Introduction to Geometry
so let’s begin our journey in geometry. Let me give you a complete overview of all chapters in geometry in just 2 minutes, and you will understand how easy and interesting it can become. So start with appoint, move the point to get a line, move the line to get a plane, move the plane like this to get a solid, so that is geometry, one way of looking at geometry, so what we are doing is started with a point, you moved the point to get a line, moved line to get a plane like this , which is two dimensional and if I move the plane like this I will get solid which are three dimensional. Now let’s simplify geometry using common language and in simple language, let’s try in building one to many. So I try building one to many in a diagram, I start with a point lot of them together to get a line, two lines to get an angle, three lines I will get a triangle, make this one to many, so two triangles together I will get a quadrilateral keep on adding triangles I will get trapeziums and pentagons and hexagons and if I do this many more times infinite number of triangles I will get a circle like this so circle is nothing but one to many of a triangle, now cut the circle, straighten this out, I will get a line, and that ‘s where we started from. So line can be taken as a circle with infinite radius, circle can be taken as a polygon with infinite lines, line to a circle and everything in between what we do in a geometry, in plane geometry. So it is very simple. So we will start with two lines together, we will call them angles, next chapter will be three lines together triangles, four of them together will be the next, we will call them quadrilaterals. And the next chapter will be infinite number of them put together we call them circles and it sounds very simple, right? It is much easier than what you think. So geometry is going to be its anyway easy but it is going to be much more interesting and you will going to find it, it’s gonna become well below any one’s level very soon. Now let’s understand geometry in a different way, but obviously in an easier way itself, so geometry can also be taken as number patterns visualized in diagrams. So let me help you understand why geometry is number pattern visualized. So if start with basic number pattern which is 1,2,3,4,5 these are natural numbers, if I take some of natural consecutive numbers, the number s which I will get is 1, 1+2 I will take, I will get 3, 1+2+3 I will take I will get 6, 1+2+3+4, I will get 10. So these numbers 1,3,6,10, we will call them actually triangular numbers. Let’s understand why we are calling them triangular numbers. So for that you just visualize 1,3,6,10 how do we visualize? Let me show you, 1 is like this 1, 3 can be taken as one, one ,two, 6 is one, one two, one, two, three like that, so it is is very simple and obvious why we are calling them triangular numbers? Because they look like triangles. Let’s see the addition of two triangular numbers , let’s visualize that. So I take a triangular number 6, which is one one, two, one two, three and previous number which is one and one, two, if I just make it upside down and if I keep it over here, I will get, what I am getting is 3,3,3 which is nothing but 9, so 9 is 3+6 and 9 is square number. so you can easily visualize that. So now I am very sure that when I mentioned that geometry is nothing but number patterns visualized, I am sure you have got an idea now. So geometry is number patterns visualized just to add on that, algebra is number patterns generally using variables, and once you represent them through words and that is what we are call the applications and that is what we mostly learn in science.
In this section, we discuss lines and angles, to start with, if I take point and I move the point freely like this you can see the path followed here, this is called curved line. Now if I take point like this and if moved this point without changing the direction like this I got a straight line. So now for a line, part of line with 2 end points like this, is called a line segment, and now if I take a line with one end point and the other one like this so I am marking an arrow mark over here, this is called a ray. So these are things all of you know, I just revising it . when there are two end points we will call it a line segment, one end point, we will call it a ray.
So let’s define, let’s look at angles next, how do get an angle? Very simple right? Two rays emerging from a point like this you will get an angle. One more way of looking at that is take the first ray, rotate like this so I will get the same angle. So angles are formed like this, very simple right? Two rays are part of it, there will be common point.
Now we will see classification of polygons, now based on number of sides or number of vertices the polygons are named like this, when there are three sides or three vertices like this, this is a triangle, when there are four sides or four vertices, it’s called quadrilateral. When there are five sides like this or five vertices, it’s a pentagon. Now if it is six sides and six vertices it is a hexagon. Seven, it will be a heptagon, 8, it will be an octagon, like this polygons can be named based on sides or the vertices. now next thing we will understand is what are diagonals. A diagonal is a line segment connecting the 2 non consecutive vertices of a polygon. This is easy to understand in a diagram, so just to help you understand, let’s take a quadrilateral ABCD, in this case A and C are 2 non consecutive vertices, so AC is diagonal. B and D are non consecutive so BD is a diagonal. So here, AC and BD are diagonals. In a pentagon, PQRST like this if you join all the nonconsecutive vertices, you will get the diagonals here, in this case, PR,PS,TR,TQ and QS they are the diagonals. Just take a look at it, PR and PS, TQ and TR and QS are diagonals for this polygon now PQRST, so now we also understand more about diagonals as part of this discussion, so diagonals are nothing but joining or connecting 2 nonconsecutive vertices of a polygon.
Next we will understand the difference between convex polygons and concave polygons. So in diagram itself if I just draw convex and concave polygons, a convex polygon can look like this an example can be this, this is hexagon or this quadrilateral these two are convex polygons, we will understand the difference. Now, a concave polygon, I can show you, this is a concave polygon. Or this is a concave polygon. Now in terms of diagonals you want me to explain what are convex and concave polygons, in aconvex polygon like thees two no portions of diagonal will lie in the exterior. So here if you consider the diagonals in this diagram, we will just take it here, all here so these diagonals are in the interior, or no part of it is actually there in the exterior. Exterior is the unshaded region here. Now in the second part of it in concave polygons, the diagonals part of diagonals, portion of diagonals out side also, as you can see in these two. So convex polygons and concave polygons are easy to understand, this part of this chapter, in this discussion, we will only deal with convex polygons. Next we will see regular polygons and irregular polygons. It is very simple, regular polygon is both equiangular and equilateral. That means they will have angles will be equal and sides will be equal. And if it is not like that, then the polygon is an irregular polygon. Just to differentiate the two. Let’s look at a square like this, where angles are equal and sides are equal is a regular polygon. But in a rectangle like this where even though
the angles are equal but the sides are not equal, it’s an irregular polygon. Now in an equilateral triangle like this, where the sides are equal and the angles are equal, it’s a regular polygon. And right angle triangle like this where the sides are not equal is an irregular polygon. Now if you see these two diagrams, this is a regular hexagon because all the sides are equal and all angles are equal and this is not a regular hexagon- this is an irregular polygon. because here all the six sides are not equal, all the six angles are not equal. So this is a regular polygon this is an irregular one. so it is very easy to differentiate, that is when the sides are equal, angles are equal, it is called a regular polygon. Next we will look at quadrilaterals specifically, lot of problems of quadrilaterals, different types of quadrilaterals