Quick Revision Notes & Formula Sheet

Class: 6th Mathematics
Chapter: Basic Geometrical Ideas
Board: MP Board

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1. Introduction to Geometry

• Geometry (ज्यामिति): The word 'Geometry' is derived from the Greek words 'Geo' (meaning Earth) and 'metron' (meaning measurement). It is a branch of mathematics that deals with shapes, sizes, position, and dimensions of things.

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2. Basic Terms and Definitions

• Point (बिंदु):

  • A point is a mark of position. It has no length, breadth, or thickness. It is represented by a fine dot made by a sharp pencil.
  • Example: The tip of a compass, the sharpened end of a pencil.

• Line Segment (रेखाखंड):

  • A line segment is the shortest distance between two points. It has a definite length and has two endpoints.
  • Representation: If endpoints are $A$ and $B$, it is denoted as line segment $AB$ (written as $\overline{AB}$).

• Line (रेखा):

  • A line segment extended endlessly in both directions is called a line. It has no endpoints and no definite length.
  • Representation: Denoted as line $AB$ (written as $\overleftrightarrow{AB}$) or by small letters like $l, m, n$.

• Ray (किरण):

  • A ray is a portion of a line that starts at one point (starting point) and goes endlessly in one direction. It has one endpoint.
  • Representation: Denoted as ray $AB$ (written as $\overrightarrow{AB}$).

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3. Intersecting and Parallel Lines

• Intersecting Lines (प्रतिच्छेद करती रेखाएँ):

  • Two lines that cross each other at a single common point are called intersecting lines. The common point is called the point of intersection.

• Parallel Lines (समांतर रेखाएँ):

  • Two lines in a plane that never meet and always remain at the same distance from each other are called parallel lines.
  • Example: Opposite edges of a ruler, railway tracks.

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4. Curves (वक्र)

• Curve: Anything that is drawn without lifting the pencil from the paper (and is not necessarily a straight line) is a curve.
• Types of Curves:
  1. Open Curve (खुला वक्र): A curve whose ends do not meet (e.g., the letter 'S').
  2. Closed Curve (बंद वक्र): A curve that starts and ends at the same point, enclosing an area (e.g., a circle, a rectangle).

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5. Polygons (बहुभुज)

• Polygon: A simple closed curve made up entirely of line segments.
• Parts of a Polygon:
  • Sides: The line segments forming the polygon.
  • Vertices: The meeting points of a pair of sides.
  • Adjacent Sides: Any two sides with a common endpoint.
  • Diagonals: A line segment joining any two non-adjacent vertices.
• Naming Polygons by Number of Sides:
  • 3 Sides: Triangle (त्रिभुज)
  • 4 Sides: Quadrilateral (चतुर्भुज)
  • 5 Sides: Pentagon (पंचभुज)
  • 6 Sides: Hexagon (षट्भुज)

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6. Angles (कोण)

• Angle: An angle is made up of two rays starting from a common endpoint.
• Parts of an Angle:
  • Vertex (शीर्ष): The common endpoint.
  • Arms / Sides (भुजाएँ): The two rays forming the angle.
• Representation: An angle formed by rays $OA$ and $OB$ is written as $\angle AOB$ or $\angle O$.

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7. Triangles (त्रिभुज)

• Triangle: A three-sided polygon. It is the polygon with the least number of sides.
• Components:
  • 3 Sides ($\overline{AB}$, $\overline{BC}$, $\overline{CA}$)
  • 3 Angles ($\angle A$, $\angle B$, $\angle C$)
  • 3 Vertices (Vertices $A$, $B$, $C$)

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8. Quadrilaterals (चतुर्भुज)

• Quadrilateral: A four-sided polygon.
• Components:
  • 4 Sides, 4 Angles, 4 Vertices, and 2 Diagonals.
• Opposite Sides: Sides that do not have a common endpoint (e.g., $\overline{AB}$ and $\overline{CD}$).
• Adjacent Sides: Sides that share a common endpoint (e.g., $\overline{AB}$ and $\overline{BC}$).

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9. Circles (वृत्त)

• Circle: A closed curve whose every point is at an equal distance from a fixed point inside it.
• Key Terms:
  • Center (केंद्र): The fixed point inside the circle from which all points on the boundary are equidistant.
  • Radius (त्रिज्या): A line segment connecting the center to any point on the circle ($\text{Radius } = r$).
  • Diameter (व्यास): A line segment passing through the center and connecting two points on the circle ($\text{Diameter } d = 2 \times r$). It is the longest chord.
  • Chord (जीव्हा): A line segment joining any two points on the circle.
  • Arc (चाप): A piece of the circle's boundary.
  • Circumference (परिधि): The total length of the boundary of a circle.
  • Interior & Exterior (अभ्यंतर और बहिर्भाग): The region inside the circle and outside the circle respectively.
  • Sector (त्रिज्यखंड): The region enclosed by an arc and two radii.
  • Segment (वृत्तखंड): The region enclosed by a chord and an arc.