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# ES 1: ENGINEERING DRAWING

Geometric Construction 2

I. Circles/Arcs Construction
II. Triangle Construction
ES 1: ENGINEERING DRAWING
Circles/Arcs Problems
A) Determining/locating center of a
circle/arc.
B) Drawing a circle/arc thru three given
points
C) Rectifying an arc.
D) Laying out a given length on an arc.

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ES 1: ENGINEERING DRAWING
DETERMINING THE CENTER OF AN ARC/
CIRCLE

Given: an arc

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ES 1: ENGINEERING DRAWING
DETERMINING THE CENTER OF AN ARC

## 1. Draw a chord, and label its endpoints as A

and B.

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ES 1: ENGINEERING DRAWING
DETERMINING THE CENTER OF AN ARC

## 2. Draw another chord, connected to point B

with the other endpoint labeled as C.

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ES 1: ENGINEERING DRAWING
DETERMINING THE CENTER OF AN ARC

## 3. Using the method outlined for bisecting a

line, bisect chords AB and BC.
Center at B
Center at B
than one-half AB.
than one-half B
AB

Center at C
than one-half
C
BC.

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ES 1: ENGINEERING DRAWING
DETERMINING THE CENTER OF AN ARC

## 4. Locate point X where the two extended

bisectors meet. Point X is the exact center of
the circle.
A

C
X

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ES 1: ENGINEERING DRAWING
DRAWING A CIRCLE/ARC THROUGH
THREE POINTS

## Given: Three points in space at random: A, B,

and C.

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ES 1: ENGINEERING DRAWING
DRAWING A CIRCLE/ARC THROUGH
THREE POINTS
1. With straight lines, lightly connect points A
to B, and B to C.

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ES 1: ENGINEERING DRAWING
DRAWING A CIRCLE/ARC THROUGH
THREE POINTS
2. Using the method outlined for bisecting a
line, bisect lines AB and BC.

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ES 1: ENGINEERING DRAWING
DRAWING A CIRCLE/ARC THROUGH
THREE POINTS
3. Locate point X where the two extended
bisectors meet. Point X is the exact center of
the arc or circle.

C
X

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ES 1: ENGINEERING DRAWING
DRAWING A CIRCLE/ARC THROUGH
THREE POINTS
4. Using X as center and radius equal to XA (or XB
or XC), draw a/an circle/arc. The circle/arc
drawn passed through the three given points.

C
X

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ES 1: ENGINEERING DRAWING
Problem
Circumscribe a circle about the given triangle
below, i.e, draw a circle such that it will contain
the vertices of the triangle.

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ES 1: ENGINEERING DRAWING
RECTIFYING AN ARC

Given: an arc

## Rectify – to determine (the length of a curve)

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ES 1: ENGINEERING DRAWING
RECTIFYING AN ARC
1. Find the center of the arc using (arbitrarily
chosen) chords AB and BC (refer to procedure for
finding the center of a circle).

A
B

X
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ES 1: ENGINEERING DRAWING
RECTIFYING AN ARC
2. Form the longest chord (12), bisect it and locate
its midpoint O.
3. Connect the center to either of the arc’s
endpoints, say point 2.
A
1
B

O
C
2

X
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ES 1: ENGINEERING DRAWING
RECTIFYING AN ARC
4. Extend the chord 12 past the drawn radius X2.
The length of the extension must be equal to O2
or one-half of the chord 12.

A
1
B

O
C
2
Line O2 = Line 23
X
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ES 1: ENGINEERING DRAWING
RECTIFYING AN ARC
5. Draw tangent line at point 2 (Draw a line
perpendicular to line segment X2 at point 2).

A
1
B

O
C
2

X
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ES 1: ENGINEERING DRAWING
RECTIFYING AN ARC
6. Using point 3 as center and radius equal to line 13, strike
an arc intersecting the tangent line at point 4.
4

A
1
B

O
Line 24 is the rectified
C length of arc 12.
2

X
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ES 1: ENGINEERING DRAWING

## Laying out a Given Length

Along an Arc,
or,
Constructing an arc of specified length
given a portion of the arc.

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ES 1: ENGINEERING DRAWING
SETTING OFF A GIVEN LENGTH ALONG
AN ARC
Given: Line AB and an arc JF

A B

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ES 1: ENGINEERING DRAWING
SETTING OFF A GIVEN LENGTH ALONG
AN ARC
1. Find the center of the given arc (see steps in
finding the center of an arc).

X 22
ES 1: ENGINEERING DRAWING
SETTING OFF A GIVEN LENGTH ALONG
AN ARC
2. Connect the center to either of the arc’s
endpoints, say F.
3. Draw tangent at F.

X 23
ES 1: ENGINEERING DRAWING
SETTING OFF A GIVEN LENGTH ALONG
AN ARC
3. Lay out the length of line AB in the tangent line (recall
steps in transferring a line). Label the length as A’F, i.e.
A’F =AB.

