Basic Geometrical Concepts

Basic Geometrical Concepts 1-1 Introduction The term “Geometry” is the English equivalent of the Greek word “Geometron”. ‘Geo’ means Earth and ‘Metron’ means measurement. It is the oldest branch of mathematics. The need of geometry was felt when man felt the need to measure their lands while buying and selling. Geometrical ideas are used in various fields. Architects and engineers use geometry in planning buildings, bridges and roads, navigators use geometry to guide boats, planes and spaceships. Thus, geometry is involved in every field of life. 1-2 Point A point is a mark of position. It has no length, no breadth and no height. It has position and location but no magnitudes. For example: A small dot made by a sharp pencil on a piece of paper. A mark obtained by pricking the finger with a needle. Top of a cone. Something on the top of English alphabets ‘i’ and ‘j’. Corners of any polygon, etc. Here

A, B, C, D, E, F, G and H are points. 1-3 Line The word ‘line’ usually refers to a ‘straight line’. A line has no width. It has length only. It is absolutely straight and can extend indefinitely in both directions. For example 1. The taught of straight thread is a physical example of a part of a line

2. The crease can be obtained by folding a piece of paper and is a good example of part of a line. There are two ways of naming a line: 1. A single small letter l , m, n, p, q, s etc as in figure 1-3. It is a denoting ‘line l ’.

suur suur 2. Two alphabets, say A and B, on a line represent ‘line AB’ written as AB or BA as in figure 1-4.

1-4 Line segment The part of a line with two definite end points is called line segment. Figure 1-5 represents a line segment

For example 1. Side of a square

2. Edge of a ruler

3. Edge of your notebook

Note:

suur 1. A line segment AB is denoted as AB . 2. A line segment has a definite length. 3. A line segment is measure in units of length, i.e., mm, cm, m, inch, foot, yard etc. 4. Important formula: If there are ' m ' points, ' n ' of which are collinear, then the number of lines joining them is given by

m(m − 1) n(n − 1) − +1 2 2 1-5 Intersecting Lines (a) Draw two lines ' l ' and ' m ' on a sheet of paper.

(b) Extend these lines indefinitely in both directions. There can be at the most two possibilities. 1. The lines cross each other. 2. Never cross each other even if extended on either side.

In fig 1-9, (i) and (ii) the lines ' l ' and ' m ' cross each other at P, i.e., they intersect each other at a common point P. Two lines ' l ' and ' m ' are called intersecting lines if they have a common point P. the common point P is called the point of intersection. 1-6 Parallel Lines When two lines do not intersect even when they are extended on either side, the lines are called parallel lines. In figure 1-9 (iii) ' l ' and ' m ' are parallel lines. Some of the examples of parallel lines are tracks of a railway line, opposite sides of a tennis court, bars on a ladder, strips used for fencing etc. NOTE: The symbol of parallel lines ‘II”

1-7 Perpendicular lines When a line stands on another line in such a way that the inclination on either side is equal, the lines are called perpendicular lines. For example: Red-cross sign, adjacent sides of a square, adjacent sides of a rectangle, cross-bars on a ladder, etc.

1-8 Comparison of line segments Comparison of two line segments means to find which of them is longer than the other. Comparison of tow (or more) lines can be done by the following methods. (a) Comparison by observation

By looking at the two lines AB and CD in fig 1-12 (i), one can easily say that AB is longer than CD. It is difficult to say which segment is longer in (ii). We can easily make out that PQ is longer than AB in (iii). Again it is difficult to guess which line segment is longer in (iv) and (v0 as well. (b) Comparison by Ruler Place the ruler with its edge along the segment PQ such that zero mark of the ruler coincides with the point P as shown in the figure 1-13. Read the mark on the ruler which is against the point Q. The reading on the ruler corresponding to the endpoint Q is the length of the line segment PQ.

(c) Comparison by tracing Suppose we want to compare two line segments AB and CD. As shown in figure 114, the tracing of the line segment AB is placed on the line segment CD, with end points C and A coinciding. Since, the other two ends points B and D are not coinciding, we can conclude that

suur suur (i) AB is not equal to CD

suur suur suur (ii) AB is shorter than CD as the end points B of AB is falling short of D, the end suur point of CD . (d) Comparison by divider

To compare the line segments AB and CD given in the figure 1-15, using a divider, place one point of the divider on A and open the other leg of the divider until the other point coincides with B. This measure the length of AB. Now, take the divider suur as it is and place one point of the divider at C and the other point along CD . If it suur exactly lies on D, both the segments are equal. If it falls beyond D, CD is shorter suur suur than AB . If it lies on CD , then AB is shorter than CD. 1-9 Measurement of line segment Measurement of line segment can be done as explained in 1-8(b) and (d) while measuring the length of a line segment using a ruler, the thickness of the ruler may cause difficulties in reading off the marks on it. This can cause errors in measurements. Therefore, using a compass or divider for measuring a length is an accurate method. 1-10 Collinear points Suppose there are three points A, B and C on the page of a notebook as shown in figure 1-16. If a line ' l ' is drawn passing though two points A and B then two possibilities arise:

(i) point C lies on the line ' l ' . (ii) point C does not lie on line ' l ' . If point C lies on the line ' l ' , then points A, B and C lie on the same line and are said to be collinear points. But, if the point C does not lie on line ' l ' , then points A, B and C do not line on the same line and are said to be non-collinear points. We conclude that: Three or more points on the page of a notebook (called a plane) are said to be collinear if they are on the same line. This line is called the line of collinearity. On the contrary, three or more points in a plane are collinear if the line passing through of the points passes through the other points also. 1-11 Concurrent Lines If three or more lines, all pass through a common point, they are called concurrent lines. The common point is called point of concurrence. In figure 1-17, O is the point of concurrence.

