Eng draw

Department of Industrial
Engineering and management ENGINEERING DRAWING
ENGINEERING DRAWING

Drawing is the fundamental means of communication in engineering. It is the method used to
impart ideas, convey information and specify correctly the shape and size of the object. Thus it
is the language of engineering and the engineer. Without the sound knowledge of drawing, an
engineer is nowhere. It is an international language and is bound by other languages by rules
and conventions. These rules may vary slightly from country to country but the underline basic
principles are common and standard. Engineering/Technical drawing is indispensable today
and shall continue to be of use in the activities of man.
It is the graphic and universal language and has its use of grammar like other systematized
languages. As it is warned against improper use of words in sentences, like that in engineering
drawing, it is also restricted against the improper use of lines, dimensions, letters, and colors in
drawings. Each line on well executed drawing has its own function. Drawings then to those
initiated into the language become the articulate vehicles of expressions between drawing
office and workshop, between text book and the students.
Drawings show how the finished parts, sub assemblies and the final products look like when
completed. These should be kept as simple as possible and be clearly drawn on standard
drawing sheets in order to facilitate their storage, filing and reproduction.
The subject ‘Engineering Drawing’ can not be learnt only by reading the book, the student
must have practice in drawing. With more practice he/she can attain not only the knowledge of
the subject but also speed. To gain proficiency in the subject, the student along with the quality
drawing instruments should pay a lot of attention to accuracy, draftsmanship (i.e, uniformity in
thickness and shade of lines according to their types), nice lettering, and above all the general
neatness in work.
It is very important to keep in mind that when only one drawing or figure is to be drawn on
sheet, it should be drawn in the centre of the working space. For more than one figure on sheet,
the working space should be divided into required blocks and each figure should be drawn in
the centre of the respective blocks. The purpose of doing so is to balance the work on sheet.
DRAWING INSTRUMENTS AND THEIR USES
Drawing instruments are used to prepare drawings easily and accurately. The accuracy of
drawings depends largely upon the quality of instruments. With the quality instruments,
desirable accuracy can be easily attained. It is, therefore, essential to procure instruments of as
superior quality as possible.
The drawing instruments and their materials which every student must possess is given as
under.
1. Drawing board
2. T-square
3. Set-squares
4. Drawing instrument box, containing:
(a) Large size compass with interchangeable pencil and pen legs
(b) Large size divider
(c) Small bow pencil
(d) Small bow pen
(e) Small bow divider
(f) Lengthening bar
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(g) Inking pen

5. Scale
6. Protractor
7. French curves
8. Drawing papers (sheets)
9. Drawing pencils
10. Drawing pins
11. Rubber eraser
12. Duster/Handkerchief/Tissue papers
DRAWING BOARD:- It is a rectangular in shape and is made of ply wood with the edges
of soft and smooth wood about 20 to 25 mm thick. The edges of the board are used as
working edges on which the T-square is made to slide. Therefore the edges of the board be
perfectly straight. In some boards, this edge is grooved throughout its length and a perfectly
straight ebony edge is fitted inside this groove to provide a true and more durable guide for
the T-square to slide on.
Drawing board is made in various sizes. Its selection depends upon the size of drawing
paper to be used. The standard sizes of drawing board are as under.

B0 1000 x 1500 mm

B1 700 x 1000 mm working
edge
B2 500 x 700 mm.

B3 250 x 500 mm.

For the students use, B2 and B3 sizes are more convenient, among which B2 size of the
drawing board is mostly recommended. Whereas large size boards are used in drawing
offices of engineers and engineering firms.
Drawing boards is placed on the table in front of the student, with its working edge on his
left side. It is more convenient if the table top is slopped downwards towards the student.
If such a table is not available, the temporary arrangements for the purpose may be made.
T-SQUARE:- The T-square should be of hard quality wood, celluloid or plastic. It consists
of two parts namely stalk and blade. These both are joined together at right angles to each
other by means of screws and pins. The stalk is placed adjoining the working edge of the
board and is made to slide on it as and when required. The blade lies on the surface of the
board. Its distant edge which is generally bevelled, is used as the working edge and hence,
it should be perfectly straight. Scale is bonded at this edge of T-square. The nearer edge of
the blade is never used. The length of the blade is selected so as to suit the size of the
drawing board
90 WORKING EDGE

BLADE

BLADE
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The T-square is used for drawing horizontal lines. The stalk of T-square is held firmly with
the left hand against the working edge of the board and the line is drawn from left to right.
The pencil should be held slightly inclined in the direction of the line (i.e. to the right)
while the pencil point should be as close as possible to the working edge of the blade.
Horizontal parallel lines are drawn by sliding the stalk to the desired positions.
The working edge of the T-square is also used as base for set-squares to draw vertical,
inclined or mutually parallel lines.
TESTING THE STRAIGHTNESS OF THE WORKING EDGE OF THE T-SQUARE
Mark any two points A and B spaced wide apart and through them, carefully draw a line
with the working edge. Turn the T-square upside down and draw an other line passing
through the same two points. If the edge is defective, the lines may not coincide. The error
can be rectified by planning or sand papering the defective edge.
SET-SQUARES:- The set-squares are made up of wood, tin, celluloid or plastic. Transparent
celluloid or plastic set-squares are most commonly used as they retain their shape and accuracy
for longer time. Two forms of set-squares are in general use. They are triangular in shape with
one corner in each a right angle. The 30 - 60 set-squares of 25 cm length and 45 set-squares
of 20 cm. length are convenient sizes for the drawing purposes.

30

45

25 cm
20 cm

90 45

90 60

Set-squares are used for drawing all straight lines except the horizontal lines which are usually
drawn with the T-square. Vertical lines can be drawn with the T-square and the set-square.
In combination with the T-square, lines at 30 or 60 angle with horizontal or vertical lines can
be drawn with 30 - 60 set square and at 45 angle with 45 set-square. The two set-squares
used simultaneously along with the T-square will produce lines making angles of 15 , 75 ,
105 , etc.

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Parallel straight lines in any position, not very far apart, as well as lines perpendicular to any
line from any given point within or outside it, can also be drawn with the two set-squares.

