1. ORTHOGRAPHIC PROJECTIONS
OF POINTS, LINES, PLANES, AND SOLIDS.
TO DRAW PROJECTIONS OF ANY OBJECT,
ONE MUST HAVE FOLLOWING INFORMATION
A) OBJECT
{ WITH IT’S DESCRIPTION, WELL DEFINED.}
B) OBSERVER
{ ALWAYS OBSERVING PERPENDICULAR TO RESP. REF.PLANE}.
C) LOCATION OF OBJECT,
{ MEANS IT’S POSITION WITH REFFERENCE TO H.P. & V.P.}
TERMS ‘ABOVE’ & ‘BELOW’ WITH RESPECTIVE TO H.P.
AND TERMS ‘INFRONT’ & ‘BEHIND’ WITH RESPECTIVE TO V.P
FORM 4 QUADRANTS.
OBJECTS CAN BE PLACED IN ANY ONE OF THESE 4 QUADRANTS.
IT IS INTERESTING TO LEARN THE EFFECT ON THE POSITIONS OF VIEWS ( FV, TV )
OF THE OBJECT WITH RESP. TO X-Y LINE, WHEN PLACED IN DIFFERENT QUADRANTS.
STUDY ILLUSTRATIONS GIVEN ON HEXT PAGES AND NOTE THE RESULTS.TO MAKE IT EASY
HERE A POINT A IS TAKEN AS AN OBJECT. BECAUSE IT’S ALL VIEWS ARE JUST POINTS.

2. FOLLOWING NOTATIONS SHOULD BE FOLLOWED WHILE NAMEING
DIFFERENT VIEWS IN ORTHOGRAPHIC PROJECTIONS.
OBJECT POINT A LINE AB
IT’S TOP VIEW a a b
IT’S FRONT VIEW a’ a’ b’
IT’S SIDE VIEW a” a” b”
SAME SYSTEM OF NOTATIONS SHOULD BE FOLLOWED
INCASE NUMBERS, LIKE 1, 2, 3 – ARE USED.

3. IF OBSERVED ALONG X-Y LINE ( IN RED ARROW DIRECTION)
2nd Quadrant VP
2nd Quad.
1ST Quad.
F.V.
1st Quadrant Y
Observer
X Y HP X
3rdQuadrant
Observer
3rd Quad.
4th Quad.
4th Quadrant
THIS QUADRANT PATTERN,
IF OBSERVED ALONG X-Y LINE ( IN RED ARROW DIRECTION)
WILL EXACTLY APPEAR AS SHOWN ON RIGHT SIDE AND HENCE,
IT IS FURTHER USED TO UNDERSTAND ILLUSTRATION PROPERLLY.

4. POINT A IN POINT A IN VP 1ST QUADRANT 2ND QUADRANT a’ VP A A a’ a HP
Point A is
Placed In
different
quadrants
and it’s Fv & Tv
are brought in
same plane for
Observer to see
clearly.
Fv is visible as
it is a view on
VP. But as Tv is
is a view on Hp,
it is rotated
downward 900,
In clockwise
direction.The
In front part of
Hp comes below
xy line and the
part behind Vp
comes above.
Observe and
note the
process.
POINT A IN
2ND QUADRANT
POINT A IN
1ST QUADRANT HP VP VP HP a’ A A a’ a
OBSERVER
OBSERVER a a HP VP HP VP
OBSERVER
OBSERVER a a’ A a’
POINT A IN
3RD QUADRANT A
POINT A IN
4TH QUADRANT

5. Basic concepts for drawing projection of point
FV & TV of a point always lie in the same vertical line
FV of a point ‘P’ is represented by p’. It shows position of the point with respect to HP only. While drawing FV, XY line represents the HP and the plane of paper represents the VP.
If the point P lies above HP, p’ lies above the XY line.
If the point P lies in the HP, p’ lies on the XY line.
If the point P lies below the HP, p’ lies below the XY line.
FV does not give any information about position of the point w.r.t. VP
TV of a point ‘P’ is represented by p. It shows position of the point with respect to VP only. While drawing TV, XY line represents the VP and the plane of paper represents the HP.
If the point P lies in front of VP, p lies below the XY line.
If the point P lies in the VP, p lies on the XY line.
If the point P lies behind the VP, p lies above the XY line.
TV does not give any information about position of the point w.r.t. HP

6. ORTHOGRAPHIC PRESENTATIONS
PROJECTIONS OF A POINT IN FIRST QUADRANT.
POINT A ABOVE HP
& INFRONT OF VP
POINT A ABOVE HP
& IN VP
POINT A IN HP
& INFRONT OF VP
For Tv
For Tv
PICTORIAL
PRESENTATION
PICTORIAL
PRESENTATION
For Tv A a’ a’ A
For Fv X Y X Y Y
For Fv a’ a a X a A
For Fv
ORTHOGRAPHIC PRESENTATIONS
OF ALL ABOVE CASES.
Fv above xy,
Tv below xy.
Fv above xy,
Tv on xy.
Fv on xy,
Tv below xy. X Y a a’ VP HP X Y a’ VP HP a X Y a VP HP a’

