As you've noticed in the title of the article, we'll deal only with 2-dimensional(plane) geometry in this article. Before moving ahead with what dimension actually is, its types and all that, let us have a brief understanding about the term 'Geometry'. Geometry is the measurements of the objects around us. The word 'geometry' came from the Greek word 'geometron' which when splitted into two terms 'geo' and 'metron' mean 'earth' and 'measurement' respectively. Let us get a bit practical, whenever we see things around ourselves, we notice many things of different shapes and sizes, we always classify these objects depending on their shapes and sizes, and this we call the geometry of the objects.
Point
The extremely basic thing when you are about study Geometry is 'a point'. A point is a zero-dimensional mathematical object which represnts a position only, in one or more dimension. A point is supposed to have a position but no magnitude and direction as well.
Curves
Curves represent a continuous and smooth collection of points in a plane, regardless of their trajectory. They can take various forms, from simple shapes like circles to more intricate designs such as parabolas and ellipses. Unlike line segments, curves do not necessarily follow a straight path and can bend and twist in different directions.
- Definition: Curves are defined by a continuous series of points.
- Types of Curves: They can be simple (non-self-intersecting) or complex (self-intersecting).
- Examples:
- Circles: A curve where all points are equidistant from a central point.
- Parabolas: A curve described by a quadratic equation, representing the path of a projectile under gravity.
- Ellipses: An elongated circle, defined by two focal points.
- Arbitrary Shapes: Any free-form path that does not adhere to a straight line.
Line Segments
A line segment is a special type of curve that is characterized by its straightness and represents the shortest distance between any two points on it. In 2D geometry, a line segment has two endpoints. While curves can have various trajectories, a line segment is always straight and finite.
- Definition: Line segments are straight paths connecting two fixed points on a curve.
- Properties:
- Straightness: Line segments have no curvature.
- Finite Length: Unlike lines that extend infinitely, line segments have a specific length determined by their endpoints.
- Formation: A line segment is formed by selecting any two fixed points on a curve and connecting them in the straightest possible path.
A line segment is mathematically denoted as \(\ \overline{AB} \). A line segment has two end-points A and B, if A is considered as starting point then B must be the ending point and vice-versa.
Conceptual Relationships
- Curves and Line Segments: Curves are general paths formed by continuous collections of points, and line segments are specific straight portions of these paths. While a curve can bend and take various shapes, a line segment is always straight and connects two specific points on the curve.
- Straight Trajectory: The unique characteristic of a line segment is its straight trajectory, which makes it the shortest distance between its two endpoints, distinguishing it from the potentially curved paths of general curves.
- "All straight lines are curves but all curves are not straight lines."
Ray
A ray can be defined as a line segment which is extended infinitely in the direction of anyone of its end points. A ray is mathematically denoted as \(\ \overrightarrow{AB} \) or \(\ \overrightarrow{BA} \). A ray has only one end-point i.e., a starting point but no ending point.
Line
A line can be defined as a line segment which is extended infinitely in the direction of both of its end points. A line is mathematically denoted as \(\ \overleftrightarrow{AB} \) or \(\ \overleftrightarrow{BA} \). A line has no end-points i.e., neither starting point nor ending point.
Simple figures
The figures in which the lines either having two end-points or being ended to a single point, do not cross themselves between their end-points are called simple figures. Practically, simple figures are those which can drawn completely without lifting the pen or pencil in between the lines.
Closed figures & Open figures
A figure in which inside-outside is clearly separated with a specific boundary, is called a closed figure.
Polygon and its components
A simple, closed (plane) geometrical figure having a number of sides is called a polygon. A polygon can also be defined as a plane figure bounded by edges that are all straight lines.
A polygon has various components - (i) Sides- the line segments which form a polygon are called its sides.
(ii) Vertices- vertices of a polygon are the meeting points of any two sides of that polygon.
(iii) Angles- "See below!"
Classsification of Polygons
A polygon can be classified into two types depending upon its appearance, viz.,
(i) Convex polygon: If all the diagonals of a polygon lie entirely inside the polygon then the polygon is said to be a convex polygon.
