾\overline{QP}.

Explanation:

A line segment is defined by its two endpoints. By picking any two points from the three available, we can name a unique segment.

Problem 2:

How many lines can pass through two distinct points AA and BB? If a third point CC is added such that it does not lie on the path of AA and BB, how many lines can be formed by joining them in pairs?

Solution:

  1. Through two distinct points AA and BB, exactly 11 line can pass (AB↔\overleftrightarrow{AB}).
  2. If point CC is non-collinear, we join the points in pairs:
  • Line 1: AB↔\overleftrightarrow{AB}
  • Line 2: BC↔\overleftrightarrow{BC}
  • Line 3: AC↔\overleftrightarrow{AC}
  1. Total lines = 33.

Explanation:

A single line is uniquely determined by two points. When three points are not in a straight line (non-collinear), they form the vertices of a triangle, resulting in three distinct lines.