Geometry is a branch in mathematics that deals with all forms of shapes and other construction. It is more than just a topic to be understood with a given set of formulae and properties. It helps learners to visualize shapes and objects that exist in our real-life. It enables people to understand and deal with objects that can be categorized into different categories like 2D shapes, three-dimensional objects, coordinate planes, lines and angles, lines, vectors, coordinate axis, etc. Let's learn more about geometry in this lesson.
|1.||Introduction to Geometry|
Introduction to Geometry
Geometry (from the Ancient Greek: γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics that is primarily concerned with questions pertaining to:
- the shape of figures
- the size of figures
- the relative position of figures
- the properties of space
You can go ahead and explore all important topics in Geometry by selecting the topics from this list below:
Euclidean Geometry refers to the study of plane and solid figures on the basis of axioms (a statement or proposition) and theorems employed by the Greek mathematician Euclid, who is often referred to as the “Father of Geometry”. There are 5 basic postulates of Euclidean geometry.
Euclid was one of the greatest thinkers of all time. We study Euclidean geometry to understand the fundamentals of geometry. In this section, you will learn about the different concepts of Euclidean geometry like points and llnes, Axioms and postulates, Euclid’s axioms and postulates, geometrical proof, and Euclid’s fifth postulate. Euclid’s geometry is introduced to students in Grade 9.
A circle is a closed figure and has no edges or corners. It is defined as the set of all points in a plane that are equidistant from a given point called the center of the circle. Circles are introduced in Grade 6. In this topic, we cover various topics related to circle starting with the definition of a circle by first understanding - what is a circle? You will learn how to find the ratio of circumference to diameter, how to measure the Area of the Circle, the concept of concentric circles, chords, and diameters, the perpendicular bisector of a chord, and the symmetry of any circle. You can also learn how to construct circles, what are equal and unequal chords, arcs, and subtended angles, concyclic points, cyclic quadrilaterals. We also define tangents by helping you understand - what is a tangent and its related concepts like tangent as a special case of secant, uniqueness of the tangent at a point, tangents from an external point, circles touching each other, alternate segment theorem, and common tangents.
A triangle is a closed figure with three sides and three vertices. Triangles are introduced in Grade 7. Now that you are familiar with the definition of a triangle, let's get you introduced to different concepts related to the triangle such as what is congruence, congruence in triangles, the SAS criterion, is there an SSA criterion, perpendicular bisectors, the ASA criterion, the ASA criterion proof, is there an AAS criterion, what is an isosceles triangle, the SSS criterion, the SSS criterion - proof, the RHS criterion, the RHS criterion - proof.
You will also get to learn more concepts like relative magnitudes of sides and angles, the triangle inequality, distance of a point from a line, angle bisector, triangles - same base, same parallels, pythagoras theorem, triangle areas - basic calculations, heron's formula, basic triangle constructions, advanced triangle constructions, angle sum property, proof of the angle sum property, exterior Angle Theorem, what is similarity, the similarity in triangles, basic proportionality theorem, internal division, angle bisector theorem, AA criterion in triangles, SSS criterion in triangles, SAS criterion in triangles, areas of similar triangles, and the pythagoras theorem.
A line is a set of points that extends in two opposite directions. Both ends of a line can be extended indefinitely. A line has no endpoints. Lines are introduced in Grade 3.
We all know what a line is. It is one of the basic concepts of geometry. But often students mix line with line segments and ray! In this topic, you will learn how to differentiate a line, line segment, and ray. We also cover the concept of intersecting and non-intersecting lines.
When two straight lines intersect at a point, they form an angle. Angles are usually measured in degrees. Angles are introduced in Grade 4. Now that you know what an angle is, let us explore more about it. In this section, we will cover concepts like pairs of angles, transversals and related angles, corresponding angles, interior angles, lines parallel to the same line, essence of geometrical constructions which include how to use a protractor, constructing angle bisectors, constructing perpendicular bisectors, constructing an angle of 90°, constructing an angle of 60°, and constructing perpendicular from point to line.
A quadrilateral is a polygon having 4 sides and 4 vertices or corners. Quadrilaterals are introduced in Grade 7. Did you know? The word Quad means four! And now you also know that a quadrilateral has four edges and four corners. A quadrilateral can be of different shapes, let’s dive in and explore more about quadrilaterals. We will be covering topics such as angle sum property in quadrilaterals, some particular types of quadrilaterals, parallelograms, properties of parallelograms, mid-point theorem, basic area concepts, same base same parallels, parallelograms - same base, same parallels and heron's formula, and quadrilaterals.