COMPASS:-Compass is used for drawing circles and arcs of circles. It consists of two legs
hinged together at upper end. A pointed needle is fitted at the lower end of one leg, while the
pencil lead is inserted at the end of the other leg. The lower part of the pencil leg is detachable
and it can be interchanged with a similar piece containing an inking pen. Both the legs are
provided with knee joints. Circles up to about 120 mm. diameter can be drawn with the legs of
the compass kept straight as shown in fig. - A. For drawing larger circles, both the legs should
be bent at the knee joints so that they are perpendicular to the surface of the paper/sheet as
indicated in fig. - B.

Fig.-A Fig. - B

To draw a circle, adjust the opening of the legs of the compass to the required radius. Hold the
compass with the thumb and the first two fingers of the right hand and place the needle point
lightly on the centre, with the help of the left hand. Bring the pencil point down on the paper
and swing the compass about the needle leg with a twist of a thumb and the two fingers in
clock-wise direction until the circle is completed. The compass should be kept slightly inclined
in the direction of its rotation. While drawing concentric circles, beginning should be made
with the smallest circle.
Circles of more than 150 mm. radius are drawn with the aid of lengthening bar. The lower part
of the pencil leg is detached and the lengthening bar is inserted in its place. The detached part
is then fitted at the end of the lengthening bar to increase the length of the pencil leg. For
drawing large circles, it is often necessary to guide the pencil leg with the other hand. For
drawing small circles and arcs of less than 25 mm. radius and particularly when a large number
of small circles of the same diameter are to be drawn, small bow compass is used.
Curves drawn with the compass should be of the same darkness as that of the straight lines. It
is difficult to exert the same amount of pressure on the lead in the compass as on a pencil. It is,
therefore, desirable to use slightly softer variety of lead (about one grade lower) in the compass
than the pencil used for drawing straight lines to maintain uniform darkness in all the lines.

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DIVIDER:- The divider has two legs hinged at the upper end and is provided with steel
points at both the lower ends, but it does has the knee joints. In most of the instrument boxes, a
needle attachment is also provided which can be interchanged with the pencil part of the
compass, thus converting it into a divider.
The dividers are used (a) to divide curved or straight lines into desired number of equal parts,
(b) to transfer dimensions from one part of the drawing to an other part, and (c) to set-off given
distances from the scale to the drawing. They are very convenient for setting-off points at equal
distances around a given point or along a given line.
Small bow divider is adjusted by a nut and is very helpful for marking minute divisions and
large number of short equal distances.

DIVIDER

SCALES:- Scales are made up of wood, steel, celluloid or plastic. Rust less steel scales are
more durable. Scales are flat or rectangular cross-section. 15 cm. long and 2 cm. wide or 30
cm. long and 3 cm. wide flat scales are in common use. They are about one mm. thick. Scales
of greater thickness have their longer edges bevelled. This helps in marking measurements
from the scale to the drawing paper accurately. Generally one of the two longer edges of the
scales are marked with divisions of inches whereas centimeters are marked on its other edge.
Centimeters are further sub divided into millimeters.

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SCALES

The scale is used to transfer true or relative dimensions of an object to the paper. It is placed
with its edge on the line on which measurements are to be marked and looking from exactly
above the required dimension. The marking is done with the fine pencil point. The scale should
never be used as a straight edge for drawing lines.
PROTRACTOR:-

Protractor, commonly called ‘D’ is made up tin, wood or celluloid. Protractors of transparent
celluloid are in common use. They are flat and circular or semi-circular in shape. The most
common type of protractor is semi-circular and about 100 mm. diameter. Its circumferential
edge is graduated to 1 divisions, numbered at every 10 interval and is readable from both the
ends. The diameter of the semi-circle (i.e., straight line 0 -180) is called the base of the
protractor and its centre ‘o’ is marked by a line perpendicular to it (i.e., at 90 ).The protractor
is used to draw or measure such angles which can not be measured by set-squares. A circle can
be divided into any number of equal parts by means of the protractor.

FRENCH CURVES:- French curves are made of wood, plastic or celluloid. They are in
various shapes. Some set-squares also have these curves cut in their middle. French curves are
used for drawing curves that can not be drawn with a compass. Faint free hand curve is first

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drawn through the known points. Longest possible curves exactly coinciding with the freehand
curve are then found out from the French curves. Finally, neat continuous curve is drawn with
the aid of French curve. Care should be taken in order to maintain the steady and uniform
flow/move of the curve drawn through the marked points.

Hint. Any three points are taken in the first attempt to connect. After that advancing for one
point by leaving one is made so that every time minimum three point should come into contact.
For example, initially the curve is drawn through the points (1 + 2 + 3), after that leaving the
first point and taking one coming point, the processing is made consists of three points again as
(2+3+4) and so on.

DRAWING PAPERS/SHEETS:- Drawing sheets are available in many varieties. For
ordinary pencil-drawings, the paper selected should be tough and strong. It should be a quality
paper with smooth surface, uniform in thickness and as white as possible. When the rubber is
used on it to erase the unwanted lines, its fibers should not disintegrate. Whereas thin and
cheep-quality paper may be used for drawings from which tracing are to be prepared. Standard
sizes of drawing papers/sheets are as under.

Designation Trimmed size (mm.) Untrimmed size (mm.)

A0 841 x 1189 880 x 1230

A1 594 x 841 625 x 880

A2 420 x 594 550 x 625

A3 297 x 420 330 x 450

A4 210 x 297 240 x 330

A5 148 x 210 165 x 240

DRAWING PENCILS:- The accuracy and appearance of a drawing depends largely upon the
quality of the pencil used. With cheap and low quality pencil, it is very difficult to draw line of
uniform shade and thickness. The grade of a pencil lead is usually shown by figures and letters
marked on one of its ends. Letters HB denotes the medium grade. The increase in hardness of
the pencil lead is shown by the value of the figure put in front of the letter H, viz. 2H, 3H, 4H
etc. Similarly, this grade becomes softer according to the figures placed in front of the letter B,
viz, 2B, 3B, 4B etc.