(ii) Concave Polygon: A polygon having at least one of its diagonals lying ouside the polygon is said to be concave polygon.
A convex polygon can further be divided into two types depending on their sizes of sides, viz.,
(i) Regular Polygon- All the sides are equal in length, e.g., equilateral triangle, square, rhombus etc.
Angles
A figure which is formed when two or more lines segments intersect. It is nothing but the measure wideness between the two line segments around the vertex they are meeting at. An angle is measured in degree \(\ (^0 )\).
An angle is completely dependent on two line segments which are meeting at a common point, called the vertex. The region within the two line segments and the vertex is called the interior part of the angle and the region left is called the exterior part of the angle.
Some standard values of angles and their names
In order to talk about angle things, you must have an idea of directions, which is nothing but a follow through from a fixed point to another. Geographically, we have four main directions named east, west, north and south. We already know a statement, 'The Sun rises in the east and sets in the west'. On following Sun if you face it in the morning, you must be facing east, hypothetically, if follow the Sun for one day, you will get yourself facing west in the evening. Now consider yourself the centre and draw two line from the point you were standing at, one towards east and the other towards west. You will see a straight line passing through the centre. And the angle formed following you movement is a 'Straight Angle\(\ (=180^0 )\) ' . The double of that much is called 'complete Angle \(\ (=360^0 )\)'. It is equivalent to 1 complete revolution, because if you would've double your movement you might have found youself facing East. This way, Straight angle is also called half revolution. Now, Half of a straight angle \(\ (=90^0 )\) is called a 'Right angle', which is one-fourth of one complete revolution. Let us now have a look at the definitions.
Right angle: If the measure of any angle is equal to \(\ 90^0 \) then the angle is called a right angle.
Obtuse angle: If the measure of any angle is greater than \(\ 90^0 \) and less than \(\ 180^0 \) then the angle is called an obtuse angle.
Straight angle: If the measure of any angle is equal to \(\ 180^0 \) then the angle is called a straight angle.
Reflex angle: If the measure of any angle is greater than \(\ 180^0 \) and less than \(\ (360^0 \) then the angle is called a reflex angle.
Complete angle: If the measure of any angle is equal to \(\ 360^0 \) then the angle is called a complete angle angle.
Classification of Lines
Intersecting Lines
When two lines meet at a certain point then the lines are called intersecting line, and the point at which they meet is called the point of intersection.Perpendicular Lines: If two intersecting lines are such that the measure of angle between them is a right angle \(\ (=90^0 )\), then the lines are called perpendicular to each other.
Parallel Lines
If two lines are such that the distance between them is always same, then the lines are said to be parallel lines.
Triangle and its types
A polygon with three sides is called a triangle. Symbolically, a triangle with vertices A, B and C is written as \(\ \triangle ABC \). A triagle has- (i) 3 vertices, (ii) 3 angles and (iii) 3 sides.
Triangles are classified as 6 types in total, 3 of which are based on their measures of sides and other 3 are based on their measures of angles.
Types of triangle based on sides
(i) Equilateral triangle: A triagle which has all the sides equal in measure (length), is called an equilateral triangle. The \(\ \triangle ABC \) in the figure above is an equilateral triagle because \(\ \overline{AB}=\overline{BC}=\overline{CA} \).(ii) Isosceles triangle: A triagle in which two of its sides equal in measure (length), is called an isosceles triangle. The \(\ \triangle ABC \) in the figure above is an isosceles triagle because \(\ \overline{AB}=\overline{AC}\neq\overline{BC} \).
(iii) Scalene triangle: A triagle which has all the sides of different measures (length), is called a scalene triangle. The \(\ \triangle ABC \) in the figure above is a scalene triagle because \(\ \overline{AB}\neq\overline{BC}\neq\overline{CA} \).