7. PROJECTIONS OF A POINT IN SECOND QUADRANT.
POINT A ABOVE HP
& BEHIND VP
For Tv
For Fv A Y X

8. PROJECTIONS OF STRAIGHT LINES.
INFORMATION REGARDING A LINE means
IT’S LENGTH,
POSITION OF IT’S ENDS WITH HP & VP
IT’S INCLINATIONS WITH HP & VP WILL BE GIVEN.
AIM:- TO DRAW IT’S PROJECTIONS - MEANS FV & TV.
SIMPLE CASES OF THE LINE
A VERTICAL LINE ( LINE PERPENDICULAR TO HP & // TO VP)
LINE PARALLEL TO BOTH HP & VP.
LINE INCLINED TO HP & PARALLEL TO VP.
LINE INCLINED TO VP & PARALLEL TO HP.
LINE INCLINED TO BOTH HP & VP.
STUDY ILLUSTRATIONS GIVEN ON NEXT PAGE
SHOWING CLEARLY THE NATURE OF FV & TV
OF LINES LISTED ABOVE AND NOTE RESULTS.

9. V.P. A Line perpendicular to Hp & // to Vp V.P. A Line // to Hp &
Orthographic Pattern
For Tv
(Pictorial Presentation)
V.P. X Y
H.P.
V.P. a’ b’
a b Fv Tv
Note:
Fv is a vertical line
Showing True Length
&
Tv is a point. a’ b’ B A 1. FV
A Line
perpendicular
to Hp
&
// to Vp X Y
For Fv
a b TV
Orthographic Pattern
(Pictorial Presentation) X Y
V.P. X Y
H.P.
V.P. a b a’ b’ Fv Tv
Note:
Fv & Tv both are
// to xy
&
both show T. L.
For Tv 2. b’ a’
A Line
// to Hp
&
// to Vp
F.V. A B
For Fv b a
T.V.

10. V.P. A Line inclined to Hp and parallel to Vp V.P.X Y
V.P. X Y
H.P.
V.P.
F.V.
T.V. a b a’ b’ 
Fv inclined to xy
Tv parallel to xy. 3. b’ a’
F.V.
A Line inclined to Hp
and
parallel to Vp
(Pictorial presentation) A B   b a
T.V.
Orthographic Projections
V.P.
Tv inclined to xy
Fv parallel to xy. X Y
H.P.
V.P. Ø a b a’ b’ Tv Fv 4.
A Line inclined to Vp
and
parallel to Hp
(Pictorial presentation) a’ b’
F.V. Ø B A Ø a b
T.V.

11. (Pictorial presentation)
For Fv
For Tv B A
For Tv X Y
V.P. a’ b’ a b  
F.V.
T.V.
V.P. 5.
A Line inclined to both
Hp and Vp
(Pictorial presentation) a’ b’
F.V. B A  X Y
On removal of object
i.e. Line AB
Fv as a image on Vp.
Tv as a image on Hp,
For Fv  a b
T.V. X Y  
H.P.
V.P. a b FV TV a’ b’
Orthographic Projections
Fv is seen on Vp clearly.
To see Tv clearly, HP is rotated 900 downwards,
Hence it comes below xy.
Note These Facts:-
Both Fv & Tv are inclined to xy.
(No view is parallel to xy)
Both Fv & Tv are reduced lengths.
(No view shows True Length)

12. H.P. V.P. H.P. V.P. H.P. V.P.  &  X Y   a b FV TV a’ b’ X Y X Y 
Note the procedure
When Fv & Tv known,
How to find True Length.
(Views are rotated to determine
True Length & it’s inclinations
with Hp & Vp).
Note the procedure
When True Length is known,
How to locate FV & TV.
(Component a’b2’ of TL is drawn
which is further rotated
to determine FV)
Orthographic Projections
Means Fv & Tv of Line AB
are shown below,
with their apparent Inclinations
 &  X Y  
H.P.
V.P. a b FV TV a’ b’ X Y
H.P.
V.P. X Y 
H.P.
V.P. a b TV a’ b’ FV
b1’
b1’  b’ Fv TL TL a a’ 
b2’  b1 b1 Tv b2 Ø TL b 
Here TV (ab) is not // to XY line
Hence it’s corresponding FV
a’ b’ is not showing
True Length &
True Inclination with Hp.
In this sketch, TV is rotated
and made // to XY line.
Hence it’s corresponding
FV a’ b1’ Is showing
True Length
&
True Inclination with Hp.
Here a’b1’ is component
of TL ab1 gives length of FV.
Hence it is brought Up to
Locus of a’ and further rotated
to get point b’. a’ b’ will be Fv.
Similarly drawing component
of other TL(a’b1‘) TV can be drawn.