Beginning of the drawing should be made with H or 2H pencil using it very lightly, so that the
lines are faint, and unnecessary or extra lines can be erased. The final fair work may be done
with harder pencils (3H and upwards). It should be kept in mind that while drawing final
figures/diagrams, the lead of the pencil must be sharp-pointed. Lines of uniform thickness
and darkness can be more easily drawn with hard-grade pencils. H and HB pencils are more
suitable for lettering, dimensioning and freehand sketching.

Great care should be taken in mending the pencil and sharpening the lead. The lead may be
sharpened to two different forms as (a) conical point, and (b) chisel edge. The conical point
lead is used in sketch work and for lettering, whereas with chisel edge, long thick lines of

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uniform thickness are drawn easily. It should also be remembered that just after using rubber
on sheet to erase the unnecessary material the waste particles be waived-off by the
handkerchief or the tissue paper. Removing waste material by bare hands will spotted the
sheet black.

CHESIL EDGE CONICAL POINT

LINES AND DIMENSIONS
LINES:- The collection of points arranged in proper sequence/manner to connect the two end points is
called a line. Various types of lines used in engineering drawing are described as under.
THICK
A

MEDIUM
B

THIN
C

THIN
D
THICK THIN THICK
E
F THIN

THIN
G

OUTLINES: The lines drawn to represent visible edges and surface boundaries of objects are called
outlines or principal lines. These are continuous and thick lines as indicated at (A).

DASHED LINES: These lines are also called dotted lines when are drawn by dots. These lines show
the interior and/or hidden edges and surfaces of the objects. These are medium thick lines (B) and made
up of short dashes of approximately equal lengths of about 2 mm. and equally spaced of about 1 mm.

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When a dashed line meets or intersects another dashed line or an outline, their point of intersection
should be clearly shown.

CENTRE LINES: Centre lines are drawn to indicate the axes of cylindrical, conical or spherical
objects or details, and also to show the centres of the circles or arcs. These are thin and long chain lines
(C) and composed of alternately long and short dashes spaced approximately 1 mm. apart. The longer
dashes are 6 to 8 times the shorter dashes which are about 1.5 mm. long. Centre lines should extend for
a short distances beyond the outlines to which they refer. For the purpose of dimensioning or to
correlate the views these lines must be extended as required. The point of intersection between the two
centre lines must always be indicated. Locus lines, extreme positions of movable parts and pitch circles
are also shown by this type of line.

DIMENSION LINES:- These lines are continuous thin lines (D). they are terminated at the outer ends
by pointed arrow-heads touching the outlines, extension lines or centre lines.

EXTENSION LINES:- These lines are continuous thin lines (D). they extend by about 3 mm beyond
the dimension lines.
CONSTRUCTION LINES:- These lines are drawn for constructing figures. They are continuous thin
lines (D), and are shown in geometrical drawings only.

HATCHING OR SECTION LLINES:- These lines are drawn to make the section evident. They are
continuous thin lines (D) and are drawn at an angle of 45 to the main outline of the

section. They are uniformly spaced about 1 mm. to 1.5 mm. apart.

LEADER OR POINTER LINES:- Leader line is drawn to connect a note with the feature to which it
applies. It is a continuous thin line (D).

BORDER LINES:- Perfectly rectangular working space is determined by drawing the border lines.
They are continuous thin lines (D).

SHORT-BREAK LINES:- These lines are continuous, thin and wavy (F). they are drawn freehand and
are used to show a short break or irregular boundaries.

LONG-BREAK LINES:- These lines are thin ruled lines with short zig-zags within them (G). They are
drawn to show long breaks.

DIMENSIONING

The technique of dimensioning and few important points useful in dimensioning the geometrical figures
are given below.
1) Dimensions should be placed outside the views except when they are clearer and more easily
readable inside.
2) Dimension lines should not cross each other.
3) As far as possible, dimensions should not be shown between dotted lines.
4) Dimension lines should be placed at least 8 mm. from the outlines and from one an other.
5) Arrow head should be pointed and filled-in partly. It should be about 3 mm. long and its
maximum width should be about 1/3 of its length. The arrow-head is drawn freehand with two
strokes made in the direction of the point.

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6) Dimension figures are usually placed perpendicular to the dimension lines and in such a manner
that they can be read from the bottom or right-hand edge of the drawing sheet. They should be
placed near the middle and above

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LEADER LINE
CENTRE LINE SECTION LINES

CUTTING-PLANE LINE

OUTLINE

HIDDEN LINE

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EXTENSION LINE
DIMENTION LINE

LETTERING

It is an important part of drawing in which writing of letters, dimensions, notes and other
important particulars/instructions about drawing are to be done on the sheet. It is very essential
that accurate and neat drawing may be drawn. A poor lettering may spoil the appearance of
drawing also sometimes impair its usefulness. It is, therefore, necessary that lettering be done
in plain and simple style, freehand and speedily. Use of instruments in lettering take
considerably more time and hence be avoided. Efficiency in the art of lettering can be achieved
by interest, patience & determination and careful & continuous practice. Lettering may be done
by Single-stroke letters or by Gothic letters.

SINGLE-STROKE LETTERS:- These are the simplest form of letters and are usually
employed in most of the engineering drawings. Single-stroke letter means that the thickness of
a line of letter should be such as is obtained in one stroke of the pencil. The horizontal lines of
the letters should be drawn from left to right, and vertical or inclined lines of letters be drawn
from top to bottom. Vertical letters lean to right. The slope of letter line being 67.5 to 75 with
the horizontal. The size of letter is described by the height of a letter. The ratio of height to
width varies but in most cases it is 6:5.

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Lettering is generally done in capital letters. Different sizes of letters are used for different
purposes. The main titles are generally written in 10 mm. to 12 mm. size. Sub titles in 3 – 6
mm. size, while notes and dimensions etc. in 3 – 4 mm. size.

Lettering should be so done as can be read from the front with the main title horizontal. All sub
titles should be placed below but not too close to the respective views. Lettering, except the
dimension figures, should be underlined to make them more prominent.

GOTHIC LETTERS:- If the stems of single-stroke letters be written by more thickness, the
letters will be called gothic. These are mostly used for main titles of ink-drawings. The outlines
of the letters are first drawn with the help of instruments and then filled-in with ink. The
thickness of the stem may vary from 1/5 to 1/10 of the height of the letters.

