Introduction

6.1 Introduction:

Geometry is that branch of mathematics which studies the properties of figures such us lines, triangles, quadrilaterals etc.

Ancient Indians used the geometrical properties for construction of fire places called ‘yajnakundas’. They also had profound knowledge of

astronomy, as even today the method of calculation of occurrences of eclipses are precise and date and times are arrived at

many years in advance of their occurrence.

1. Can you divide the below mentioned line segment in to two exact parts without using the scale?

2. What should be the size of underground tank to store petrol/milk and water supplied through tankers?

3. Why do people who prepare sweets, makers cut the ‘burfi’ in the shape of parallelograms rather than rectangulars?

In geometry we find answers/solutions to above mentioned concepts/problems.

6.1.0 Definitions and some basic constructions:

Definition:

‘Angle’ is a figure formed by two rays with a common end point. They are measured in degrees (0⁰ to 360⁰)

The common end point is called ‘vertex’. In the above figure B is a vertex; ABC is an angle and is denoted by ABC. When we measure with protractor, we get

ABC = 50⁰. Note ABC =CBA = 50⁰

No	Type of angle	Measure of angle is	Example in the adjacent figure
1	Acute Angle	Between 0⁰ and 90⁰	AOC
2	Right Angle	= 90⁰
3	Obtuse Angle	Between 90⁰ and 180⁰	smaller COB (clock wise direction)
4	Straight Angle	= 180⁰( Angle on straight line)	AOB
5	Reflex Angle	Between 180⁰ and 360⁰	Larger BOC ( clock wise direction)

6.1.1. Construction of Perpendicular to a line from a point.

Step 1: Draw the line AB of given measurement and C be an external point.

Step 2: With C as center, draw an arc of suitable radius to cut the line AB at 2 points

(If necessary extend the line) X, Y as shown in the figure.

Step 3: With X and Y as centers, draw arcs (of radius more than half of XY) to cut at

point Z on the other side of AB.

Step 4: Join C, Z to intersect the line AB at L

CL is perpendicular to AB.

Note: The procedure for construction of perpendicular at a point on the line is same as above. (C could be a point like L on AB)

Note: Using SSS and SAS postulate (Refer Section 6.4.3), you can prove that CLY =CLX = 90⁰)

6.1.2. Construction of Perpendicular bisector to a line.

Step1 : Draw the line AB of given measurement.

Step2: With A, B as centers, draw arcs of radius more than half the length of AB on both sides of AB. Let these arcs meet at X and Y.

Step3: Join XY to meet AB at L.

(Note that XY bisects AB and XY is perpendicular to AB and L is mid point of AB)

Note: Using SSS and then SAS postulate (Refer Section 6.4.3), you can prove that AL=BL and ALY =YLB = 90⁰)

6.1.3. Construction of Angular bisector.

Step1: construct CAB of given measurement using protractor.

Step2: With A as center, draw arcs of same radius to cut the sides AB and AC at P and Q respectively.

Step3: With P and Q as centers as and with more than half of PQ as radius, draw arcs to intersect at R.

Step4: Join AR, AR is the angular bisector of CAB. (CAR =RAB)

Note: Using SSS postulate (Refer Section 6.4.3), you can prove that CAR =RAB)

Locus is the path traced by a moving point, which moves so as to satisfy the given condition/conditions.

An example is the circumference of a circle. Any point on a circumference is at equal distance from the center of the circle. Thus circumference is locus of circle.

6.1 Summary of learning

No	Points learnt
1	Definitions and basic constructions