13. Type-I:Given projections (FV & TV) of the line
Type-I:Given projections (FV & TV) of the line. To find True length & true inclination of the line with HP (θ) and with VP(Φ).
End A of a line AB is 20mm above HP & 20mm in front of VP while its end B is 55mm above HP and 75mm in front of VP. The distance between end projectors of the line is 50mm. Draw projections of the line and find its true length and true inclination with the principal planes. Also mark its traces.
Given, b’
b1’
A: 20mm ↑HP,20 mm →VP
θ: True inclination of the line with HP = 26º
B: 55mm ↑HP,75 mm →VP
a0 b0 = 50 mm
α : Inclination of FV of the line with HP/XY
TL=83
To draw FV & TV, 55
To find TL, θ ,Ø, HT & VT a’
b2’ θ α h v’ 20 a0 b0 Y
Ø: True inclination of the line with VP = 41º X h’ v 50 20
β : Inclination of TV of the line with VP/XY Φ β a b1
h’v’a’b’ are always collinear.
h v a b are always collinear. 75
TL=83 b b2
List of questions of type I : 10.13,10.7,5,11,12, 10.28,18,23,26,10.18,10.23

14. Problems for finding TL, θ,Ø, HT & VT
Problem1: End A of a line AB is in the HP and 12 mm in front of VP, while its end B is 45mm above HP and 75mm in front of VP. The distance between their end projectors is 30 mm. Draw its projections and find its true length and true inclinations with the principle planes. Also mark its traces.
Problem 2: End A of a line AB is 25mm below the HP and 20 mm behind VP, while its end B is 45 mm below HP and 65mm behind VP. The distance between their end projectors is 45 mm. Draw its projections and find its true length and true inclinations with the principle planes. Also mark its traces.
Problem 3: End A of a line AB is 70 mm above HP and 20 mm in front of VP, while its end B is 20mm above HP and 50mm in front of VP. The distance between their end projectors is 60 mm. Draw its projections and find its true length and true inclinations with the principle planes. Also mark its traces.
Problem 4: End A of a line AB is 25mm below the HP and 20 mm behind VP, while its end B is 40mm below HP and 65mm behind VP. The distance between their end projectors is 45 mm. Draw its projections and find its true length and true inclinations with the principle planes. Also mark its traces.

15. PROBLEM Line inclined to both HP & VP
Type –II Given (i) T.L., θ and Ø,
(ii) T.L., F.V., T.V.
to draw projections, find α, β,H.T. and V.T.
A line AB, 70mm long, has its end A 25 mm above HP and 25mm in front of VP. It is inclined at 30° to HP and 45°to VP. Draw its projections and mark its traces. b’
b1’
Given,
A: 25mm ↑HP,25 mm →VP
TL=70mm, 70
θ=30º,
Ø=45º a’
b2’
To draw/find,
30°
FV, TV, HT & VT 25 h v’ X Y h’ v
List of questions of type II : 10.8,10.11,10.12,10.14,10.17, 10.10,1,4,6,8, 25
45° b1 a 70 b b2

16. Let a’b’ and ab are FV & TV of a line.
Trapezoidal Method
This method is used to determine true length, true inclination, θ and Ø when projections of the line (FV & TV) are already given.
Let a’b’ and ab are FV & TV of a line. B2
Now draw perpendiculars at a’ and b’
For solving the problem by trapezoidal method draw perpendiculars at a and b
Measure distance of a from the XY line and mark it on the perpendicular drawn from a’.
Measure distance of a’ from the XY line and mark it on the perpendicular drawn from a. b’
Similarly measure the distance of b from the XY line and mark it on the perpendicular drawn from b’.
Similarly measure the distance of b’ from the XY line and mark it on the perpendicular drawn from b. A2 a’
Mark the points as A1and B1 respectively and join them. This will give True length of the line. X Ø Y
Mark the points as A2and B2 respectively and join them. This will give True length of the line. v’ h a
Extend this line to intersect the top view. The point of intersection gives HT and the angle between TL and TV will be θ. A1 θ
Extend this line to intersect the front view. The point of intersection gives VT and the angle between TL and FV will be Ø. b
Note: If the distances of the points are measured on the opposite sides of XY line, then perpendiculars are drawn on opposite sides and, points are also marked on opposite sides. B1

17. Compulsory problems in projection of Straight Lines
Type I 10.13, 5, 10.17,10.18, 10.23
Type II 10.11,10.12,10.14, 6, 8,10.10
Type III 10.19