It should be noted carefully that we need at least three lines to pass through a common point to be called concurrent lines. If two lines pass through a common point, they are called intersecting lines and not concurrent lines.

In figure 1-18, line l and m are intersecting lines and not concurrent lines. Point O is their point of intersection.

1-12 Ray A straight line extending indefinitely in one direction from a fixed point is called a ray. A ray is shown by a solid dot on one end and an arrow on the other. Figure 1-19 uuur shown as ray OP with initial point O and extending indefinitely in the direction of point P.

It should be carefully noted that in naming the rays, order of letters plays a very uuur uuur uuur uuur important role. In figure 1-20, ray OA is different from ray OB . OA and OB extend in opposite directions, and hence are opposite rays.

1-13 Curves Take a pencil and a paper; put the sharp tip of the pencil on the paper and without lifting it move it aimlessly from one point to the other. The figure traced on the paper is called a curve. Examples of curves:

In figure 1-21, curves (iii) and (iv) are ‘closed curves’ because they begin and end at the same point. Curves (i), (v) and (vi) are ‘open curves’. Now, look at the following closed curves:

In the above closed curves, (i) and (iv) are called ‘simple closed curves’ because they do not cross themselves as in curves (ii), (iii) and (v) 1-14 Interior and exterior of a simple closed curve The boundary line is a tennis court divides it into three parts: Inside the line, on the line and outside the line. You cannot enter inside without crossing the line. In the same way, in a close curve, there are three disjoint parts: 1. Interior (‘inside’) of the curve 2. Boundary (‘on’) of the curve and 3. Exterior (‘outside’) of the curve In figure 1-23, A is in the interior, C is in the exterior and B is on the curve.

The interior of a curve together with its boundary is called its ‘region’.

1-15 Polygons Look at the following figures:

All of them are simple closed figures. Figures (i) and (iii) are made up of line segments only, whereas figures (ii) and (iv) are made up of curves. Figures of the type (i) and (iii) are called polygons. Thus, a polygon is closed figure (curve) formed by the line segments such that no two line segments intersect except at their end points. It may be noted that the following figure is not a polygon because the line segments intersect other than their end points.

The line segments BF and GD intersect at C. Therefore, it is not a polygon. 1-16 Terms related to polygons 1. Sides The line segments used in forming a polygon are called its sides. The sides of the given polygon are AB, BC, CD and DA. Look at the sides AB and BC. AB has an end point B and BC has an end point B. This shows that AB and BC have a common end point B. We call AB and BC adjacent sides of a polygon.

Thus, any two sides of a polygon having a common end pint are called adjacent sides. The other pairs of adjacent sides of the polygon are BC and CD; CD and DA; DA and AB. 2. Vertices The point of intersection of two adjacent sides of a polygon is called its vertex.

(Note: Plural of vertex is vertices) In figure 1-27, sides DA and AB meet at A, hence A is the vertex of the polygon. Similarly, B, C and D are other vertices of the polygon. Observe the vertices B and C. They are the end points of the sides BC. These are called adjacent vertices. The other pair of adjacent vertices of the polygon are C and D; D and A; and A and B. 3. Diagonals In figure 1-28, take any two vertices of the polygon which are not adjacent and join them. The line segment obtained in this way is called a diagonal of the polygon.

Thus, if we join two vertices of a polygon which are not adjacent, the line segment so formed is called a diagonal of the polygon. In figure 1-28, the diagonals of the polygons are AC, AD, AE, BD, BE, BF, CE, CF n(n − 1) and DF. (Number of diagonals = − n , where n is the number of sides). It 2 should be noted that AC and CA is one diagonal, DF and FD is the same diagonal. So we mention them once only. 1-17 Convex polygon A polygon is a convex polygon if the line segment joining any two points inside it lies completely inside the polygon. In figure 1-29 (i), is the convex polygon but (ii) is not as the line segment joining the two points P and Q does not lie completely in it. It is called a concave polygon.

1-18 Regular polygon A polygon is called a regular polygon if all its sides are equal and all its angles are equal.

LIST OF FORMULAS 1. If there are ' m ' points on a plane and no three of them are collinear, the m(m − 1) number of lines passing through them joining in pairs is . 2

2. If there be ' m ' points on a plane, ' n ' of which are collinear, then the number of lines one can have by joining them is

m(m − 1) n(n − 1) − +1 2 2

3. The number of diagonals of a rectangular figure bounded by ' n ' sides is n(n − 1) given by −n 2

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