**Chapter Notes and Summary**

• Practical Geometry : Geometry is to draw various shapes with tool like rulers, compasses, dividers, set-squares and protractors. It is study of shapes. Some of these tools are given below :

A tool used to rule straight lines and measure distances. Usually, a ruler is marked in cm along top and inches** **along bottom.A pair of compasses simply known as compass, is a technical drawing instrument that can used for inscribing

**circle or arcs.**

**can be used for mathematics, drafting, navigation and other purposes.**

Compasses

Compasses

**are one of earliest and most basic type of mathematical instrument. In their simplest form, divider**

Dividers

Dividers

**consist of a jointed pair of legs, each with a sharp point. They can be used for geometrical operations such as**

**scribing circles but not for taking off and transferring dimensions.**

**This can be used for drawing vertical lines. 30/60 degree set square also has a 90° degree angle. This set**

**square can be used to draw 30, 60 or 90 degrees angles. Set squares are only accurate if they are used along with**

**a**

**T-square.**

A protractor is a tool used to draw and angles. It is a semi-circle device graduated into 180° degree-parts. measure start from 0° on right hand side and ends with 180° on left hand side and vice-versa.

**•**

**Circle :**

Circle is closed plane figure consisting of all points which are at a constant distance from a fixed point. This

**fixed point is called centre and distance of fixed point from points on circle is called radius of circle.**

**A line segment joining two points of a circle is called chord of circle.**

Chord :

Chord :

**chord passing through centre is called diameter of a circle.**

Diameter :

Diameter :

**Let us draw a circle of radius 3 cm. We need to use our**

Construction of a circle when its radius is known :

Construction of a circle when its radius is known :

**compasses. Here are steps to follow.**

**Open compass for required radius of 3 cm.**

Step I :

Step I :

**Mark a point with a sharp pencil.**

Step II :

Step II :

**Place pointer of compass on O.**

Step III :

Step III :

**Turn compass slowly to draw circle. Be careful to complete movement amount in one instant.**

Step IV :

Step IV :

**In geometry, line segment is a part of line that is bounded by two distinct end points and contains**

Line Segment :

Line Segment :

**every point on line between its end points.**

**Draw a line I. Mark a point A on a line I.**

Construction of a line segment of a given length using ruler and compass : Let us draw a line segment of length 4.7 cm. We can use ruler and mark two points A and B and get (AB) . While marking points A and B, we should look straight down at ruler. Here, are steps to following :

Step I :

Construction of a line segment of a given length using ruler and compass : Let us draw a line segment of length 4.7 cm. We can use ruler and mark two points A and B and get (AB) . While marking points A and B, we should look straight down at ruler. Here, are steps to following :

Step I :

**Place compass pointer on zero mark of ruler. Open it to place pencil point upto 4.7 cm**

Step II :

Step II :

**mark.**

**Taking caution that opening of compasses has not changed, place pointer on A and swing an**

Step III :

Step III :

**arc to cut I at B.**

**AB is a line segment of require length.**

Step IV :

Step IV :

**Given, a line segment AB whose length is not known.**

Constructing a copy of a given line segment using ruler and compass : Let us draw a line segment whose length is equal to that of a given line segment AB To make a copy of a line segment AB , here are steps to follow :

Step I :

Constructing a copy of a given line segment using ruler and compass : Let us draw a line segment whose length is equal to that of a given line segment AB To make a copy of a line segment AB , here are steps to follow :

Step I :

**Fix compass pointer on A and pencil end on B. opening of instrument now gives length of line segment AB .**

Step II :

Step II :

**Draw any line I. Choose a point C on I. Without changing compasses setting, place pointer on C.**

Step III :

Step III :

**Swing an arc that cuts I at a point, say D. Now, line segment CD is a copy of line segment AB .**

Step IV :

Step IV :

**•**

**Perpendicular :**In geometry, property of being perpendicular (perpendicularity) is relationship between

**how lines which meet at right angle (90 degrees). property extends to other related objects. A line is said to**

**be perpendicular to another line if two lines interest at right angle.**

**Here, line segment AB is perpendicular to line segment CD.**

**A line I and a point P are given as shown alongside.**

Drawing perpendicular to a line through a point on it, using ruler and a set-square.

