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.
Compasses can be used for mathematics, drafting, navigation and other purposes.
Dividers are one of earliest and most basic type of mathematical instrument. In their simplest form, divider 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.
Chord : A line segment joining two points of a circle is called chord of circle.
Diameter : chord passing through centre is called diameter of a circle.
Construction of a circle when its radius is known : Let us draw a circle of radius 3 cm. We need to use our compasses. Here are steps to follow.
Step I : Open compass for required radius of 3 cm.
Step II : Mark a point with a sharp pencil.
Step III : Place pointer of compass on O.
Step IV : Turn compass slowly to draw circle. Be careful to complete movement amount in one instant.
Line Segment : In geometry, line segment is a part of line that is bounded by two distinct end points and contains every point on line between its end points.
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 : Draw a line I. Mark a point A on a line I.
Step II : Place compass pointer on zero mark of ruler. Open it to place pencil point upto 4.7 cm mark.
Step III : Taking caution that opening of compasses has not changed, place pointer on A and swing an arc to cut I at B.
Step IV : AB is a line segment of require length.
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 : Given, a line segment AB whose length is not known.
Step II : Fix compass pointer on A and pencil end on B. opening of instrument now gives length of line segment AB .
Step III : Draw any line I. Choose a point C on I. Without changing compasses setting, place pointer on C.
Step IV : Swing an arc that cuts I at a point, say D. Now, line segment CD is a copy of line segment AB .
• 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.
Drawing perpendicular to a line through a point on it, using ruler and a set-square.
Here, and steps to follow :
Step I : A line I and a point P are given as shown alongside.
Step II : Place a ruler with one of its edges along I. Hold this firmly.
Step III : Place a set-square with one of its edges along already aligned edge of ruler such that right angled corner is in contact with ruler.
Step IV : Slide set-square along edge of ruler until its right angled corner coincides with P.
Step V : Hold set-square firmly in this position. Draw PQ along edge of set-square, where PQ is perpendicular to I.
Drawing perpendicular to a line through a point on it, using ruler and compass.
Here, are steps to follow :
Step I : Let a point P on a line I.
Step II : With P as centre and a convenient radius, construct an arc intersecting line I at two points A and B.
Step III : With A and B as centres and a radius greater than AP construct two arcs, which cut each other at Q.
Step VI : Join PQ . Then, is perpendicular to I. We write PQ 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 : Let I be given line and P be a point outside I.
Step II : Place a set-square on I such that one arm of its right angle aligns along I.
Step III : Place a ruler along edge opposite to right angle of set-square.
Step IV : Hold ruler fixed. Slide set-square along ruler till point P touches other arm of setsquare.
Step V : Join PM along edge through P meeting I at M. Now, PM I.
Drawing a perpendicular to a line through a point not on it, using ruler and compass.
Here, one steps to follows :
Step I : Given a line I and a point P not on it.
Step II : With P as centre, draw an arc, which intersects line I at two points A and B.
Step III : Using same radius and with A and B as centres, construct two arcs that intersect at a point, (say Q) on other side.
Step IV : Join PQ. Thus, PQ is perpendicular to I.
• 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.
Step II : With A as centre, using compasses, draw a circle. radius of your circle should be more than half length of AB .
Step III : With same radius and with B as centre, draw another circle using compasses. Let it cut previous circle at C and D.
Step IV : Join CD which cuts AB at O, use divider to verify that O is mid-point of AB . Also, Verify that ∠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.
Some methods of constructing an angle are as follow :
Constructing an angle of a given measure Suppose, we want to make an angle of measure 40°.
Some steps are as follows :
Step I : Draw AB of any length.
Step II : Place centre of protractor at A and zero edge along AB .
Step III : Start with zero near B. Mark point C at 40°.
Step IV : Join AC. ∠ BAC is required angle.
Constructing a copy of an angle of unknown measure. Suppose an angle (whose measure is not given) is given and we want to make a copy of this angle. We will have to use only a straight edge and compasses.
Let ∠ A be an angle whose measure is not know. Here, we follow steps as given below :
Step I : Draw a line I and choose a point P on it.
Step II : Place compasses at A and draw an arc to cut rays of ∠ A at B and C.
Step III : Use same compasses setting to draw an arc with P as centre, cutting I at Q.
Step IV : Set your compasses to length BC with same radius.
Step V : Place compasses pointer at Q and draw arc, to cut arc drawn earlier in R.
Step VI : Join PR. This gives us ∠ P. It has same measure as ∠ A.
Bisector of an angle—the bisector of an angle is a ray whose end point is a vertex of angle and which divides 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.
Construction of bisector of an angle using ruler and compasses : For constructing bisector of an angle, we use ruler and compasses in this method.
Here, are steps to follow : Let an angle, (say ∠ A) given.
Step I : With A as centre and using compasses, draw an arc that cuts both rays of ∠ A. Label points of intersection as B and C.
Step II : With B as centre, draw (in interior angle of ∠ A) an arc, whose radius is more than half length of BC.
Step III : With same radius and with C as centre, draw another arc in interior angle of ∠ A.
Angles of special measures : There are some elegant and accurate methods to construct some angles of special measures, which do not require use of protractor.
Some of them are as follow :
Constructing a 60° angle Step I : Draw a line I and mark a point O on it.
Step II : Place pointer of compasses at O and draw an arc of convenient radius, which cuts line PQ ____ at a point, say (A).
Step III : With pointer at A (as centre), now draw an arc that passes through O.
Step IV : Let two arcs intersect at B. Join OB. We get ∠ BOA, whose measure is 60°.
Constructing a 30° angle : Construct an angle of 60° as shown in above construction. Now, bisect this angle. Each angle is 30° and verify it by using a protractor.
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 : Draw any line PQ ____ and take a point O on it.
Step II : Place pointer of compasses at O and draw an arc of convenient radius which cuts line at A.
Step III : Without disturbing radius on compasses, draw an arc with A as centre which cuts first arc at B.
Step IV : Again without disturbing radius on compasses and with B as centre, draw an arc which cuts first arc at C.
Step V : Join OC, ∠ COA is required angle, whose measure is 120°.
Constructing a 90° angle— Construct a perpendicular to a line from a point lying on it, as discussed earlier. This is required 90° angle.