18. Q The top view of a 75mm long line AB measures 65mm,while its front view measures 50mm. Its one end A is in HP and12mm in front of VP. Draw the projections of AB and determine its inclination with HP and VP
To draw FV &TV of the line AB
To find θ & Ø
Given,
TL=75mm,TV=65mm,FV=50mm
A is in HP & 12mm→VP
Hint: Draw ab1=65mm // to XY. Because when TV is // to XY, FV gives TL. b’
b1’
Ans. θ=30º 50 75
Ans. Ø=48º a’ X Y
30º 12 65 b1 a
48º 75 65 b b2

19. Q 5. The end A of a line AB is in the HP and 25 mm behind the VP
Q 5. The end A of a line AB is in the HP and 25 mm behind the VP. The end B is in the VP and 50 mm above the HP. The distance between the end projectors is 75 mm. draw the projections of AB and determine its true length, traces and inclination with two planes.
Given,
A is in HP & 25mm ←V.P B is in VP & 50 ↑ V.P. a0b0=75mm
To find, True Length, θ,Ø, H.T. and V.T. 75 v’ b’
b1’
33º h a b1 50 93
Ans. TL= a’b1’=93 mm 25 93
θ=33º
b2’
Ø=15º X Y a’ b
15º b2 h’ v

20. Q10. 12 A line AB, 65mm long has its end A 20mm above H. P
Q A line AB, 65mm long has its end A 20mm above H.P. and 25mm in front of VP. The end B is 40mm above H.P. and 65mm in front of V.P. Draw the projections of AB and show its inclination with H.P. and V.P.
Given,
TL=65mm
A is 20mm ↑ HP & 25mm →V.P B is 40mm ↑ & 65mm → V.P.
To draw FV &TV of the line AB
To find θ & Ø
Hint1:Mark a’ 20mm above H.P & a 25mm below XY
Hint2:Draw locus of b’ 40mm above XY & locus of b 65 mm below XY b’
b1’ a’ 65
b2’ 40
18º 20
Ans. θ=18º X Y
Ans. Ø=38º 25
38º b1 a 65 65 b b2

21. Q10. 13:The projectors of the ends of a line AB are 5cm apart
Q10.13:The projectors of the ends of a line AB are 5cm apart. The end A is 2cm above the H.P and 3cm in front of V.P. The end B is1cm below H.P. and 4cm behind the V.P. Determine the true length and traces of AB, and its inclination with the two planes
Given,
a0b0=50mm A is 20mm ↑ HP & 30mm →V.P B is 10mm ↓ & 40mm ← V.P.
To find, True Length, θ,Ø, H.T. and V.T. b b2 a’ 40 h
b2’ 20 v’ 91 X Y v 50 h’ 10
b1’ 30
20º b’
Ans. θ=20º a
50º b1
Ans. Ø=50º

22. Q10. 13:The projectors of the ends of a line AB are 5cm apart
Q10.13:The projectors of the ends of a line AB are 5cm apart. The end A is 2cm above the H.P and 3cm in front of V.P. The end B is1cm below H.P. and 4cm behind the V.P. Determine the true length and traces of AB, and its inclination with the two planes
Given,
A0B0=50mm A is 20mm ↑ HP & 30mm →V.P B is 10mm ↓ & 40mm ← V.P.
To find, True Length, θ,Ø, H.T. and V.T. B1 A2 b θ a’ TL 40 h Ø 20 91 v’ X Y 50 10 30 b’ TL
Ans. θ=20º
TL=91 a
Ans. Ø=50º A1 B2

23. Q10. 18: The ends of a line PQ are on the same projector
Q10.18: The ends of a line PQ are on the same projector. The end P is 30 mm below the HP and 12 mm behind the VP. The Q is 55 mm above the HP and 45mm in front of the VP. Determine the true length and traces of PQ and its inclination with two planes.
Given, p0q0= 0mm P is 30mm ↓ HP & 12mm ←V.P Q is 55mm ↑ & 45mm → V.P.
To find, True Length, θ,Ø, H.T. and V.T.
For solving the problem by trapezoidal method, first draw perpendiculars at p & q but on opposite sides as p’ & q’ are on the opposite sides of the xy line. Q2 q’ 55
T.L. P1 p 12 Ø X h Y 30 v’
T.L.
T.L.=102 mm θ 45
θ= 34º p’
ø= 56º P2 Q1 q

24. Q10. 14:A line AB, 90mm long, is inclined at 45º to the H. P
Q10.14:A line AB, 90mm 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 12mm in front of V.P. Draw its front view and find its true inclination with the V.P. b’
Given,
T.L.=90mm, θ=45º, β=60º A is in the H.P. & 12mm→V.P.
b1’
To find/draw,
F.V.,T.V. & Ø 90
Ans. Ø = 38º a’ X Y
45º 12 b1
60º
38º a 90 b2 b

