Ch. 04. Basic Geometrical Ideas
लेखक
ॐ जितेन्द्र सिंह तोमर
M.A., B.Ed., DNYS, MASSCOM.
10/4/10/2/2022
Geometry
* Point
* Line
Line segment
Curve Line
Straight Line
Standing line (Vertical Lines)
Sloping line
Sleeping line (Horizontal Line)
* Figure
Interior Of the Figure, On the Figure And Exterior Of the Figure
* Curved Figure
simple curve
close curve
* Polygon
Triangle (Trigon)
Quadrilateral (Quardigon)
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
* Parts of Polygon
Sides, Opposite Sides And Adjacent Sides
Vertex Opposite Vertex And Adjacent Vertex
Angle Opposite Angle And Adjacent Angle
* Diagonal
* Interior, On and Exterior of the ......
Triangle
Quadrilateral
Circle
* Parts Of The Circle
Centre
Radius
Diameter
Circumference
Arc
Chord
Segment
Sector
Chapter 4
Basic Geometrical Ideas
Points to remember:
Point: Point represents a definite location. It is drawn as a dot (.). It is denoted by capital alphabets.
• •
P T
Line Segment: It represents the shortest distance between two points. It has a definite length.
Line: It is a line segment that extends indefinitely in both the directions. It doesn't have definite length.
Line l or PQ or QP
Ray: It is a line segment that extends in only one direction.
Ray ST
Collinear Points: Three or more points are said to be collinear if they lie on a single straight line.
F, G and H are collinear points.
Non-Collinear Points: Three or more points which don't lie on same line are known as non-collinear points.
A, B and C are non-collinear points.
Angle: It is formed when two rays have a common starting point, this common point is called the vertex of the angle and the rays are called the arms of the angle.MN and ML are two arms of the angle ∠LMN.
∠M=∠LMN = ∠NMA
Intersecting Lines: Two distinct lines meeting (or appearing to meet) at a point are called intersecting lines.
Parallel Lines: Two lines in a plane are said to be parallel if they never meet. Bere. All and CD are parallel to each other. It is denoted by AB | CD.
Curve: A curve is a smooth flowing line with no shape changes. In mathematics, a line is also a curve.
Simple Curve: A simple curve is a curve that doesn't cut itself.
For example:
are simple curve.
Closed and Open Curve: A curve is said to be closed if its ends are joined otherwise it is said to be an open curve.
OPEN CURVE CLOSED CURVE
Simple Closed Curve: A closed curve which doesn't intersect itself at any point.
Simple Closed Curve
Polygon: A simple closed curve made up of line segments is called a polygon.
Regular Polygon: A polygon whose all sides and all angles are equal.
Triangle (Trigon): A triangle is a three-sided polygon. Triangle has three vertices, three sides and three angles.
Angles-∠ABC, ∠BCA and ∠CAB
Sides-AB = BA, BC = CB and CA = AC
Vertices- A, B and C
Quadrilateral (Quardigon) : A quadrilateral is a four-sided polygon. It has four sides AB, BC, CD and DA, four vertices A, B, C and D, four angles ∠ABC, ∠BCD, ∠CDA and ∠DAB, two diagonals AC and BD. AB and DC are opposite sides. ∠A and ∠C are opposite angles. AB and BC are adjacent sides.i
Circle: A circle is a path taken a point sach that its distance from a fixed point is always constant. The fixed distance is called the radius and the fixed point is the center of the circle.
Parts of Circle
(1)
Chord: A line segment joining any two points located on the circumference of the circle.
Radius: A line segment joining a point on the circumference of the circle to the centre of the circle.
Diameter: A chord passing through the centre of the circle. It is twice the radius.
Sector: It is the region in the interior of the circle enclosed by an are and a pair of radil.
Segment: A chord of a circle divides the circle into two regions, which are called the segments of the circle.
Questions:
1. How many points are shown in the given figure?
2. How many lines can pass through one given point?
3. How many lines can pass through two given points?
4. Find the number of diagonals in a regular hexagon.
5. Determine the maximum number of points where three lines can intersect.
Chapter 5
Understanding Elementary Shapes
The length of a line segment is the distance between its end points.
