ICSE Class 6 Mathematics Chapter 23 Selina Concise Solutions - Free PDF Download

Selina Concise Maths Class 6 Solutions Chapter 23

Here is the introduction to what you are going to read in this chapter. You can also get pdf of notes from the Vedantu platform Let us look into some of the important topics of the Fundamental Concepts chapter:

Introduction to Geometry

• Basic Terminology

• Definition of point.

• Properties of Points

• Definition of line.

• Properties of Line

• Definition of the line segment.

• Properties of the line segment

• Definition of ray.

• Properties of Ray

• Definition of the plane.

• Properties of the plane 

• Definition of space.

• Definition of angle.

• Properties of Angle

• Types of lines.

• Concurrent Lines 

• Angle of Concurrence

• Types of angles.

• Condition for parallelism.

Let us briefly discuss each topic of ICSE Mathematics Class 6 Solutions Chapter 23.

Geometry

Geometry word is taken from the Greek language. The meaning of Geometry is the " measurements of the earth ". In other words, Geometry is the branch of mathematics that deals with the shapes and sizes of solids. It is the study of the position, shape, size, and other properties of different figures. Geometry deals with the study of lines, points, angles, shapes, and planes. In geometry, terms like point, line, and plane are the fundamental units of geometry.

In geometry, we learn about different shapes and their dimensions. The applications of Geometry are in architecture and designing of different types of objects. So it becomes important to us to learn about such an important branch of mathematics. 

Here is the basic terminology of geometry.

Basic Terminology

Degree 

Degree is the measure of rotation. It is the unit of angle measurement. 

Vertices

A vertex is the point where generally lines meet. Its plural is vertices. For example, a Pentagon ABCDE has five vertices which are A, B, C, D, and E.

Points, lines, and planes are the basics of about every concept of geometry. In geometry, a shape can be two-dimensional or even three-dimensional. For example, a line and a rectangle are examples of two-dimensional geometry while a cube and a sphere are examples of three-dimensional geometry.

Midpoints 

The midpoint is the point on the line segment which divides it into two equal parts. In other words, The distance of both ends of the A-line segment remains the same from the midpoint.

Point

A point is a 0-dimensional mathematical entity that can be represented in n-dimensional space by an n-tuple of n coordinates. A point in Euclidean geometry is a basic principle upon which the rest of the geometry is centered, which means that it cannot be described in terms of previously defined objects. That is, a point is defined solely by the properties that it must satisfy, known as axioms. The geometric points, in particular, have no length, area, volume, or other dimensional attributes. The definition of a point, according to one interpretation, is intended to capture the idea of a specific position in Euclidean space.

In other terms, a point is a geometrical shape that has no shape and size. It is generally represented by a single dot. To denote it, we generally use capital letters.

It has the following properties :

1. It does not have any fixed dimension.

2. It is the fundamental unit of geometry.

3. A point is used to mark a position in Euclidean geometry.

4. The address of a point in a three-dimensional plane is denoted as (x,y,z). Here x denotes x-coordinate, y denotes y coordinate and z denotes z-coordinate.

Line

A line is a one-dimensional figure that has no thickness and can stretch in both directions indefinitely. A line in the plane is frequently defined in analytic geometry as a set of points whose coordinates satisfy a given linear equation.

In other words, a line is a geometrical shape that has only one dimension.  A line has the following properties:

1. It has only one dimension.

2. It has no endpoints and so it can increase to infinity.

3. To define a single line, we are required to have a minimum of two points.i.e.only a single line can pass through two given points.

4. If a line passes more than two given points then it means that they are collinear I.e. in a single line.

5. A line is named AB if it passes through points A and B.

Line Segment

A line segment is a section of a line that is bounded by two distinct endpoints and includes any point on the line between them in geometry. A closed line segment contains both endpoints, while an open line segment does not; a half-open line segment contains only one endpoint. A line segment is often represented in geometry by a line above the symbols for the two endpoints. If the endpoints of the line segment are A and B then it is denoted as \[\overline{AB}\].

Properties of a Line Segment :

1. It is a type of line whose both endpoints are fixed.

2. In comparison to a line, a line segment has a fixed length.

3. It is a type of a line as well as a ray.

4. A line segment is called AB if its two endpoints are A and B respectively.

Ray

A ray is a segment of a line that has a fixed starting point but no endpoint in geometry. It can stretch in one direction forever. A ray can pass through multiple points on its way to infinity. The starting point of the rays is the vertex of the angles. 

