Maharashtra Board Class 9 Maths Part II Chapter 1 Basic concepts in Geometry PDF Download

Basic Concepts In Geometry

Did you recognise the adjacent picture? It is a picture of pyramids in Egypt, built 3000 years before Christian Era. How the people were able to build such huge structures in so old time? It is not possible to build such huge structures without developed knowledge of Geometry and Engineering.

The word Geometry itself suggests the origin of the subject. It is generated from the Greek words Geo (Earth) and Metria (measuring). So it can be guessed that the subject must have evolved from the need of measuring the Earth, that is land.

Geometry was developed in many nations in different periods and for different constructions. The first Greek mathematician, Thales had gone to Egypt. It is said that he determined height of a pyramid by measuring its shadow and using properties of similar triangles.

Ancient Indians also had deep knowledge of Geometry. In vedic period, people used geometrical properties to build altars. The book shulba-sutra describes how to build different shapes by taking measurements with the help of a string. In course of time, the mathematicians Aaryabhat, Varahamihir, Bramhagupta, Bhaskaracharya and many others have given valuable contribution to the subject of Geometry.

Basic Concepts In Geometry (Point, Line And Plane)

We do not define numbers. Similarly we do not define a point, line and plane also. These are some basic concepts in Geometry. Lines and planes are sets of points. Keep in mind that the word 'line' is used in the sense 'straight line'.

Point, line and plane

Co-ordinates of a points and distance

Betweenness

Conditional statements

Proof

Teacher's Note

Geometry is used in building houses and roads in India. When we see a building, we use geometry to make it straight and strong.

Exam Trick

Remember: A point has no size. A line has no thickness. These are basic ideas we accept without defining them. Just like we don't define numbers as 1, 2, 3.

Points To Remember

A point is a location with no size.
A line is made of infinite points and goes on forever.
A plane is a flat surface with no edges.
We use coordinates to show where a point is on a number line.
Distance is always a positive number.

Co-Ordinates Of Points And Distance

Observe the following number line.

Here, the point D on the number line denotes the number 1. So, it is said that 1 is the co-ordinate of point D. The point B denotes the number -3 on the line. Hence the co-ordinate of point B is -3. Similarly the co-ordinates of point A and E are -5 and 3 respectively.

The point E is 2 unit away from point D. It means the distance between points D and E is 2. Thus, we can find the distance between two points on a number line by counting number of units. The distance between points A and B on the above number line is also 2.

Now let us see how to find distance with the help of co-ordinates of points.

To find the distance between two points, consider their co-ordinates and subtract the smaller co-ordinate from the larger.

The co-ordinates of points D and E are 1 and 3 respectively. We know that 3 > 1.

Therefore, distance between points E and D = 3 - 1 = 2

The distance between points E and D is denoted as d (E,D). This is the same as l(ED), that is, the length of the segment ED.

d (E, D) = 3 - 1 = 2

Therefore l(ED) = 2

d (E, D) = l(ED) = 2

Similarly d (D, E) = 2

d (C, D) = 1 - (-2) = 1 + 2 = 3

Therefore d (C, D) = l(CD) = 3

Similarly d (D, C) = 3

Now let us find d(A,B). The co-ordinate of A is -5 and that of B is -3; -3 > -5

Therefore d (A, B) = -3 - (-5) = -3 + 5 = 2.

From the above examples it is clear that the distance between two distinct points is always a positive number.

Note that, if the two points are not distinct then the distance between them is zero.

The distance between two points is obtained by subtracting the smaller co-ordinate from the larger co-ordinate.

The distance between any two points is a non-negative real number.

Teacher's Note

On a straight road between two villages, the distance between them is calculated this way. If one village is at kilometer 5 and another at kilometer 12, the distance is 12 - 5 = 7 kilometers.

Exam Trick

Remember: Always subtract the smaller coordinate from the larger one. Distance is never negative. Just like the distance between two houses is always positive.

Points To Remember

Co-ordinate tells us where a point is on the number line.
Distance is found by subtracting smaller coordinate from larger coordinate.
Distance is always positive or zero.
The symbol d(A,B) means distance between point A and point B.
d(A,B) is same as l(AB) which means length of segment AB.

Betweenness

If P, Q, R are three distinct collinear points, there are three possibilities.

Point Q is between P and R when Q is in the middle of P and R.

Point R is between P and Q when R is in the middle of P and Q.

Point P is between R and Q when P is in the middle of R and Q.

If d (P, Q) + d (Q, R) = d (P, R) then it is said that point Q is between P and R. The betweeness is shown as P - Q - R.

Solved Examples

Ex (1) On a number line, points A, B and C are such that d (A, B) = 5, d (B,C) = 11 and d (A, C) = 6. Which of the points is between the other two?

Solution: Which of the points A, B and C is between the other two, can be decided as follows.

d(B,C) = 11 . . . . (I)

d(A,B) + d(A,C) = 5 + 6 = 11 . . . . (II)

Therefore d (B, C) = d (A, B) + d (A, C) . . . . [from (I) and (II)]

Point A is between point B and point C.

Ex (2) U, V and A are three cities on a straight road. The distance between U and A is 215 km, between V and A is 140 km and between U and V is 75 km. Which of them is between the other two?

Solution: d (U,A) = 215; d (V,A) = 140; d (U,V) = 75

d (U,V) + d (V,A) = 75 + 140 = 215; d (U,A) = 215

Therefore d (U,A) = d (U,V) + d (V,A)

Therefore The city V is between the cities U and A.

Teacher's Note

Three villages on a straight road can be checked this way. If village V is between U and A, then distance from U to V plus distance from V to A equals distance from U to A.

Exam Trick

Remember: If d(P,Q) + d(Q,R) = d(P,R), then Q is between P and R. This is the magic formula to find which point is in the middle.

Points To Remember

Three collinear points means three points on the same line.
Betweenness means one point is in the middle of two other points.
If d(P,Q) + d(Q,R) = d(P,R), then Q is between P and R.
We write P - Q - R to show Q is between P and R.
If the three points are not on the same line, we cannot find betweenness.

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