FILL IN THE BLANKS

1. The edges of the board on which T-square is made to slide is called its working
edge.

2. To prevent warping of the board Battens are cleated at its back.

3. The two parts of the T-square are called Stalk and Blade.

4. The T-square is used for drawing horizontal lines.

5. Angles in multiple of 15 are constructed by the combined use of T-square and Set-
squares.

6. To draw or measure angles, Protractor is used.

7. For drawing large size circles, lengthening bar is attached to the compass.

8. Circles of small radii are drawn by means of a Bow compass.

9. Measurements from the scale to the drawing paper are transferred with the aid of a
Divider.

10. The scale should never be used as a Straight edge for drawing straight lines.

11. Bow divider is used for setting-off short equal distances.

12. For drawing thin lines of equal thickness, the pencil should be sharpened in the
form of Chisel edge.

13. Pencil of Soft grade sharpened in the form of conical point is used for sketching and
lettering.

14. French curves are used for drawing curves which can not be drawn by the compass.

15. To remove unnecessary lines, the Eraser is used.
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16. Uses of T-square, Set-squares, Protractor and Scale are combined in the Drafting
machine.

17. Circles and arcs of circle are drawn by means of a Compass.

18. Inking pen is used for drawing straight lines in ink.

19. Set-squares are used for drawing Vertical, inclined and parallel lines.
____________________________________________________________________________
a) In _________ projection, the ____________ are perpendicular to the ________ of
projection
b) In first-angle projection method,
(i) the _________ comes between the __________ and the ___________.
(ii) the _________ view is always ___________ the ______________ view.

c) In third-angle projection method,
(i) the ____________ comes between the ___________ and the ______________.
(ii) the _______________ view is always ____________ the ___________ view.

LIST OF WORDS

1. above 5. object 9. projectors
2. below 6. orthographic 10. Plane
3. front 7. observer 11. side
4. left 8. right 12. top

ANSWERS

(a) 6,9 & 10 (b) (i) 5,7 & 10 (b) (ii) 12, 2 & 3 (c) (i) 10, 5 & 7 (c) (ii) 12, 1
&3

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LOCUS OF THE POINT (PLURAL IS LOCI)

A locus is a path of point which moves in space. The locus of a point P moving in a plane
about an other point O in such a way that its distance from it is constant, is a circle of a radius
equal to OP as shown in figure (a).

P P
P

O

B
A B A O
(b) (c)
(a)

The locus of a point P moving in a plane in such a way that its distance from a fixed line AB is
constant is a line through P, parallel to the fixed line as indicated in figure (b).

When a fixed line is an arc of a circle, the locus will be another arc drawn through P with the
same centre point as shown in figure (c).

A
B
D
P
A

B
C C D
(d) (e)

The locus of a point equidistant from two fixed points A and B in the same plane, is the
perpendicular bisector of the line joining the two points as shown in figure (d).

The locus of a point equidistant from two fixed non-parallel straight lines AB and CD will be a
straight line bisecting the angle between them as indicated in figure (e).

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CURVES

CONIC SECTION:- The sections obtained by the intersection of a right circular cone by a
plane in different positions relative to the axis of the cone are called conics or conic sections.

When a section plane is inclined to the axis and cuts all the generators on one side of the apex,
the section is called an ellipse.

MINOR DIAMETER F
D
C

MAJOR DIAMETER

A B
ELLIPSE
E
PARABOLIC CURVE

When the section plane is inclined to the axis and is parallel to one of the generators, the
section is called parabola. The parabolic curves are mainly used in arches, bridges, sound &
light reflectors and etc.

When the section plane cuts both the parts of the double cone on one side of the axis, the
section is said to be the hyperbola.

The conic may be defined as the locus of the point movement in a plane in such a way that the
ratio of its distances from a fixed point and a fixed straight line is always constant. The fixed
point is called the focus and the fixed line is called the directrix.

The ratio between the distance of the point from the focus to the distance of the point from the
directrix is called eccentricity. It is always less than 1 for an ellipse, equal to 1 for parabola and
greater than 1 for hyperbola.

The line passing through the focus and perpendicular to the directrix is called the axis. The
point at which the conic cuts its axis is called the vertex.

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CYCLOIDAL CURVES

A curve generated by a point on the circumference of a circle which rolls along a straight line,
is called cycloid.

Cycloidal curves are generated by a fixed point on the circumference of a circle, which rolls
without slipping along a fixed straight line or a circle. These curves are used in the profile of
teeth of gear wheels.

The curve generated by a point on the circumference of a circle, which rolls without slipping
along another circle outside it, is called epicycloid. And when the circle rolls along another
circle inside it, the curve is known as hypocycloid.

A curve generated by a point fixed to a circle, within or outside its circumference, as the circle
rolls along a straight line, is termed as trochoid. When the point is within the circle, the
trochoid is called inferior trochoid, and when the point is outside the circle, it is termed as
superior trochoid .

INVOLUTE:- The involute is a curve traced out by an end of the piece of thread unwounded
from a circle or a polygon, the thread being kept tight. It may also be defined as a curve traced
out by a point in a straight line which rolls without slipping along a circle or a polygon.
Involute of a circle is used teeth profile of gear wheels.

TICK ( ) THE CORRECT ANSWER FROM THOSE GIVEN IN THE BRACKETS.