Types of triangle based on angles
(i) Acute angled triangle: A triagle in which each of the angles has value less than \(\ 90^0 \), is called an acute angled triangle. The \(\ \triangle ABC \) in the figure above is an acute angled triagle because \(\ m\angle A < 90^0, m\angle B <90^0, m\angle C < 90^0 \).(ii) Right angled triangle: A triagle in which anyone of its angles is equal to \(\ 90^0 \), is called a right angled triangle. The \(\ \triangle ABC \) in the figure above is a right angled triagle because \(\ m\angle B=90^0 \).
(iii) Obtuse angled triangle: A triagle in which anyone of its angles is greater than \(\ 90^0 \), is called a obtuse angled triangle. The \(\ \triangle ABC \) in the figure above is a obtuse angled triagle because \(\ m\angle B>90^0 \).
Quadrilaterals
A polygon with four sides is called a quadrilateral. In the following figure ABCD is a quarilateral, which has-
4 vertices (and their equivalent angles):
\(\ A = \angle{BAD} = \angle{DAB} \)
\(\ B = \angle{ABC} = \angle{CBA} \)
\(\ C = \angle{BCD} = \angle{DCB} \)
\(\ D = \angle{ADC} = \angle{CDA} \)
Adjacent components:
Sides:
\(\ \overline{AB} \) is adjacent to \(\ \overline{BC} \) and \(\ \overline{AD} \)
\(\ \overline{BC} \) is adjacent to \(\ \overline{AB} \) and \(\ \overline{CD} \)
\(\ \overline{CD} \) is adjacent to \(\ \overline{BC} \) and \(\ \overline{AD} \)
\(\ \overline{AD} \) is adjacent to \(\ \overline{AB} \) and \(\ \overline{DC} \)
Vertices:
A is adjacent to B and D
B is adjacent to A and C
C is adjacent to B and D
D is adjacent to A and C
Angles:
\(\ \angle{BAD} \) is adjacent to \(\ \angle{ABC} \) and \(\ \angle{ADC} \)
\(\ \angle{ABC} \) is adjacent to \(\ \angle{DAB} \) and \(\ \angle{BCD} \)
\(\ \angle{BCD} \) is adjacent to \(\ \angle{ABC} \) and \(\ \angle{ADC} \)
\(\ \angle{CDA} \) is adjacent to \(\ \angle{BAD} \) and \(\ \angle{BCD} \)
Opposite components:
Sides:
\(\ \overline{AB} \leftrightarrow \overline{CD} \)
\(\ \overline{BC} \leftrightarrow \overline{AD} \)
Vertices:
\(\ A \leftrightarrow C \)
\(\ B \leftrightarrow D \)
Angles:
\(\ \angle{ABC} \leftrightarrow \angle{ADC} \)
\(\ \angle{DAB} \leftrightarrow \angle{BCD} \)
N.B.: The bidirectional symbol (\(\ \leftrightarrow \)) here meant to represent vice versa. For example, \(\ A \leftrightarrow C \) is supposed to be read as A is opposite to C and C is opposite to A.
Circles and its components
A circle can be defined as a 2D geometric figure which is a curve consisting of all those points in a plane which are equidistant from a fixed point, called the centre. A cicle has various components, let us have a quick look,
Radius: A radius is a line segment connecting the circle to its centre. The measure of all the radii of a circle are always equal.
Chord: A chord is a line segment connecting any two points on a circle.
Diameter: A chord which passes through the centre of a circle is called the diameter of the circle. The measure of all the diameters of a circle are always equal. It is the longest chord of a circle.
Arc: A continuous portion from the circumference of a circle is called the arc of a circle.
But when two end points of the diameter of a cicle divides the circle into two parts, then no longer the major or minor arcs are formed, the arcs thus formed are equal and they are called the semi circles of the cicle.
Sector: A sector is the portion of the circle which is extended to the centre of the circle. It can also be defined as the part of circular region which is enclosed by an arc on one side and a pair of radii on the other two sides.
Segment: A portion of a circle between its circumference and a chord which is not supposed to be the diameter of the circle.
Soon we well be discussing with 3D shapes, their properties and all in any other article.
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