The document provides instructions for using drawing tools and applying geometry concepts. It covers preparing tools like paper, pencils, and compasses. It then explains how to use tools to draw various shapes, including straight lines, arcs, circles, and inclined lines. It also covers applying geometry concepts like bisecting lines, drawing parallels and perpendiculars, and drawing tangent lines and curves. The final section outlines general problem solving steps.

1. Chapter 2
Using drawing tools
& applied geometry

2. Contents
Applied Geometry
(or geometrical constructions)
Using of tools
Preparation of tools
Problem solving steps

3. Preparation of
Tools
Contents

4. 1. Paper1. Paper
2. Pencils2. Pencils
3. Compass3. Compass
Fastening a sheet to a drafting board
Sharpening the lead
Sharpening the lead
Tools to be prepared

5. Paper
1. Place a paper close to the left edge of a
table where a drafter can work conveniently.
3. Move the paper until its lower edge lies
close to the top edge of a T-square.
4. Align the top edge of the paper with
T-square blade.
5. Attach the paper’s corners with tape.
6. Move T-square down to smooth the paper.
7. Attach the remaining paper’s corners
with tape.
2. Place a T-square.

6. 1. Remove the wood with penknife while expose a lead
about 8-10 mm.
2. Polish the lead into a conical shape with a sandpaper.
3. Clean the lead with tissue paper.
Pencil

7. Compass
2. Adjust the needle and the lead so that the tip of
the needle extends slightly more than the lead.
1. Sharpen the lead with a sandpaper.
needle lead

8. Using the Tools
Contents

9. Straight line
Arc, Circle
4. Circle template
1. T-square
2. Triangles
3. Compass
Tools Shape to be drawn
Function of the tools
T-square and triangles can be used together to draw
an inclined line with 15o
increment, i.e. 15o
, 30o
, 45o
,
60o
, 75o
, 90o
, 105o
, 120o
, 135o
, 150o
, 165o
, 180o
etc.

10. To keep your drawing clean
Do Don’t

11. Using a compass
1. Locate the center of the circle to be drawn.
Draw two intersecting lines.
2. Adjust the distance between a needle and a lead
to be a radius of the circle.
3. Set the needle point at the circle’s center.

12. 4. Start circle. Apply enough pressure to the needle,
holding the compass handle between thumb and index
fingers.
5. Complete circle. Revolve the handle clockwise.
Using a compass

13. 1. Draw two perpendicular lines that pass through center of
a circle to be drawn.
Construction line
Visible line
2. Align all markings on template with the center lines.
3. Tracing the circle.
Given
Center of a circle to be drawn
Using a template

14. A
B
Draw a line through the given points
1. Place the pencil tip at one of the
given points.
2. Place the triangle against the
pencil tip.
3. Swing the triangle around the
pencil tip until its edge aligns with
the second point.
4. Draw a line.
Given
play
Explanations

15. Draw a horizontal line
1. Press the T-square head against the left edge
of the table.
2. Smooth the blade to the right.

16. Draw a horizontal line
3. Lean the pencil at an angle about 60o
with the paper
in the direction of the line and slightly “toed in”.
4. Rotate the pencil slowly while moving the pencil
from left to right.

17. Draw a horizontal line
5. Move T-square up or down to draw another
horizontal line.

18. Draw a vertical line
1. Set T-square as before.
Place any triangle on T-square edge.
2. Use your left hand to hold both T-square and triangle
in position.

19. Draw a vertical line
3. Lean the pencil to the triangle.
4. Draw the line upward while rotating the pencil slowly.

20. 1. Place 30o
-60o
triangle on the T-square edge and press
them firmly against the paper.
2. Draw the line in the direction as shown below.
Draw a line at 30o
with horizontal

21. Draw a line at 45o
with horizontal
2. Draw the line in the direction as shown below.
1. Place 45o
triangle on the T-square edge and press
them firmly against the paper.

22. Draw a line at 60o
with horizontal
1. Place 30o
-60o
triangle on the T-square edge and press
them firmly against the paper.
2. Draw the line in the direction as shown below.

23. Draw a line at 15o
with horizontal
1
2
+-30o
45o = 15o
CCW
+60o
(-45o
) = 15o
CCW

24. Draw a line at 75o
with horizontal
1
2
+30o
45o = 75o
CCW
+45o
30o = 75o
CCW

25. Draw a line at 105o
with horizontal
1
2
+60o
45o = 105o
CCW
+45o
60o = 105o
CCW

26. Practice by Yourself
Arrange the triangles to draw a line at
a) 120o
b) 135o
c) 150o
with a horizontal.

27. Applied Geometry
Contents
r
Bisecting Parallel line
Perpendicular
line
Inclined line
Tangent line
Tangent arc

28. Bisecting a line
and an angle
Applied Geometry Contents

29. To bisect a given line
1. Swing two arcs having a radius
greater than half-length of the
line with the centers at the
ends of the line.
2. Join the intersection points of
the arcs with a line.
A
B
r1
r1
3. Locate the midpoint.
Given
play
Explanations
Applied Geometry

30. To bisect a given angle
2. Swing the arcs of any radius
from the intersection points
between the previous arc and
the lines.
3. Draw the line.
1. Swing an arc of any radius
whose centers at the vertex.
r1
A
B
C
r2
r2
Given
play
Applied Geometry
Explanations

31. Drawing a parallel line
Applied Geometry Contents

32. +
C
Line parallel to a given line
through a given point
Given
play
Applied Geometry
Explanations
3. Slide the first triangle until its
edge passes through the
given point.
1. Line an edge of a triangle up
to a given line.
2. Support the triangle with
another one.
4. Draw a line.

