Quick Revision Notes & Formula Sheet

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

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.

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 $\overrightarrow{AB}$).

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.

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).

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)

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 $\angle O$.

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 $A$, $B$, $C$)

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 $\overline{BC}$).

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.

Basic Geometrical Ideas