1. The ratio of the length of the drawing of the object to the actual length of the object is
called ……..
{ (a) resulting fraction (b) representative figure (c) representative fraction}

2. When the drawing is drawn of the same size as that of the object, the scale used is
…………... { (a) diagonal scale (b) full-size scale (c) vernier scale }

3. For drawing of small instruments, watches etc, ……………… scale is always used.
{ (a) reducing (b) full-size (c) enlarging }

4. Drawing of buildings are drawn using ……………….
{ (a) full-size scale (b) reducing scale (c) scale of chords }

5. When measurements are required in three units, ………………… scale is used.
{ (a) diagonal (b) plain (c) comparative }

6. The scale of chord is used to set out or measure …… { (a) chords (b) lines (c) angles}

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FILL IN THE BLANKS

a) When a cone is cut by planes at different angles, the curves of intersection are called
____________.

b) When the plane makes the same angle with the axis as do the generators, the curve is a
__________.

c) When the plane is perpendicular to the axis, the curve is a ____________________.

d) when the plane is parallel to the axis, the curve is a _________________________.

e) when the plane makes an angle with the axis greater than what do the generators, the
curve is a _________________________.

f) A conic is a locus of a point moving in such a way that the ratio of its distance from the
_________ and its distance from the ____________ is always constant. The ratio is
called the _______________. It is _________________ in case of parabola,
________________ in case of hyperbola, and _______________________ in case of
ellipse.

g) In a conic the line passing through the fixed point and perpendicular to the fixed line is
called the ____________________________.

h) The vertex is a point at which the ____________________ cuts the
___________________.

i) The sum of the distances of any point on the ______________ from its two foci is
always the same and equal to the _____________________.

j) The distance of the ends of the ________________ of an ellipse from the
______________ is equal to the half the __________________.

k) In a ____________________ the product of the distances of any point on it from two
fixed lines at right angles to each other is always constant. The fixed lines are called
___________.

l) Curves generated by a fixed point on the circumference of a circle rolling along a fixed
line or circle are called _______________________.

m) The curve generated by a point on the circumference of a circle rolling along another
circle inside it, is called a ________________________.

n) The curve generated by a point on the circumference of a circle rolling along a straight
line, is called a ________________________.

o) The curve generated by a point on the circumference of a circle rolling along another
circle outside it, is called a ________________________.

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p) The curve generated by a point fixed to a circle outside its circumference, as it rolls
along a straight line is called _______________________.

q) The curve generated by a point fixed to a circle inside its circumference as it rolls along
a circle inside it is called ______________________.

r) The curve generated by a point fixed to a circle outside its circumference as it rolls
along a circle outside it is called ______________________.

s) The curve traced out by a point on a straight line which rolls, without slipping, along a
circle or a polygon, is called _________________________.

t) The curve traced out by a point moving in a plane in one direction towards a fixed point
while moving around it, is called a __________________.

u) The line joining any point on the spiral with the pole is called ___________________.

v) In __________________, the ratio of the lengths of consecutive radius vectors
enclosing equal angles is always constant.

LIST OF WORDS

1. Asymptotes 12. Eccentricity 23. Parabola
2. Axis 13. Focus 24. Radius vector
3. Cycloidal 14. Greater than 1 25. Rectangular
4. Conic 15. Hyperbola 26. Smaller than 1
5. Circle 16. Hypocycloid 27. Superior
6. Cycloid 17. Hypotrochoid 28. Spiral
7. Directrix 18. Involute 29. Trochoid
8. Epicycloid 19. Inferior 30. Conics
9. Equal to 1 20. Logarithmic 31. Curves
10. Epitrochoid 21. Minor axis
11. Ellipse 22. Major axis

ANSWERS

(a) 30, (b) 23 (c) 5 (d) 15 (e) 11 (f) 13,7.12,9.14 & 26 (g) 2 (h) 4 &2 (i) 11 & 22
(j) 21, 13 & 22 (k) 25,15 &1 (l) 3 & 31 (m) 16 (n) 6 (o) 8 (p) 27 & 29 (q) 19 & 17
(r) 27 & 10 (s) 18 (t) 28 (u) 24 (v) 20 & 28.

17

ORTHOGRAPHIC PROJECTION

If the straight lines are drawn from the various points on the contour of an object to meet the
plane, the object is said to be projected on that plane. The figure formed by joining in correct
sequence the points at which these lines meet the plane, is called the projection of the object.
The lines from the object to the plane are called projectors.

When the projectors are parallel to each other and also perpendicular to the plane, the
projection so formed is called Orthographic Projection.

FIG. - A PLANE

PROJECTORS FIG. - B
PROJECTION

OBJECT V.P

H
E

W

T RAYS OF SIGHT

Referring fig.-A, assume that a person looks at the block (object) from a infinite distance so
that the rays of sight from his eyes are parallel to one another and perpendicular to the front
surface (F). The shaded view of this block shows its front view in its true shape and projection.

If these rays of sight are extended further to meet perpendicularly a plane (marked V.P.)set up
behind the block, and the points at which they meet the plane are joined improper sequence,
the resulting figure (marked E)will also be exactly similar to the front surface. This figure is
the projection of the block. The lines from the block to the plane are the projectors. As the
projectors are perpendicular to the plane on which the projection is obtained, it is the
orthographic projection. The projection is shown separately in fig.-B, it shows only two
dimensions of the block viz. the height H and the width W, but it does not show the thickness.
Thus, we find that only one projection is insufficient for complete description of the block.

Let us further assume that another plane marked H.P. (as shown in blow given fig.- C ) is
hinged at right angles to the first plane, so as the block is in the front of the V.P. and above the
H.P. The projection on the H.P.(fig. P) shows the top surfaces of the block. If a person looks at

18

the block at the above, he will obtain the same view as the fig. –P. It however does not show
the height of the block (object).
One of the planes is now rotated or turned around on the hinges so that it lies in extension of
the other plane. This can be done in two ways: (1) by turning V.P in the direction of arrows A
or (2) by turning the H.P in the direction of arrows B. The H.P. when turned and brought in
line with the V.P. is shown by the dashed lines. The two projections can now be drawn on a
flat sheet of paper, in correct relationship with each other, as shown in fig.- D. When studied
together, they supply all information regarding the shape and the size of the object.

FIG.-.C
ABOVE
FIG.- D V.P

H

W

H.P

19


FOUR QUADRANTS OF TWO PLANES

2ND
QUADRANT
V.P

H.P

1ST QUADRANT
ALWAYS BE OPENED

Y

X
RD
3 QUADRANT
ALWAYS BE OPENED

H.P

V.P

4TH
QUADRANT
When the planes of projections are extended beyond the line of intersection, they form four
quadrants or dihedral angles as mentioned in figure above. The object may be situated in any
one of the quadrants, its position relative to the planes being described as above or below the
H.P. and in front of or behind the V.P. The planes are assumed to transparent. The projections
are obtained by drawing perpendiculars from the object to the planes (by looking from the front
and from the above). They are then shown on a flat surface by rotating one of the planes. It
should be remembered that the first and the third quadrants are always opened out while
rotating the planes. The positions of the views with respect to the reference line will be
changed according to the quadrant in which the object may be situated. Different positions of
the views of an object in various quadrants are mentioned as under.