25. Q 4:A line AB, 75mm long, is in the second quadrant with the end A in the H.P. and the end B in the V.P. The line is inclined at 30º to the H.P. and at 45º to the V.P. Draw the projections of AB and determine its traces. b’
b1’
Given,
To draw F.V. and T.V.
TL=75mm,
To find H.T. and V.T. TL
A: in H.P.,
B: in V.P., a’
θ=30º, θ
b2’ X Y
Ø=45º ø a TL h b b2 a v’ b’
b1’ 75 75 75 a’ v X Y h’
30º
45º b b2
b2’
45º

26. 10. 17:A line AB, 90mm long, is inclined at 30 to the HP
10.17:A line AB, 90mm long, is inclined at 30 to the HP. Its end A is 12mm above the HP and 20mm in front of the VP. Its FV measures 65mm. Draw the TV of AB and determine its inclination with the VP
Given,
T.L.=90mm θ = 30º A is 12mm ↑H.P. &20mm→V.P. F.V.=65mm b’
b1’
To find/draw,
T.V. & Ø 90 65 a’ 12
30° X Y 20 b1
44° a
Ans: Ø = 44º 90 b b2

27. Q10.23:Two lines AB & AC make an angle of 120 between them in their FV & TV. AB is parallel to both the HP & VP. Determine the real angle between AB & AC.
Given,
AB parallel to both HP and VP,
a’c’ & ac inclined at 120 with
a’b’ and ab respectively C
To find,
True angle between AB & AC
c1’
c2’ c’
Assume any arbitrary length of
AB & AC. Also assume any
distances of A & B from HP & VP
112°
Ans. 112º a’ b’
120° X Y a b c2 c1
120° c

28. Y X v’ b1’ b’ b2’ a’ b0 a0 v h’ b2 b a b1 h
Q10.28: The distance between the end projectors of a line AB is 70 mm and the projectors through traces are 110 mm apart. The end A of the line is 10 mm above the H.P. If the top view and the front view of the line make 30° and 60° with XY line respectively. Draw the projections of the line and determine (i) the traces, (ii) the angles with the H.P. and the V.P. and (iii) the true length of the line.
Assume that the line is in the first quadrant. v’
Given, a0b0= 70 mm, h’v= 110 mm, A: 10 mm ↑ H.P., α =60º, β=30º
b1’ b’
Ans: HT is 63 mm in front of V.P.
VT is 190 mm above the H.P.
θ = 56º
To find, True Length, θ, Ø, H.T. and V.T.
190
T.L.= 146 mm
ø = 16º
T.L. a’
60º
b2’
56º b0 10
30º X Y a0 v h’ b b2 63
T.L. a
16º b1 h 70
110

29. Q18: The projectors drawn from H. T. and V. T
Q18: The projectors drawn from H.T. and V.T. of a straight line AB are 80 mm apart while those drawn from its ends are 50 mm apart. The H.T. of the line 35 mm in front of V.P. ,the V.T. is 55 mm above the H.P and the end A is 10 mm above the H.P. Draw the projections of AB and determine its length and inclinations with the reference planes.
Given, a0b0= 50 mm, h’v= 80 mm, A: 10 mm ↑ H.P.,
To find, True Length, θ, Ø v’
32º b’
b1’
T.L. 55
b2’ a’ 10 b0 v X Y h’ a0 b b2
Ans. : T.L.= 65 mm 35
T.L.
θ = 32º a
ø = 20º
20º b1 50 h 80

30. Q23: The front view of a line makes an angle of 30º with the xy. The H
Q23: The front view of a line makes an angle of 30º with the xy. The H.T. of the line is 45 mm in front of the V.P., while its V.T. is 30 mm below the H.P. One end of the line is 10 mm above the H.P. and the other end is 100 mm in front of the V.P.
Draw the projections of the line and determine (i) its true length, and (ii) its inclinations with the H.P. and the V.P. b’
b1’
Given, α =30º, H.T.: 45 mm→V.P., V.T. : 30 mm ↓H.P A: 10 mm ↑ H.P., B: 100 mm→V.P.
To find, True Length, θ, Ø
T.L. a’
b2’
23º 10
30º v h’ X Y 30 45 v’ h
100
Ans. : T.L.= 66mm a
37º
θ = 23º b1
ø = 37º
T.L. b b2

31. Q26: The projectors of the ends of a line PQ are 90 mm apart
Q26: The projectors of the ends of a line PQ are 90 mm apart. P is 20 mm above the H.P. while Q is 45 mm behind the V.P. The H.T. and the V.T. of the line coincide with each other on xy, between the two end projectors and 35 mm away from the projector of the end P. Draw the projections of PQ and determine its true length and inclinations with the two planes.
Given, p0q0= 90, h’v= 0 mm, P: 20 mm ↑ H.P., Q: 45 mm←V.P., p0h’=35 mm
To find, True Length, θ, Ø q q2 p’ 45
q2’
T.L. 20
h’v h v’ X Y p0 q0
T.L.
35º q1
Ans. : T.L.= 127 mm p q’
24º
q1’
θ = 24º 35
ø = 35º 90