A’
Length of line AB

X 24
ES 1: ENGINEERING DRAWING
SETTING OFF A GIVEN LENGTH ALONG
AN ARC
4. Divide line A’F into four equal segments. Label the
points at equal intervals as 1, 2, and 3. [Actually,
bisect A’F with its midpoint labeled 2, then bisect 2F
with its own midpoint labeled 1.]
A’

3
J
2

X 25
ES 1: ENGINEERING DRAWING
SETTING OFF A GIVEN LENGTH ALONG
AN ARC
4. Using point 1 as center and radius equal to line
1A’, strike an arc intersecting the given arc. Label
the intersection as C.
A’

3
C
J
2

X 26
ES 1: ENGINEERING DRAWING
SETTING OFF A GIVEN LENGTH ALONG
AN ARC
Arc FC is approximately equal to line AB.

A’

3
C
J
2

X 27
ES 1: ENGINEERING DRAWING
TRIANGLE PROBLEMS

## 1. Draw a triangle given its 3 sides.

2. Draw a right triangle given its
hypotenuse and a leg.
3. Inscribe a circle in a given triangle.

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ES 1: ENGINEERING DRAWING
DRAWING A TRIANGLE GIVEN THE
LENGTH OF THE THREE SIDES
• Given: length of three sides
Side A

1 2
Side B
1 3
Side C
2 3

Location of a triangle’s
side

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ES 1: ENGINEERING DRAWING
DRAWING A TRIANGLE GIVEN THE
LENGTH OF THE THREE SIDES
1. Layout Side A in the desired position.

1 2

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ES 1: ENGINEERING DRAWING
DRAWING A TRIANGLE GIVEN THE
LENGTH OF THE THREE SIDES
2. Using endpoint 1 of side A as center and the
length of side B as radius, draw an arc above
side A.

Center at 1

1 2

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ES 1: ENGINEERING DRAWING
DRAWING A TRIANGLE GIVEN THE
LENGTH OF THE THREE SIDES
3. Using endpoint 2 of side A and the length of
side C as radius, draw a second arc
intersecting the first arc at point 3.
3

Center at 2

1 2

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ES 1: ENGINEERING DRAWING
DRAWING A TRIANGLE GIVEN THE
LENGTH OF THE THREE SIDES
4. Connecting point 3 with points 1 and 2
establishes triangle 123.

1 2

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ES 1: ENGINEERING DRAWING
DRAWING A RIGHT TRIANGLE GIVEN ITS
HYPOTENUSE AND A LEG
• Given: - length of one side
- length of hypothenuse
C B

A Hypotenuse B
The key idea
here is the
application of
the inscribed
angle
Location of the triangle
theorem.

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ES 1: ENGINEERING DRAWING
DRAWING A TRIANGLE GIVEN THE
HYPOTENUSE AND A GIVEN LEG
1. Using the length of the given hypotenuse as
diameter, draw a semi-circle.

A B

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ES 1: ENGINEERING DRAWING
DRAWING A TRIANGLE GIVEN THE
HYPOTENUSE AND A GIVEN LEG
2. Using one endpoint of the hypotenuse as
center and the length of the side BC as
radius, draw an arc intersecting the semi-
circle at point C.

A B

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ES 1: ENGINEERING DRAWING
DRAWING A TRIANGLE GIVEN THE
HYPOTENUSE AND A GIVEN LEG
3. Connecting point C with endpoints A and B
establishes the desired Right Triangle ABC

A B

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ES 1: ENGINEERING DRAWING
INSCRIBING A CIRCLE IN TRIANGLE

## Given triangle ABC, draw a circle which is

tangent to all 3 sides of the triangle.

## The key idea

here is that
the angle A
bisector of
any angle
contains the
center of the B
inscribed-to-
be circle, and
that the circle
is tangent to
all 3 sides.
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ES 1: ENGINEERING DRAWING
INSCRIBING A CIRCLE IN TRIANGLE ABC

## 1. Bisect angle A by line AD extending this

beyond the middle of the triangle.

A
Center at point 2 with 2
Center at point 1 with
B
D

C 39
ES 1: ENGINEERING DRAWING
INSCRIBING A CIRCLE IN TRIANGLE ABC

at point O.

3
B E
4 D O

## Center at point 4 with

C 40
ES 1: ENGINEERING DRAWING
INSCRIBING A CIRCLE IN TRIANGLE ABC

## 3. Draw line FG through point O perpendicular

to side AB at point H.

A
H

B E
D O

C 41
ES 1: ENGINEERING DRAWING
INSCRIBING A CIRCLE IN TRIANGLE ABC

## 4. Using point O as center and radius equal to

OH, draw the desired circle.

A
H

B E
D O

C 42
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