20

QUADRANT POSITION OF VIEWS OF AN OBJECT

First Above the H.P. and In front of the V.P.
Second Above the H.P. and Behind the V.P.
Third Below the H.P. and Behind the V.P.
Fourth Below the H.P. and In front of the V.P.

In the H.P. means the elevation of the object lies in the reference line (xy-line), in the V.P.
means the plan of the object lies in xy-line, whereas in the H.P. and the V.P. means both
elevation and the plan of the object lie in the xy-line.

It should also be remembered that the object is denoted by capital alphabetic letter viz
A,B,C,D, etc. whereas its elevation (front view) and the plan (top view) are labeled with the
same but the small alphabetic letters with the difference that the elevation is represented by the
small letter with apostrophe over it (i.e, a’, b’, c’, d’, etc.), whereas its plan is symbolized by
the same letter having no mark on it (i.e, a, b, c, d, etc.).

21

PROJECTION OF POINTS

A point may be situated in space in any one of the four quadrants formed by the two principal
planes of projection or may lie in any one or both of them. Its projections are obtained by
extending projectors perpendicular to the planes. One of the planes is then rotated so that the
first and thirds quadrants are opened out. The projections are shown on a flat surface in their
respective positions either above or below the XY line or in XY line.

v.p
Observer
1ST QUADRANT
H.P
a’
T. VIEW Above the H.P and
In front of the V.P
a’
h
A Observer
h

X Y a

F. VIEW
d a

a

T.VIEW 2ND QUADRANT

Above the H.P and
B
Behind the V.P

b F. VIEW

b
o
b’
d
h

X Y

22

T. VIEW
3RD QUADRANT

Below the H.P and
C Behind the V.P

C C

d

X Y
h C

C’ C’
F.VIEW

T. VIEW
4TH QUADRANT

V.P Below the H.P and
In front of the V.P

H.P

X Y

Height ‘h’
Depth ‘d’
d’

F. VIEW
d
D

23


FOUR QUADRANTS

V.P

H.P
Elevetion
Top View

h

Object
a’

Front View

h d

Plan
X Y

d Plan H.P

V.P
a

24


PROBLEM
Draw the projections of the following points on the same ground line keeping the projectors 25 mm
apart:
i) A point A is in the H.P. and 20 mm behind the V.P.
ii) A point B is 40 mm above the H.P. and 25 mm in front of the V.P.
iii) A point C is in the V.P. and 40 mm above the H.P.
iv) A point D is 25 mm below the H.P. and 25 mm behind the V.P.
v) A point E is 15 mm above the H.P. and 50 mm behind the V.P.

vi) A point F is 40 mm below the H.P and 25 mm in front of the V.P.
vii) A point G is in both, the H.P. and the V.P.

50
40
40
25
20 15

X Y
25 25 25 25 25 25
25 25 25

40

25


PROBLEM
A point P is 50 mm from both the reference planes. Draw its projections in all possible positions.

(i) (ii) (iii) (iv)

50 50 50

X Y

50 50 50

RESULT
(i) A point P is 50 mm above the H.P. and 50 mm in front of the V.P.
(ii) A point P is 50 mm above the H.P. and 50 mm behind the V.P.
(iii) A point P is 50 mm below the H.P. and 50 mm behind the V.P.
(iv) A point P is 50 mm below the H.P. and 50 mm in front of the V.P.

PROBLEM
State the quadrants in which following points are situated:
(a) A point P; its top view is 40 mm above the xy; and the front view 20 mm below the top view.
(b) A point Q; its projections coincide with each other and 40 mm below xy.

20
40

X Y

RESULT:-

(a) A point P is in 3rd quadrant; (i.e., 40
above the H.P. and behind the V.P.)
(b) A point Q is in 4th quadrant; (i.e.,
below the H.P. and in front of the V.P

26


PROBLEM:-
Projections of various points are given in below given figure. State the position of each point with
respect to the planes of projection, giving the distances in centimeters.

RESULT
A point A is 2 cm below the 4
H.P & 5 cm in front of the 3
V.P. 2 1.5
A point B is in the V.P &
4 cm below the H.P. X Y
A point C is 3 cm below the
H.P & 2 cm behind the V.P. 2
A point D is in the H.P. & 3
4
3 cm behind the V.P.
A point E is 4 cm above the 5
H.P. & 1.5 cm behind the
V.P.

PROBLEM:-
A point P is 15 mm above the H.P. and 20 mm in front of the V.P. An other point Q is 25 mm
behind the V.P. and 40 mm below the H.P. Draw projections of P and Q keeping the distance
between their projectors equal to 90 mm. Draw straight lines joining (i) their top views and (ii) their
front views.

27


90 mm

25 mm
15 mm

X Y

20 mm

40 mm
RESULT pp’ and qq’ are the required projections.

Pq and p’q’ are the straight lines joining the top views and
the front viewa respectively.

PROBLEM
Two points A and B are in the H.P. The point A is 30 mm in front of the V.P., while B is behind the
V.P. The distance between their projectors is 75 mm and the line joining their top vies makes an
angle of 45º with xy. Find the distance of the point B from the V.P.

b

??

a’ 45º
X Y
b’

30 mm

45º
m n
a

RESULT the distance of the point B from the V.P. = ……….. mm.