Solid shapes are three-dimensional in nature. The three dimensions that are taken into consideration are:
Solid Shapes are introduced in Grade 2.
Solid shapes are different from plain shapes. Solid figures can also be called 3D figures. There are different types of solid figures like a cylinder, cube, sphere, cone, etc, and these figures acquire some space, unlike the 2D figures. In this section, we will learn various concepts about symmetry such as line of symmetry, rotational symmetry, order of symmetry, and mirror symmetry.
Coordinate Geometry is a branch of geometry where the position of any given point on a plane is defined with the help of an ordered pair of numbers, or coordinates. Coordinate Geometry is introduced in Grade 9. It is a fun and interesting concept of mathematics. In coordinate geometry, we use a graph and plot points along the x-axis and y-axis. Let's go ahead and understand the concepts of coordinate geometry which includes topics starting from the basics such as labeling points on a number line, labeling points in a plane, cartesian system, straight-line distance, the distance formula, internal division, external division, area of a triangle, area of a quadrilateral, collinearity, lines parallel to the axes, slope-intercept form of a line, point-slope form, two-point form, general equation of a line, the intersection of two lines, circles centered at the origin, circles with arbitrary centers, and general equations of a circle.
In the above example, point A is defined as (4, 3) Point B is defined as (-3, 1).
An object that has both magnitude and direction is referred to as a vector. Geometrically, it can be visualized as a directed line segment: the length of the line segment becomes the magnitude of the vector and the arrow indicates the direction of the vector.
Vectors are introduced in Grade 10. Now that you are familiar with what a vector is, you can visualize the direction and magnitude of a vector easily. Vector quantities are often represented by scaled vector diagrams. There is a lot more to learn about vectors. We will cover topics such as vector addition as net effect, adding two vectors in general, subtracting two vectors, the triangle inequality, multiplication of a vector by a scalar, the i-j system, resolving a vector into components, and handling vectors specified in the i-j form.
Solved Examples on Geometry
Example 1: Carlos will begin attending classes for the sixth grade from next week. He suddenly realized that he doesn't have either a notebook or a pen. He goes to a shop to purchase them and realizes that the cost of a notebook is twice more than $5 for a pen. Represent this information using the equation of a straight line.
Let's assume the following, the cost of a pen = $x and the cost of a notebook = $y. According to the given information, the equation of the straight line is y=2x+5
Example 2: James has to find the area of a regular hexagon. If the value of one side is 3 units, what would be the hexagon area?
Applying the formula of area of a regular hexagon, A=(3√3s2)/2. Now, as s = 3 units, A=(3√3×32)/2. So, we get, A=23.382 units2. Therefore, the area of the hexagon is 23.382 square units.
FAQs on Geometry
What is Geometry Divided Into?
Geometry is the branch in mathematics that is further divided into various sub-branches that are given in the list below:
- Euclid’s Geometry
- Two-Dimensional Shapes
- Solid Shapes
- Coordinate Geometry
What are the Basics of Geometry?
The basics of geometry are the proper understanding of points, lines, and planes. It then helps in building all other concepts in geometry that are based on these basic concepts.
Who is the Father of Geometry?
Euclid is the father of Geometry.
What are the 2 Types of Geometry?
Two types of geometry are plane geometry and solid geometry. Plane geometry deals with the two-dimensional shapes and planes (x-axis and y-axis), while solid geometry deals with three-dimensional objects and 3D planes. These are the two types of geometry.
How is Geometry Used Today?
Real-life applications of geometry include construction activities, designing, architecture, in making machines, nanotechnology, etc.
How do you Teach Geometry?
Teaching geometry is not an easy task that can be done with a textbook and drawing some shapes on the board. It requires a range of activities that demand learners' involvement to understand this concept with much more clarity. Some of the teaching-learning activities are listed below:
- Visualization- It involves learning through real-life experiences. We can take learners outside the classroom and help them to observe different shapes of objects, x-axis and y-axis on the floor or any other flat surface, etc.
- Demonstration- It includes bringing some real-life objects to represent a concept in geometry. For example, it is always better to bring a dice or any other object that represents cubical shape to make learners understand the properties of a cube.
- DIY activities- We can ask learners to participate in some Do-It-Yourself activities that help them to work and play around with shapes and other concepts in geometry.
- Introducing Scientific Concepts- After all these activities, we can introduce the names and properties used in geometry using scientific terms. It includes introducing the terms like cartesian plane, polyhedrons, quadrilaterals, etc.