An angle is formed by the hands of a clock when they move from one position to another.
A reflex angle is larger than a straight angle (180°) and less than a complete angle (360°).
Two intersecting lines are perpendicular if the angle between them is 90°.
Types of triangles:
(A) Based on sides:
Equilateral triangle:- All sides are equal.
Isosceles triangle:- Any two sides are equal.
Scalene triangle:- Three unequal sides.
AAA Equilateral Triangle
Isosceles Triangle
Scalene Triangle
(B) Based on angles:
Acute angled triangle:- All angles are acute (less than 90°).
Obtuse angled triangle:- One angle is obtuse (greater than 90°).
Right angled triangle:- One angle is right angle (90°).
Acute Angled Triangle
Obtuse Angled Triangle
Right Angled Triangle
Note:- A triangle can't have two obtuse or two right angles.
CHAPTER - 10
Mensuration
Perimeter
(i) Perimeter of any closed figure is the distance covered along its boundary.
(ii) Perimeter of ABCDE-AB+BC+CD+DE+EA
(iii) Unit of Perimeter is mm (millimetre), cm (centimetre), m (metre), km (kilometre) etc.
● Area
(i) Area of any closed figure is the surface enclosed by its boundary.
(ii) Area of PQRS-Shaded portion
(ii) Unit of area is mm² (square millimetre), cm' (square centimetre). m² (square metre), km² (square Kilometre) etc.
Regular Polygons
(i) Figures, in which all sides and all angles are equal, are called regular polygons.
(ii) Perimeter of regular closed figure = n × length of side where 'n' is number of sides of the regular closed figure
(iii) Perimeter of Equilateral triangle = 3 × length of side
(iv) Perimeter of square= 4 × length of side
(v) Perimeter of regular pentagon=5 × length of side
(vi) Perimeter of regular hexagon-6 x length of side
Perimeter of Rectangle = 2 x (Length + Breadth)
Area of Rectangle-Length x Breadth
Area of Square - Side x Side
If an area of floor/wall is to be covered by tiles,
then number of tiles = (Area of floor or wall) / (Area of 1 tile)
CHAPTER 14
PRACTICAL GEOMETRY
Circle
* A Circle is the set of all those points in a plane whose distance from a fixed point remains constant
* Fixed point is called Centre of the circle. In figure, O is the centre.
* Fixed distance is called radius.
In figure, radius OA=radius OB=radius OC
* Line segment joining any two points on the circumference of circle is called chord. In figure, EF is a chord > Diameter is the line segment joining two points on the circumference of the circle passing through the Centre. Diameter of a circle is the longest chord. A C is diameter.
* An arc is a part of circumference of a circle. FGE is an arc of circle
* A circle can be constructed by taking the measurement of radius with the help of compass.
Diameter of circle = 2 × radius of circle
Line segment:
* Line segment is a part of line AB or I'.
* A line segment has two end points.
* A line segment has a definite length.
* A line segment of given length can be constructed using a ruler
* A line segment whose length is sum of two lines segments can be constructed
* A line segment equal in measurement of a given line segment can be constructed with the help of compass
Perpendiculari
* Perpendicular lines Two lines are said to be perpendicular if they intersect each other at an angle of 90⁰
In figure, CD ⊥ AB
* Perpendicular to a line through a point on it can be constructed
* Perpendicular to a line through a point outside the line can also be constructed.
Perpendicular bisector:
* It is also known as axis of symmetry of a line segment.
* It divides the line segment into two equal parts.
Angles:
* An angle is a figure formed by two rays with the same initial points.
* line OP and line OQ are forming an angle POQ, where O is the vertex of angle
* Any angle having measure of a multiple of 15⁰ can be constructed using a compass like 15⁰, 30°, 45°, 60°, 75° etc.
Angle bisector:
* It is a ray which divides the angle in two equal parts.
The ray OY is an angle bisector of angle XOZ
∠XOY = ∠YOZ
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