33. r
Line parallel to a given line
at a given distance
Given
play
Explanations
3. Draw a line parallel to a given
line and tangent to the arc.
1. Choose a convenient point on
a given line.
2. Use that point as center of an
arc with a radius equal to a
given distance.
Applied Geometry
r

34. Drawing a perpendicular line
Applied Geometry Contents

35. +
C
Line perpendicular to a point in a line
Revolve method
Given
play
Explanations
3. Flip the first triangle and slide
until its edge passes through
the given point.
1. Line an opposite edge of a 45o
triangle up to a given line.
2. Support the triangle with another
one.
4. Draw a line.
Applied Geometry

36. +
C
Adjacent-sides method
Given
play
Line perpendicular to a point in a line
Explanations
3. Slide the first triangle until
another adjacent edge passes
through the given point.
1. Line an adjacent edge of a 45o
triangle up to a given line.
2. Support the triangle with another
one.
4. Draw a line.
Applied Geometry

37. r1
r2
+
C
r2
r2 > r1
A
B
D
Given
play
Line perpendicular to a point in a line
Compass method
Explanations
1. Use a given point as center,
draw the arc with any radius.
2. Bisect the distance between the
intersection points between an
arc and a given line.
3. Draw a line.
Applied Geometry

38. Line perpendicular to a given line
through a point outside the line
+
C
Given
play
Adjacent-sides method
Explanations
3. Slide the first triangle until
another adjacent edge passes
through the given point.
1. Line an adjacent edge of a 45o
triangle up to a given line.
2. Support the triangle with another
one.
4. Draw a line.
Applied Geometry

39. Given
play
r2
+ C
r2
r1
B
A
D
Line perpendicular to a given line
through a point outside the line
Compass method
Explanations
1. Use a given point as a center,
draw the arc with any radius that
intersect the given line.
2. Bisect the distance between the
intersection points between an
arc and a given line.
3. Draw a line.
Applied Geometry

40. Practice by Yourself
Draw a line perpendicular to a given line and pass
through a point lies outside using revolved method.
Applied Geometry

41. Drawing an inclined line
Applied Geometry Contents

42. Line making 15o
with a given line
through a given point
+
C
+
C
Given
play
Given
play
Applied Geometry

43. Line making 30o
with a given line
through a given point
+
C
+
C
Given
play
Given
play
Applied Geometry

44. Line making 75o
with a given line
through a given point
+
C
Given
play
Given
play
+
C
Applied Geometry

45. Drawing a Tangent line
to an arc (or a circle)
Applied Geometry Contents

46. C
Tangent line to a given arc (or circle)
play
Case 1 : A given point lies on an arc
Applied Geometry
Given Explanations
3. Slide the first triangle until
another adjacent edge passes
through the given point.
1. Line an adjacent edge of a 45o
triangle up to the center of an
arc and a given given.
2. Support the triangle with another
one.
4. Draw a line.

47. Tangent line to a given arc (or circle)
play
C
play
C
Applied Geometry
Case 2 : A given point lies outside an arc
Given Given1st
method 2nd
method

48. Drawing a tangent curve
to the given lines
Applied Geometry Contents

49. Key Concept
1. its center, C.
To draw a tangent arc (of a specified radius, R),
it is necessary to locate
2. the start and end points
(or tangent points) of the arc.
It places outside a line for a distance
equal to a radius of an arc.
It lies on a given line in the way
that the line passing through this
point and the center of an arc be
perpendicular to a given line.
R
R
R

50. Tangent arc to the given lines
R
R
play
Given 1. Locate the center of an arc
Applied Geometry
Continue

51. TP.1
TP.2
Tangent arc to the given lines
2. Locate the tangent points
Applied Geometry
Replay

52. Drawing a tangent curve
to the given curves
Applied Geometry Contents

53. Tangent point lies on the line passes through the centers
of each arc (or circle).
Key Concept
R1
R2
R3

54. Tangent arc to a given arcs (or circles)
C2C1
C
C1
C2
C
1. its center, C.
To draw a tangent arc (of a specified radius, R),
it is necessary to locate
Case 1 : External Case 2 : Internal
2. the start and end points (or tangent points) of the arc.
R1
R R
R2 R1
R-R1 R-R2
R2

55. + +C1
C2
R + R1
R + R2
R1
R2
C
External tangent arc
R
play
Applied Geometry
Given

56. + +
C1
C2
R – R2
Internal tangent arc (Type 1)
R – R1
R1
R2
C
R
play
Applied Geometry
Given

57. R + R2
R – R1
C
Internal tangent arc (Type 2)
+ +C1
C2
R1
R2
play
R
Applied Geometry
Given

58. Problem solving steps
1. Calculate the required space.
2. Layout the drawing steps.
3. Match the construction techniques to each drawing step.
4. Start drawing.
Always use a construction line if the information to draw
a line or a curve is incomplete.

59. Example
Drawing steps VDO