Geometry

Geometry is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a 🎉geometer. 

Space

Space: space refers to a set of points that form a particular type of structure. 

Plane

A plane is a flat surface that extends infinitely in all directions

Point

Point is an exact position or location in space with no dimensions.

Mathematically, a point is defined as a circle with zero radius.

Since it is not possible its is represented by a very small dot.

A point is usually represented by a capital letter.

In mathematical terms, pont is a cirlce with no radius. It does mean that a very very small circle.

Lines And Its Types

Ray

A Ray is a straight path that stars at a point and extends infinitely in one direction.

Note: A ray is a portion of line starting at a point and extends in one direction endlessly. A ray has only one endpoint (Initial point).

Line or Straight Line

A line is a straight path that extends infinitely in two opposite directions. It can be treated as a combination of two rays starting from the same point but extending in the opposite directions.

Note: A line has no end points.

Line Segment:

A line segment is the part of a line between two points. (Segment means part).

The length of a line segment is the shortest length between two end points.

The line segment has two end points. Note: A line Segment has two endpoints. (both Initial and end points). (Scroll down to continue …).

Parallel Lines

Lines that are always the same distance apart from each other and that never meet are called Parallel lines.

Note: Parallel lines do not have any common point.

Intersecting Lines

 Lines that meet at a point with each other are called intersecting lines.

Intersecting lines meet at a point.

Angle

An angle is made up of two rays starting from a common end point.

An angle leads to three divisions of a region:

On the angle, the interior of the angle and the exterior of the angle.

Curve

Curve is a drawing (straight or non-straight) made without lifting the pencil may be called a curve.

Note: In this sense, a line is also a curve.

Types of Curves

Cureves are two types. They are (i) Simple or Open curve (ii) Closed curve.

Simple Curve

Simple or open curve is a curve that does not cross (intersect) itself.

Closed Curve

Closed curve is a curve that crosses (intersects) itself.

Curves–Concave And Convex

In geometry, a curve is a line or shape that is drawn smoothly and continuously in a plane with bends or turns. 

Mathematicians define a curve as any shape that can be drawn without lifting the pen.

In Mathematics, A curve is a continuous and smooth line that is defined by a mathematical function or parametric equations.

Concave And Convex curves:

Curves are of two types. They are concave curve and convex curve.

Concave Curve:

A curve is concave is a curve that curves inward, resembling a cave.

Examples: 

   – The interior of a circle.

   – The graph of a concave function like y = -x².

Convex Curves:

A curve is convex if it curves outward.

Examples:

   – The exterior of a circle.

   – The graph of a convex function like y = x².

Polygon

Polygon:

Polygon is a closed figure bounded by three or more line segments such that each line segment intersects the other two line segements at exactly two other points (vertices) as shown in the following figures.

Polygons are classified into two types on the basis of interior angles: as

(i) Convex polygon (ii) Concave polygon.

Part of Polygon

(i) Sides Of The Polygon

The line segments of a polygone are called sides of the polygon.

(ii) Adjacent Sides Of Polygon

Adjacent sides of a polygon are thesides of a polygon with a common end point.

(iii) Vertex Of Polygon

Vertext of a polygon is a point at which a pair of sides meet.

(iv) Adjacent Vertices

Adjacent vertices of polygon are the end points of the same side of the polygon.

(v) diagonal

Diagonal of a polygone is a line segment that joins the non-adjacent vertices of the polygon.

(a) Convex Polygon: In this case, each angle is either acute or obtuse (angle < 180 ^o) as shown in the following figures.

Concave Polygon: In this case, any one angle is reflex (angle > 180°) and one diagonal is outside the polygon as shown in the following figures.

On the basis of sides, there are two types of polygons :

(a) Regular Polygon: A convex polygon is called a regular polygon, if all its sides and angles are equal as shown in the following figures.

Each angle of a regular polygon of n-sides =

Important results on polygon :

For a regular polygon of n sides (n > 2).

(b) Irregular Polygon: A polygon in which all the sides are unequal as shown in the following figures,

Triangle :

A simple closed figure bounded by three line segments is called a triangle, it has three sides as AB, BC and AC; three vertices as A, B and C and three interior angles A, ZB and ZCand the sum of all angles is 180°.

i.e., <A + <B + <C = 180°

Triangle

A triangle is a three-sided polygon.

In other terms triangle is a three sided closed figure.

Quadrilateral

A quadrilateral is a four-sided polygon. (It shouldbe named cyclically).

In any

similar relations exist for the other three angles.

Circle And Its Parts

A circle is the path of a point moving at the same distance from a fixed point.

Centre Of Circle

Centre Of Circle is a point that is equidistant from any point on the boundary of the circle.

In other words the centre of the circles is a centre point of the circle.

Radius Of Circle

Radius of circle is the distance between the centre of the circle and any point on the boundary of the circle.

Circumference Of Circle

Circumference of circle is the length of the boundary of the circle.

Chord Of Circle

A chord of a circle is a line segment joining any two points on the circle.

A diameter is a chord passing throughthe Centre of the circle.

Sector Of Circle

A sector is the region in the interior of a circle enclosed by an arc on one side and a pair of radii on the other two sides.

Segment Of Circle

A segment of a circle is a region in the interior of the circle enclosed by an arc and a chord.

The diameter of a circle divides it into two semi-circles.

The diameter of a circle divides it into two semi-circles.