32. Q10. 19 A line AB, inclined at 40º to the V. P
Q10.19 A line AB, inclined at 40º to the V.P. has its end 50mm and 20mm above the H.P. the length of its front view is 65mm and its V.T. is 10mm above the H.P. determine .the true length of AB its inclination with the H.P. and its H.T.
Given,
Ø = 40º, A is 20mm↑HP, B is 50 mm ↑ HP, FV=65mm, VT is 10mm ↑ HP
To find,
TL, θ & HT
Note: Here we do not know position of a and b in top view, but we know v’ and v. So for time being we consider the line as v’b’ instead of a’b’. Now if we make v’b’(FV) parallel to xy, we will get TL of VB in top view at angle Ø.
b1’ b’ 85 50 a’ h
b3’
21º v’ 20 10 X Y v
40º h’ b1
Ans,
TL = 85 mm, θ = 21º & HT is 17 mm behind VP a b3 b

33. Q10. 16:The end A of a line AB is 25 mm behind the V. P
Q10.16:The end A of a line AB is 25 mm behind the V.P. and is below the H.P. The end B is 12 mm in front of the VP and is above the HP The distance between the projectors is 65mm. The line is inclined at 40 to the HP and its HT is 20 mm behind the VP. Draw the projections of the line and determine its true length and the VT
Given,
A0B0=65mm A is 25mm ←V.P.& is ↓H.P B is 12mm →V.P. & is above HP θ = 40º
To find/draw,
F.V., T.V., T.L., v’
b3’ b’
b1’
Ans. TL= a’b1’=98 mm v’ a b1 h b3 25 20 X h’ Y
40º v 12 a’ b 65

34. Q10. A line AB is in the first quadrant
Q10. A line AB is in the first quadrant. Its end A and B are 20 mm and 60 mm in front of the V.P. respectively. The distance between the end projectors is 75 mm. The line is inclined at 30º to the H.P. and its H.T. is 10 mm above the xy. Draw the projections of AB and determine its true length and the V.T. b’
b3’
Given, AB is in first quadrant, A is 20mm→V.P., B is 60 mm → V.P., a0b0= 75 mm, θ = 30º H.T. is 10mm↑ xy
To draw F.V. and T.V to find,TL, θ & V.T.
Note: Here we do not know position of a’ and b’ in front view, but we know h’ and h. So for time being we consider the line as hb instead of ab. Now if we make hb(TV) parallel to xy, we will get TL of HB in front view at angle θ.
T.L. a’
Note: As b3’ and b’ lies on the same locus, similarly a3’ and a’ will also lie on the same locus. And a3’b3’ will be true length of AB
a3’ v’ h b3 10 X Y h’
30º v a0 b0 20
Ans. TL= a3’b3’=98mm 75
VT is 13 mm above the H.P. a 60 b

35. Q9. The front view of a line AB measures 65 mm and makes an angle of 45º with xy. A is in the H.P. and V.T. of the line is 15 mm below the H.P. The line is inclined at 30º to the V.P. draw the projections of AB and find its true length and inclination with the H.P. Also locate its H.T.
Given, FV=65mm, α = 45º, A is inHP, VT is 15 mm ↓HP Ø = 30º,
To find,
TL, θ & HT
Note: Here we do not know position of a and b in top view, but we know v’ and v. So for time being we consider the line as v’b’ instead of a’b’. Now if we make v’b’(FV) parallel to xy, we will get TL of VB in top view at angle Ø. b’
b1’ 65
T.L. v
30º a’ X Y h’
45º
38º 12 15 b1 h a v’
b3’
Ans: TL=75mm
θ = 38º
HT is 12 mm in front of VP b b3

36. Q 33. The front view of a line AB measures 70 mm and makes an angle of 45º with xy. A is in the H.P. and V.T. of the line is 15 mm below the H.P. The line is inclined at 30º to the V.P. draw the projections of AB, and find its true length and inclination with the H.P. Also locate its H.T.
Given, FV=70 mm, α = 45º, A is in HP, VT is 15mm ↓HP Ø = 30º,
To find,
TL, θ & HT
Note: Here we do not know position of a and b in top view, but we know v’ and v. So for time being we consider the line as v’b’ instead of a’b’. Now if we make v’b’(FV) parallel to xy, we will get TL of VB in top view at angle Ø. b’
b1’
T.L. h’ v X
30º a’ Y
45º
38º 15 b1 a h v’
b3’
Ans: TL=81mm
θ = 38º b b3