Here, and steps to follow :

Step I :

Drawing perpendicular to a line through a point on it, using ruler and a set-square.

Here, and steps to follow :

Step I :

**Place a ruler with one of its edges along I. Hold this firmly.**

Step II :

Step II :

**Place a set-square with one of its edges along already aligned edge of ruler such that right**

Step III :

Step III :

**angled corner is in contact with ruler.**

**Slide set-square along edge of ruler until its right angled corner coincides with P.**

Step IV :

Step IV :

**Hold set-square firmly in this position. Draw PQ along edge of set-square, where PQ is**

Step V :

Step V :

**perpendicular to I.**

**Let a point P on a line I.**

Drawing perpendicular to a line through a point on it, using ruler and compass.

Here, are steps to follow :

Step I :

Drawing perpendicular to a line through a point on it, using ruler and compass.

Here, are steps to follow :

Step I :

**With P as centre and a convenient radius, construct an arc intersecting line I at two points A and B.**

Step II :

Step II :

**With A and B as centres and a radius greater than AP construct two arcs, which cut each other at Q.**

Step III :

Step III :

**Join PQ . Then, is perpendicular to I. We write PQ I.**

Step VI :

Step VI :

**Let I be given line and P be a point outside I.**

Drawing a perpendicular to a line through a point not on it, using ruler and set-square.

Here, are steps to follow :

Step I :

Drawing a perpendicular to a line through a point not on it, using ruler and set-square.

Here, are steps to follow :

Step I :

**Place a set-square on I such that one arm of its right angle aligns along I.**

Step II :

Step II :

**Place a ruler along edge opposite to right angle of set-square.**

Step III :

Step III :

**Hold ruler fixed. Slide set-square along ruler till point P touches other arm of setsquare.**

Step IV :

Step IV :

**Join PM along edge through P meeting I at M. Now, PM I.**

Step V :

Step V :

**Given a line I and a point P not on it.**

Drawing a perpendicular to a line through a point not on it, using ruler and compass.

Here, one steps to follows :

Step I :

Drawing a perpendicular to a line through a point not on it, using ruler and compass.

Here, one steps to follows :

Step I :

**With P as centre, draw an arc, which intersects line I at two points A and B.**

Step II :

Step II :

**Using same radius and with A and B as centres, construct two arcs that intersect at a point, (say Q)**

Step III :

Step III :

**on other side.**

**Join PQ. Thus, PQ is perpendicular to I.**

Step IV :

Step IV :

**•**

**Perpendicular bisector of a line segment :**A perpendicular bisector of a line segment is a line segment perpendicular

**to and passing through mid point. perpendicular bisector of a line segment can be constructed using a**

**compass by drawing circles centred at and with radius and connecting their two intersections.**

**Method of constructing perpendicular bisector by using ruler and compasses In construction using ruler and compasses, some steps are as follow:**

**Step I :**Draw a line segment AB of any length.

**With A as centre, using compasses, draw a circle. radius of your circle should be more than half length of AB .**

Step II :

Step II :

**With same radius and with B as centre, draw another circle using compasses. Let it cut previous**

Step III :

Step III :

**circle at C and D.**

**Join CD which cuts AB at O, use divider to verify that O is mid-point of AB . Also, Verify that**

Step IV :

Step IV :

**∠COA and ∠COB are right angles. Therefore, CD is perpendicular bisector of AB .**

**•**

**Angle :**In geometry an angle is a figure formed by two rays, called sides of angle, sharing a common end

**point, called vertex of angle.**

**Constructing an angle of a given measure Suppose, we want to make an angle of measure 40°.**

Some methods of constructing an angle are as follow :

Some methods of constructing an angle are as follow :

**Draw AB of any length.**

Some steps are as follows :

Step I :

Some steps are as follows :

Step I :

**Place centre of protractor at A and zero edge along AB .**

Step II :

Step II :

**Start with zero near B. Mark point C at 40°.**

Step III :

Step III :

**Join AC. ∠ BAC is required angle.**

Step IV :

Step IV :

**Suppose an angle (whose measure is not given) is given**

Constructing a copy of an angle of unknown measure.