The example of a Ray you can observe in your daily life. For instance, a light source emits the light in a straight line and it is also an example of Ray. It emits from a light source and goes to infinity.

Properties of a Ray :

1. It is a type of line.

2. It has one end fixed and its other end has infinite length.

3. One endpoint is free while another endpoint is fixed.

4. A ray is named AB if A is one of the endpoints of the ray and B is a point that is present on ray A.

Surface

A surface is a geometrical shape that has length and breadth but does not have any thickness.

The surface can be of both the types of Plane surfaces and curved surfaces.

The flat surfaces are plain while the surfaces which are not flat are curved.

Plane

In geometry, a plane is a flat, two-dimensional surface that extends indefinitely. The two-dimensional equivalent of a point, a line, and three-dimensional space is a plane. Planes can exist as subspaces of higher-dimensional space, such as one of a room's infinitely extended walls, or they can exist independently, such as in the setting of Euclidean geometry.

The page of your notebook is an example of a plane.

Properties of a plane are :

1. A plane is made up of an infinite number of points taken together.

2. A plane has length and breadth, but no thickness.

3. A plane is a surface that can extend in any of the directions to infinity.

4. When a single straight line has two of its points on a plane then it means that it entirely coincides with the plane.

Curved Surfaces 

Curved surfaces are those surfaces that have no flat surface. These surfaces are generally bent inwards or outwards. For example, the surface of the ball is curved.

Space

In geometry, space is a set with some additional structure. Space is made up of mathematical objects that are viewed as points and the relationships that exist between them.

Space is a fixed area having specified boundaries.

Angle

In Euclidean geometry, an angle is a figure formed by two rays, called the sides of the angle, that share a common endpoint, called the vertex of the angle. Angles created by two rays lie in the plane in which the rays are contained. Angles are often created when two planes collide which are known as Dihedral angles.

Properties of Angles : 

1. The sum of angles on a straight line is 180°.

2. Two angles are equal if they are vertically opposite.

3. Two angles are equal if they are alternate angles.

4. Angle is the unit for the physical measurements of the extent of rotation.

5. Two corresponding angles i.e. the angles on the same side of an interesting line are equal.

Types of Points 

1. Collinear Points 

A group of points is said to be collinear if a single line through them can be drawn. In other words, if all points are in a straight line then they are called Collinear points.

2. Non-Collinear Points

A group of points that are not in a single line is said to be non-Collinear Points. In other words, if a group of points is not collinear, then it is called non-collinear.

Types of Lines

There are four types of lines in geometry:

1. Horizontal Lines

In a plane coordinate system, a horizontal line is a straight line that is mapped from left to right and parallel to the X-axis. In other words, a horizontal line is a straight line that has no intercept on the X-axis but may have an intercept on the Y-axis.

2. Vertical Lines

In a coordinate plane, a vertical line is a line that runs parallel to the y-axis and goes straight up and down.

3. Parallel Lines

Parallel lines in geometry are lines in a plane that do not overlap that is two straight lines in a plane that do not intersect at any point are said to be parallel.

4. Perpendicular Lines

If two lines meet at a right angle, then the two lines are said to be perpendicular.

Based on intersection we can divide lines into two categories :

1. Parallel Lines: Parallel lines are the lines that never intersect each other. This type of line cuts a single line at the same angle.  As they are parallel, so the distance between these two lines always remains the same. For example, the two ends of a road end of a river, etc.

2. Intersecting Lines: These lines intersect each other at a point. When two lines cut each other then the vertically opposite angles become equal. The examples of intersecting Lines are two different roads intersecting each other at any point.

5. Concurrent Lines 

If three or more straight lines pass through a single point then these types of lines are called concurrent lines.

6. Point of Concurrence

The point through which three or more lines pass is called the point of Concurrence.

Types of Angles

There are seven basic angle types based on their degree measurement:

1. Zero Angle

If all of the angel's arms are in the same place, the angle is considered a zero angle.

2. Acute Angle

An acute angle is less than 90 degrees in length.

3. Right Angle

The right angle is exactly 90 degrees.

4. Obtuse Angle

The measurement of an obtuse angle is greater than 90 degrees but less than 180 degrees.

5. Straight Angle

A 180-degree angle is referred to as a straight angle.

6. Reflex Angle

An angle that is greater than 180 degrees but less than 360 degrees is known as a reflex angle.

7. Complete Angle

360 degrees equals a complete angle.