37. Q34. A line AB measures 100 mm. The projectors through its V. T
Q34. A line AB measures 100 mm. The projectors through its V.T. and the end A are 40 mm apart. The point A is 30 mm below the H.P. and 20 mm behind the V.P. The V.T. is 10 mm above the H.P. Draw the projections of the line and determine its H.T. and inclinations with the H.P. and the V.P.
Given, TL=100 mm, a0v= 40 mm, A: 30mm↓ H.P., 20mm ← V.P VT is 10mm ↑HP
To draw FV & TV and find, HT,θ, & ø b b2
100 a
20º v’ h 20 10 5 X Y v a3 a0 h’ 30
a3’ 40 a’
42º
θ = 42º
ø = 20º
100
HT is 5 mm behind the VP
b3’ b’

38. Q9. The front view of a line AB measures 65 mm and makes an angle of 45º with xy. A is in the H.P. and V.T. of the line is 15 mm below the H.P. The line is inclined at 30º to the V.P. draw the projections of AB and find its true length and inclination with the H.P. Also locate its H.T.
Q10. A line AB is in the first quadrant. Its end A and B are 20 mm and 60 mm in front of the V.P. respectively. The distance between the end projectors is 75 mm. The line is inclined at 30º to the H.P. and its H.T. is 10 mm above the xy. Draw the projections of AB and determine its true length and the H.T.
Q33. The front view of a line AB measures 70 mm and makes an angle of 45º with xy. A is in the H.P. and V.T. of the line is 15 mm below the H.P. The line is inclined at 30º to the V.P. draw the projections of AB, and find its true length and inclination with the H.P. Also locate its H.T.
Q34. A line AB measures 100 mm. The projectors through its V.T. and the end A are 40 mm apart. The point A is 30 mm below the H.P. and 20 mm behind the V.P. The V.T. is 10 mm above the H.P. Draw the projections of the line and determine its H.T. and inclinations with the H.P. and the V.P.
Q10.9. A line PQ 75 mm long, has its end P in the V.P. and the end Q in the H.P. The line is inclined at 30 to the H.P. and at 60 to the V.P. Draw its projections.

39. Q10. 19 A line AB, inclined at 40º to the V. P
Q10.19 A line AB, inclined at 40º to the V.P. has its end 50mm and 20mm above the H.P. the length of its front view is 65mm and its V.T. is 10mm above the H.P. determine .the true length of AB its inclination with the H.P. and its H.T.
B1’
Step1: For solving the problem by trapezoidal method, draw a line at 40º(Ø) from VT’. Then draw perpendiculars from a’ and b’ on this line.
Given,
Ø = 40º, A is 20mm↑HP, B is 50 mm ↑ HP, FV=65mm, VT is 10mm ↑ HP
To find,
TL, θ & HT
Step2: Then draw projectors from a’ and b’ and mark the distance of b’B1’ on the projector of b’ below XY. Similarly mark the distance a’A1’ on the projector of a’ below XY b’
A1’ 65 50 a’
40º v’ h
21º 20 10 X Y h’ v a
Ans: A1’B1’=TL=85mm
Ans:HT is 17 mm behind VP
Ans:θ = 21º b

40. Q10. 25 The straight line AB is inclined at 30º to H. P
Q10.25 The straight line AB is inclined at 30º to H.P., while its top view at 45º to XY line. The end A is 25 mm in front of the V.P. and it is below the H.P. The end B is 75 mm behind the V.P. and it is above the H.P. Draw the projections of the line when its V.T. is 40 mm below the H.P. Find the true length of the portion of the straight line which is in the second quadrant and locate its H.T..
Given, θ=30º , β = 45º, A is 25mm →V.P. & ↓ H.P B is 75mm ← V.P. & ↑ H.P V.T is 40 mm ↓ H.P.
To draw/find, F.V. and T.V, TL of portion in second quadrant H.T. b h b2
HBis the portion of the line in II quadrant as its FV and TV are above XY line 75 b’
b2’
b1’ 42 v h’ b1 X Y 25 40 a
β=45º
Ans: HT= 49 mm behind the V.P. v’
θ=30º
TL of HB=42mm a’

41. Q10.26 The front view of a line PQ makes an angle of 30º with XY line (ground line). The H.T. of the line is 45 mm behind the V.P. while its V.T. is 30 mm above H.P. The end P of the line is 10 mm below H.P. and the end Q is in the first quadrant. The line is 150 mm long. Draw the projections of the line and determine the true length of the portion of the line in second quadrant. Also find the angle of the line with H.P. and V.P.
Given, α = 30º, H.T.: 45 mm←VP, in the HP V.T.: 30 mm↑HP. In the VP P:10 mm ↓HP Q; in first quadrant. TL=150 mm
To draw/find, F.V. and T.V, TL of portion in second quadrant θ. ø
HV is the portion of the line in II quadrant as its FV and TV are above XY line p v1 q’ h
q1’ v’ 45 75
v1’
150 30
v3’
θ=24º X Y
α=30º
ø=37º 10 h’ v v3 p’
Ans: θ=24º
TL of HV=75 mm q
ø=37º