Constructing a copy of an angle of unknown measure.

**and we want to make a copy of this angle. We will have to use only a straight edge and compasses.**

**Draw a line I and choose a point P on it.**

Let ∠ A be an angle whose measure is not know. Here, we follow steps as given below :

Step I :

Let ∠ A be an angle whose measure is not know. Here, we follow steps as given below :

Step I :

**Place compasses at A and draw an arc to cut rays of ∠ A at B and C.**

Step II :

Step II :

**Use same compasses setting to draw an arc with P as centre, cutting I at Q.**

Step III :

Step III :

**Set your compasses to length BC with same radius.**

Step IV :

Step IV :

**Place compasses pointer at Q and draw arc, to cut arc drawn earlier in R.**

Step V :

Step V :

**Join PR. This gives us ∠ P. It has same measure as ∠ A.**

Step VI :

Step VI :

**—the bisector of an angle is a ray whose end point is a vertex of angle and which divides**

Bisector of an angle

Bisector of an angle

**angle into two equal angles. As shown in figure, ray OC is bisector of angle AOB if and only if angles AOC and BOC have equal measures.**

**For constructing bisector of an angle, we use**

Construction of bisector of an angle using ruler and compasses :

Construction of bisector of an angle using ruler and compasses :

**ruler and compasses in this method.**

**Let an angle, (say ∠ A) given.**

Here, are steps to follow :

Here, are steps to follow :

**With A as centre and using compasses, draw an arc that cuts both rays of ∠ A. Label points of**

Step I :

Step I :

**intersection as B and C.**

**With B as centre, draw (in interior angle of ∠ A) an arc, whose radius is more than half length of**

Step II :

Step II :

**BC.**

**With same radius and with C as centre, draw another arc in interior angle of ∠ A.**

Step III :

Step III :

**There are some elegant and accurate methods to construct some angles of special**

Angles of special measures :

Angles of special measures :

**measures, which do not require use of protractor.**

**Draw a line I and mark a point O on it.**

Some of them are as follow :

Constructing a 60° angle Step I :

Some of them are as follow :

Constructing a 60° angle Step I :

**Place pointer of compasses at O and draw an arc of convenient radius, which cuts line PQ**

Step II :

Step II :

**____**

**at**

**a point, say (A).**

**With pointer at A (as centre), now draw an arc that passes through O.**

Step III :

Step III :

**Let two arcs intersect at B. Join OB. We get ∠ BOA, whose measure is 60°.**

Step IV :

Step IV :

**Construct an angle of 60° as shown in above construction. Now, bisect this angle. Each**

Constructing a 30° angle :

Constructing a 30° angle :

**angle is 30° and verify it by using a protractor.**

**Draw any line PQ**

Constructing a 120° angle : An angle of 120° is nothing but twice of an angle of 60°. Therefore, it can be constructed as follows :

Step I :

Constructing a 120° angle : An angle of 120° is nothing but twice of an angle of 60°. Therefore, it can be constructed as follows :

Step I :

**____**

**and take a point O on it.**

**Place pointer of compasses at O and draw an arc of convenient radius which cuts line at A.**

Step II :

Step II :

**Without disturbing radius on compasses, draw an arc with A as centre which cuts first arc**

Step III :

Step III :

**at B.**

**Again without disturbing radius on compasses and with B as centre, draw an arc which cuts first arc at C.**

Step IV :

Step IV :

**Join OC, ∠ COA is required angle, whose measure is 120°.**

Step V :

Step V :

**— Construct a perpendicular to a line from a point lying on it, as discussed earlier. This**

Constructing a 90° angle

Constructing a 90° angle

**is required 90° angle.**