28

PROJECTION OF STRAIGHT LINES
Straight line is the shortest distance between two points. Hence the projection of straight line
may be drawn by joining the respective projections of its end points. The position of straight
line may be described with respect to the two reference planes i.e, horizontal plane (H.P) and
vertical plane (V.P). It may be:
(i) Parallel to one or both the planes.
When a lines is parallel to one plane, its projection on that plane is equal to its true length;
while its projection on the other plane is parallel to the reference line (XY-Line) as shown in
below given figure.

d’
c’
e’ f’
a’ b’

X Y

a c d
e f

b

ab, c’d’ and ef/e’f’ are the true lengths of lines AB, CD, EF respectively.
(ii) Contained by one or both the planes.
When a line is contained by a plane, its projection on that plane is equal to its true length, while
its projection on the other plane is in reference line as shown under.

d’
c’

a’ b’ e’ f’
X Y
c d e f
a

b

(iii) Perpendicular to one of the planes.
When a line is perpendicular to one reference plane, it will be parallel to the other plane.
When a line is perpendicular to a plane, its projection on that plane is a point; while its
projection on the other plane is a line equal to its true length and perpendicular to the reference
line.
29

In first-angle projection method, when top views of two or more points coincide, the point
which is comparatively farther from XY-Line in the front view will be visible; and when their
front views coincide, that which is farther from XY-Line in the top view will be visible.
In third-angle projection method, it is just the reverse. When the top views of two or more
points coincide, the point which is comparatively nearer to the XY-Line in the front view will
be visible; and when their front views coincide, the point which is nearer to XY-Line in the
top view will be visible as shown in figure below.

a’ c’

d’
a
d’
b’
X Y X Y
a’
c
d
a

b’
d
Line AB is perpendicular to the H.P. The top views of its ends coincide in the point ‘a’. Hence,
the top view of the line AB is the point ‘a’. Its front view a’b’ is equal to AB and perpendicular
to XY-Line.
Line CD is perpendicular to the V.P. The point d’ is its front view. cd is the top view and is
equal to line CD which is perpendicular to XY-Line.
(iv) Inclined to one plane and parallel to the other.
The inclination of a line to a plane is the angle which the line makes with its projection on that
plane.
When a line is inclined to one plane and is parallel to the other, its projection on the plane to which it is inclined,
is a line shorter than its true length but parallel to the reference line; its projection on the plane to which it is
parallel, is a line equal to its true length and inclined to the reference line at its true inclination.
q’1
r’ s’1 s’
p’
q’

X Y

r s

p q1 q

S1

30

It is clear from the above that when a line is inclined to the H.P. and parallel to the V.P., its top
view is shorter than its true length, but parallel to XY; its front view is equal to its true length
and is inclined to X at its true inclination with the H.P. And when the line is inclined to the
V.P. and parallel to the H.P., its front view is shorter than its true length but parallel to XY-
Line; its top view is equal to its true length and is parallel to XY at its true inclination with the
V.P.
(v) Line inclined to both the planes.
When a line is inclined to both the planes, its projections are shorter than the true length and
inclined to XY-Line at angles greater than the true inclinations. These angles are termed as
apparent angles of inclinations and are denoted by the symbols (alpha) and (beta), as
indicated in figure.

b1’
b1’ b’

a’ b’
a a’
X Y

a b

b1 b
b1

TRUE LENGTH OF A STRAIGHT LINE AND ITS INCLINATIONS WITH THE REFERENCE
PLANES.

When the projections of a line are given, its true length and inclinations with the planes are
determined by the application of the following rule:
When a line is parallel to a plane, its projection on that plane will show its ture length and the
true inclination with the other plane. The line may be parallel to the reference plane, and its
true length obtained by one of the following methods.
1) Making each view parallel to the reference line and projecting the other view from it.
2) Rotating the line about its projections till it lies in the H.P. or in the V.P.
3) Projecting the views on auxiliary planes parallel to each view.

31

b’ b’1
m n

a’ b’2
e f
X Y

a b1
g h

j k
b b2

TRACES OF A LINE.
When a line is inclined to a plane, it will meet that plane (produced if necessary). The point in
which the line or line produced meets the plane is called its trace. The point of intersection of
the line with the H.P. is called the horizontal trace and is denoted by the symbol H.T., whereas,
that with the V.P. is called the vertical trace or V.T.
(ii) NO V.T
(i ) (iii) (iv) (v)
c’ NO V.T p’
a’ b’ d’ e’ f’ V.T V.T
s’
q
h
X Y
v
r
p
a b c d H.T f NO H.T
H.T s
NO TRACE NO H.T
e
(vi) h’ m’
V.T (vii)

g’
V.T v n’
X Y
H.T m h
g

H.T
n
h

A line AB as shown in fig.-I is parallel to both the planes. It has no trace.
A line CD (fig.-II) is inclined to the H.P. and parallel to the V.P. It has only H.T. but has no
V.T.
A line EF (fig.-III) is inclined to the V.P. and parallel to the H.P. It has only V.T. but has no
H.T.
32

Hence, when a line is parallel to a plane, it has no trace on that plane.

A line PQ (fig.-IV) is perpendicular to the H.P. Its H.T. coincides with its top-view which is a
point. It has only H.T. but has no V.T.
A line RS (fig.-V) is perpendicular to the V.P. Its V.T. coincides with its front-view which is a
point. It has only V.T but has no H.T.
Thus, when a line is perpendicular to a plane, its trace on that plane coincides with its
projection on that plane. It has no trace on the other plane.
A line GH (fig.-VI) has its end G is in both the H.P. and the V.P. Its H.T and V.T. coincide
with c and c’ in XY-Line.
A line MN (fig.-VII) has its end M in the H.P. and the end N in the V.P. Its H.T. coincides
with m the top-view of M and the V.T. coincides with n’ the front-view of N.
Hence, when a line has an end in a plane, its trace upon that plane coincides with the
projection of that end on that plane.

EXERCISE (IX-A)
PR-01
A 100 mm long line is parallel to and 40 mm above the H.P. Its two ends are 25
mm and 50 mm in front of the V.P. respectively. Draw its projections and find its inclination
with the V.P.

a’ 100
b’

40

X Y

25

m n 50
a

b
RESULT: (i) a’b’ and ab are the required projections of
a line.
(ii) Inclination of a line with the V.P. = = …… .

33

02
A 90 mm long line is parallel to and 25 mm in front of the V.P. Its one end is in the
H.P while the other is 50 mm above the H.P. Draw its projections and find its inclination with
the H.P.

b’

50

a’
X Y

25

90
a b

RESULT: (i) a’b’ and ab are the required projections of a line.
(ii) Angle measures ……. .