42. Q10.27 The end P of a line PQ 130 mm long, is 55 mm in front of the VP. The H.T. of the line is 40 mm in front of the V.P. and the V.T. is 50 mm above the HP. The distance between H.T. and V.T. is 110 mm. Draw the projections of the line PQ and determine its angles with the H.P. and the V.P.
Given, T.L.= 130 mm, P:55 mm →V.P., H.T.: 40 mm→V.P. , V.T.: 50 mm↑H.P., h’v = 110 mm
To draw/find, F.V. and T.V,θ. ø
110 v’
v1’ q’
q1’ 50
130 h’ v X Y
θ=23º q p’ q2 40 55
130 h
ø=19º p v1

43. Q6. The top view of a 75mm long line CD measures 50 mm
Q6. The top view of a 75mm long line CD measures 50 mm. C is 50 mm in front of the VP & 15mm below the HP. D is 15 mm in front of the VP & is above the HP. Draw the FV of CD & find its inclinations with the HP and the VP. Show also its traces.
Given,
TL = 75 mm, TV =50 mm, C is 15mm ↓ HP & 50 mm → VP, D is 15 mm → VP
To draw,
FV & to find θ & Ø
To mark HT & VT v’
Hint 1: Cut an arc of 50 mm from c on locus of D to get the TV of the line d’
d1’
Hint 2: Make TV (cd), // to XY so that FV will give TL 75 X h’ Y v 15 d
d2’ d2 c’
θ=48º
Ans: θ=48º 50
Locus of D 50 75
Ans: Ø=28º h c d1
Ø=28º

44. Q10. 10 A line PQ 100 mm long is inclined at 30º to the H. P
Q 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 the third quadrant and Q is in the first quadrant.
Given,
TL = 100, θ = 30º,Ø=45º, Mid point M is 20mm↑HP & in the VP End P in third quadrant & End Q in first quadrant
To draw,
FV & TV q’
q1’ p2 p 50
p2’ m’
q2’ 50
30º 50 20 p1 X
45º Y q1 m
p1’ p’ 50 q q2

45. Problem 3: The front view of a 125 mm long line PQ measures 75 mm while its top view measures 100 mm. Its end Q and the mid point M are in the first quadrant. M being 20 mm from both the planes. Draw the projections of line PQ.
Given,
TL = 125mm, FV=75mm, TV=100mm, Mid point M is 20mm↑HP &20mm in front of the VP End Q and mid point M are in first quadrant
To draw,
FV & TV q’
q1’
62.5
37.5 p m’ 20 X
37.5 Y
p1’ 20 p’ m q1 p1 50 q

46. VP b b2 b a’ b2’ a’ a” HP X Y b1 a a b’ b’ b1’ b”
Q8:A line AB 65 mm long has its end A in the H.P. & 15 mm in front of the V.P. The end B is in the third quadrant. The line is inclined at 30 to the H.P. and at 60 to the V.P. Draw its projections.
Given,
TL =65mm,
A: in the HP & 15mm →VP
B: in third quadrant
θ = 30º,
Ø=60º
To draw,
FV & TV VP b b2 b 65 15 a’
b2’ a’
X Y a” HP X Y
30º
30º 15
60º b1
60º 65 a a b’ b’
b1’ b”

47. Q11:A room is 4.8 m X 4.2 m X 3.6 m high, determine graphically the distance between a top corner and the bottom corner diagonally opposite to it.
Taking scale 1cm=0.5m
4.8m b’
3.6m B
3.6m
b2’ b X Y
4.2m a’ b2
4.2m
T.L. A
4.8m a
Ans. T.L.= 7.32m

48. PROBLEM 12 :- Two oranges on a tree are respectively 1. 8 m and 3
PROBLEM 12 :- Two oranges on a tree are respectively 1.8 m and 3.0 m above ground, and those are 1.2 m & 2.1 m from a 0.3 m thick wall, but on the opposite sides of it. The distance between the oranges, measured along the ground and parallel to the wall is 2.7m. Determine the real distance between the oranges.
As the dimensions are in metres, we will use a scale 1cm = 0.5m or 1:50
0.3m
T.V. b’
b1’
0.3 m
T.L. a’
3.0m B
1.8m
2.1m
1.2m X Y
1.2m
2.1m A
1.8m
3.0m a b1
2.7m
2.7 m
F.V. b
Ans. T.L.= 4.65m