The Slope of a Line 

The slope of a line is the difference in y-coordinate concerning its x-coordinate.

The slope of a line = \[\frac{∆y}{∆x}\]

Where ∆y is the change in y-coordinate whereas  ∆x is the change in x-coordinate.

 It is generally denoted by m.

The slope of a line is also equal to the angle which it makes with the x-axis. 

m = tan x = \[\frac{∆y}{∆x}\]

Where the x is depicting the angle which it makes with the x-axis.

Condition for Parallelism

The two lines are said to be parallel when they are inclined at the same angle θ to the positive x-axis direction and their slopes are the same.

For example, if the two lines have the slopes as m1 and m2, then they are said to be parallel if m1 = m2. 

Condition for Perpendicular Lines 

Two lines are said to be perpendicular to each other if the angle at which they are inclined has a difference of 90°. 

If one of the lines has slope m1 and the other one m2, then they are said to be perpendicular lines if \[m_1= \frac{-1}{m_2}\] 

Plane Closed Figure 

A plane Closed Figure is generally any of the plane shapes which is closed. In other words, it should have a fixed boundary and area. The open figure has no boundaries. Some examples of plane Closed Figures are circles, triangles, rectangles, squares, etc.

Triangle 

The closed figure bounded by three line segments is called a triangle. A triangle is named ABC if its three vertices are A, B, and C.

We can also divide triangles into three categories 

1. Equilateral Triangle 

Equilateral Triangles are those triangles of which all the sides are equal. Every angle of a triangle will be 60°. 

2. Isosceles Triangle 

Isosceles Triangles are those triangles that have two or three sides equal. So, there also the opposite angles of equal sides become equal.

Hence the isosceles Triangle has two equal sides and angles.

3. Scalene Triangle

Unlike equilateral and isosceles Triangles, the scalene Triangle has all of their side different from each other.

Drawing a scalene Triangle. We should keep in mind that no sum of two sides of any of the triangles can be less than of the third side.

Rectangle

A rectangle is a type of quadrilateral. It has the following properties:

1. Opposite sides are equal

2. All angles are right angles i.e. of 90°

3. Diagonals of a rectangle bisect each other.

4. Diagonals intersect each other at right angles.

Square

A square is a type of rectangle. It has the following properties:

1. All sides are equal.

2. All angles are right angles i.e. of 90°

3. Diagonals of a rectangle bisect each other.

4. Diagonals intersect each other at right angles.

Circle 

The circle is a closed shape on which the distance of every point from its center remains the same.

Properties of circles are as follows :

1. The distance of every point from the center is equal.

2. The Center is the midpoint of the diameter of circles.

3. The circles drawn from a common center are called concentric circles.

Centre of the Circle 

The Center is the midpoint of the circle. It is the point from which a circle is drawn by putting a compass. The distance of every point from the center of the circle remains the same.

Radius of Circle

The distance from any of the points on the circle to the center of the circle is called the radius of the circle. The radius of a circle remains the same at every point on the circle.

Diameter of Circle

The diameter of a circle is the line that meets two points on the circle and passes through the center of the circle. Infinite no. of diameters can be drawn for a circle. But all of them are equal in length.

Chord of Circle 

The chord of a circle is the line that meets two points on the circle. A diameter is also a type of chord, which passes through the center of a circle.

Perpendicular Line of a Line Segment

The perpendicular line drawn from any of the points on the line is called the perpendicular line of a line segment. By measuring one of the angles as 90° we can easily draw a perpendicular line of a line segment.

Perpendicular Bisector of a Line Segment 

The Perpendicular line drawn through the midpoint of the line segment is called the perpendicular Bisector of a line segment. To draw a perpendicular Bisector, we have to draw a perpendicular through the midpoint of the line segment.

Properties of a perpendicular Bisector of a line segment 

1. Any line segment can only have one perpendicular Bisector maximum.

2. It passes through the midpoint of the line segment.

3. It is perpendicular to the given line segment.

Conclusion

The Selina Concise Mathematics Class 6 Solutions Chapter 23 is developed to provide a comprehensive learning experience for the students. The solutions to the fundamental concepts are carefully designed by the experts by explaining all the important concepts in a detailed manner to make the students understand the concepts without any doubts.

Students can download the free PDF of ICSE Mathematics Class 6 Solutions Chapter 23 on the Vedantu platform. Students can also download the free PDF of different chapters of Class 6 solutions of the ICSE board from the Vedantu platform.