03
The top view of 75 mm long line measures 55 mm. The line is in the V.P., its one end
being 25 mm above the H.P. Draw its projections.

b’1
m n

a’ b’
55
25

X Y
a
RESULT:
a’b’ and ab are the required projections
75
of a line.

e b f
b1

34

04
The front view of a line, inclined at 30 to the V.P. is 65 mm long. Draw the
projections of a line when it is parallel to 40 mm above the H.P., its one end being 30 mm in
front of the V.P.

65mm

40

30

05. A vertical line AB, 75 mm long has its end A in the H.P. and 25 mm in front of the V.P.
A line AC, 100 mm long is in the H.P and parallel to V.P. Draw the projections of the line
joining B and C, and determine its inclination with the H.P.

b’

75

a =? c’
X Y
25
100
b c

RESULT: bc and b’c’ are the required projections.
Angle = …….. .

35

06
Two pegs fixed on a wall are 4.5 metres apart. The distance between the pegs measured
parallel to the floor is 3.6 metres. If one peg is 1.5 metres above the floor, find the height of the
second peg and the inclination of the line joining the two pegs with the floor.

b’ SCALE:
1 metre = 1 cm.
4.5

a’ ?
=?
k l

1.5
3.6
X Y
a b

RESULT: The height of the second peg to the floor is ………. metres.
The inclination of line joining the two pegs = = ……… .

PROBLEM
A line AB 50 mm long, has its end A in both the H.P. and the V.P. It is inclined at 30º to the
H.P. and at 45º to the V.P. Draw its projections.

= 30º
= 45º

X Y

RESULT : ab2 and ab’2 are the required projections. and are the apparent angles

36


PROBLEM
The top view of 75 mm long line AB measures 65 mm, while the length of its front view is 50
mm. Its one end A is in the H.P. and 12 mm in front of the V.P. Draw the projections of AB
and determine its inclinations with the H.P. and the V.P.

75
= ??
= ??

X Y
65
12
50

RESULT: ab2 and a’b2’ are
the required projections.
and are the true
inclinations

PROBLEM
A line AB 65 mm long, has its end A 20 mm above the H.P. and 25 mm in front of the V.P.
The end B is 40 mm above the H.P. and 65 mm in front of the V.P. Draw the projections of
AB and show its inclinations with the H.P. and the V.P. (both the planes).

65

40
20

25
65

37


PROBLEM
A line AB 90 mm long, is inclined at 45º to the H.P. and its top view makes an angle of 60º
with the V.P. The end A is in the H.P. and 12 mm in front of the V.P. Draw its front view and
find its true inclination with the V.P.

= 45º
90 = ??
= 60º

X Y
12

??
Angle = ……..º is the true
inclination with the V.P.

PROBLEM
Incomplete projections of a line PQ, inclined at 30º to the H.P. are given in fig. (a). Complete
the projections and determine the true length of line PQ and its inclination with the V.P.

Fig. (a)
=30º

30º

15
X Y X Y
15

45º
=45º
65

RESULT: p’q’ is the
required front view, &
Angle is the inclination
of line PQ with the V.P.

38

PROBLEM
A line AB 90 mm long, is inclined at 30º to the H.P. Its end A is 12 mm above the H.P. and
20 mm in front of the V.P. Its front view measures 65 mm. Draw the top view of AB and
determine its inclination with the V.P.

65
90

=30º

12
X Y
20

= ??
RESULT: ab1 is the required
top view.
Angle = ….º is the required
true inclination of the line AB
with the V.P.

PROBLEM
A straight road going uphill from a point A, due east to an other point B, is 4 km long and has a
slope of 15º. An other straight road from point B, due 30º east of the north, to a point C is also
4 km long but is on ground level. Determine the length and slope of the straight road joining
the points A and C. SCALE 1 km = 2.5 cm.
b’ c’ c1’
4 km

a’

RESULT: =15º =??

The straight
road joining the
Points A & C is
…… km long.

Slope of the
road = =……º 30º 4 km

a b c1
X Y

39


PROBLEM
An object O is placed 1.2 m above the ground and in the centre of a room 4.2 m x 3.6 m x 3.6
m high. Determine graphically its distance from one of the corners between the roof and two
adjacent walls. SCALE: 1 m = 1”

3.6

3.6

1.2
1.8
X Y

2.1

RESULT: O’C1’ (True length) is
4.2 a distance of the object from one
of the top corners of the room.

PROBLEM
A line AB, inclined at 40º to the V.P., has its ends 50 mm and 20 mm above the H.P. The
length of its front view is 65 mm and its V.T. is 10 mm above the H.P. Determine the true
length of AB, its inclination with the H.P. and its H.T.

40


65

V.T H.T

v h

40º
RESULT: a1v is the true length of a line
AB.
The inclination of the line with the H.P.

= = ………º
H.T. is shown in the fig.

PROBLEM
A line PQ 100 mm long, is inclined at 30º to the H.P. and at 45º to the V.P. Its mid-point is in
the V.P. and 20 mm above the H.P. Draw its projections if its end P is in third quadrant and Q
in the first quadrant.

= 30º
= 45º

X Y

RESULT: p3q3 p3‘q3‘ are the
required projections.

PROBLEM
41

The projectors of the ends of a line AB are 5 cm apart. The end A is 2 cm above the H.P. and 3
cm in front of the V.P. The end B is 1 cm below the H.P. and 4 cm behind the V.P. Determine
the true length and traces of AB, and its inclinations with the two planes.

5

4
2

1
3

PROBLEM
A line AB 50 mm long, has its end A in both the H.P. and the V.P. It is inclined at 30º to the H.P. and at 45º
to the V.P. Draw its projections.

= 30º
= 45º

X Y

RESULT : ab2 and ab’2 are the required projections. and are the apparent angles

42

PROBLEM
The top view of 75 mm long line AB measures 65 mm, while the length of its front view is 50 mm. Its one end A
is in the H.P. and 12 mm in front of the V.P. Draw the projections of AB and determine its inclinations with the
H.P. and the V.P.

75
= ??
= ??

X Y
65
12
50

and are the true
inclinations

PROBLEM
A line AB 65 mm long, has its end A 20 mm above the H.P. and 25 mm in front of the V.P. The end B is 40 mm
above the H.P. and 65 mm in front of the V.P. Draw the projections of AB and show its inclinations with the H.P.
and the V.P. (both the planes).

65

40
20

25
65

43

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