1. Chapter 2
Applied geometry

2. Applied Geometry Bisecting Parallel line Inclined line Tangent arc
Perpendicular
line
Tangent line
Contents

3. Bisecting a line and an angle
Applied Geometry
Contents

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

5. To bisect a given angle Explanations Given Ar1
1. Swing an arc of any radius
whose centers at the vertex. r2
2. Swing the arcs of any radius from the intersection points
between the previous arc and
the lines. r2
3. Draw the line.
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Applied Geometry

6. Drawing a parallel line
Applied Geometry
Contents

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

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

9. Drawing a perpendicular line
Applied Geometry
Contents

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

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

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

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

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

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

16. Drawing an inclined line
Applied Geometry
Contents

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

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

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

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

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

22. Tangent line to a given arc (or circle)
Case 2 : A given point lies outside an arc
1st method
2nd method
Given
Given C C
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Applied Geometry

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

24. Key Concept To draw a tangent arc (of a specified radius, R),
it is necessary to locate
1. its center, C.
It places outside a line for a distance
equal to a radius of an arc. R R
2. the start and end points
(or tangent points) of the arc. R
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.

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

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

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

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

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

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

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

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

33. 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.

34. Example
Drawing steps VDO