Geometrical Constructions and Tangents Creation
ب
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Bisect a Line (or Arc)
1. Draw two arcs of any 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.
(
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3. Locate the midpoint.
Bisect an Angle
2. Draw the arcs of any radius from the intersection
points between the previous arc and the lines.
3. Draw the line.
1. Draw an arc of any radius whose centers at the vertex.
(not to scale)
Dividing a Line into a Number of Equal
Parts
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arc radius r
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Construct an Arc Tangent to a Line
and an Arc
Given line
AB
and arc
CD.
Strike arcs
R
1
(given radius).
Draw construction arc parallel to
given arc, with center
O.
Draw construction line parallel to
given line
AB.
From intersection
E
, draw
EO
to get
tangent point
T
1
, and drop
perpendicular to given line to get
point of tangency
T
2
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with center
E.
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1
Geometric Constructions
R=44
R=18
R=16+19=35
R=16
R=8
R=22-5=17
R=22+5=27
R=19
R=18+22=40
R=44+22=66
R=19
R=19
END
Construct a Hexagon
given distance Across Flats (Circumscribed)
Given distance across the
flats of a hexagon, draw
centerlines and a circle
with a diameter equal to
the distance across flats
With parallel edge and
30
°
– 60
°
triangle, draw
the tangents
Construct a Hexagon
given distance Across Corners (Inscribed)
Given distance
AB
across the corners, draw a circle
with
AB
as the diameter
With
A
and
B
as centers and
the same radius, draw arcs
to intersect the circle at
points
C
,
D
,
E
, and
F
Connect the points to
complete the hexagon
Construct an Octagon
Across Flats (Circumscribed)
Given the distance across the flats,
draw centerlines and a circle with a
diameter equal to the distance across
flats
1
2
3
4
5
6
7
8
With a parallel edge and 45
triangle, draw lines tangent to
the circle in the order shown to
complete the octagon
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(
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)
Given the distance across the
corners, draw centerlines
AB
and
CD
and a circle with a
diameter equal to the distance
across corners
Connect the points to
complete the octagon
With the T-square and 45
°
triangle, draw diagonals
EF
and
GH
General Method to Draw any Polygon
Note:
See Page (29) of Your Textbook
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Note:(Ellipse Drawing)
Also see Figure(3.24), page(43)
from textbook
END
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In this collection, you will find detailed explanations and visual guides on various geometrical constructions such as bisecting lines and angles, dividing lines into equal parts, drawing perpendicular lines, constructing tangents to circles, and filleting sharp corners. Explore step-by-step instructions and diagrams to enhance your geometric skills and understanding.
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Presentation Transcript
Bisect a Line (or Arc) 1. Draw two arcs of any 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. 3. Locate the midpoint. Given A A r1 r1 B B (not to scale)
Bisect an Angle 1. Draw an arc of any radius whose centers at the vertex. 2. Draw the arcs of any radius from the intersection points between the previous arc and the lines. 3. Draw the line. A (not to scale) Given A B r2 r1 B r2 C C
To draw the line perpendicular to a given line from a point not on the line Adjacent-sides method +C
To draw the line perpendicular to a given line from a point not on the line Adjacent-sides method +C Repeat
To draw the line perpendicular to a given line from a point not on the line Using compass + C r2 D r2 A r1 B Repeat
Tangents- Construction Straight Line Tangents to a Circle from an External point
Tangents- Construction Common Parallel Straight Line Tangents to Two Circles of Radius R and r r R
Tangents- Construction Common Cross Straight Line Tangents to Two Circles of Radius R and r r R
FILLET AND ROUND Round Sharp corner Fillet Round
Tangents- Construction Circular Tangent of Radius R Between a Point to a Straight Line R R R R
Tangents- Construction Circular Tangent of Radius R Between Two Straight Lines at an Angle R R R R R
To draw an arc of given radius tangent to two lines Given arc radius r T.P.1 T.P.2
Tangents- Construction Internal Circular Tangent of Radius R Between a Straight Line and a Circle of Radius r R+r r R R R R R R
Tangents- Construction External Circular Tangent of Radius R Between a Straight Line and a Circle of Radius r R-r r R-r R R R R R R
FILLET AND ROUND To draw the arc, we must find the location of the center of that arc. How do we find the center of the arc?
Draw an arc of given radius tangent to two perpendicular lines Given arc radius r r r
Draw an arc of given radius tangent to two perpendicular lines Given arc radius r center of the arc Starting point Ending point
Construct an Arc Tangent to a Line and an Arc Given line AB and arc CD. Strike arcs R1(given radius). Draw construction arc parallel to given arc, with center O. Draw construction line parallel to given line AB. From intersection E, draw EO to get tangent point T1, and drop perpendicular to given line to get point of tangency T2. Draw tangent arc R from T1 to T2with center E. C E T1 R1 O A B T2 D
When circle tangent to other circle Tangent point R1 C1 R2 C2 The center of two circles and tangent point lie on the same straight line !!!
Draw a circle tangent to two circles I Given C+ + C2 + C1 Example
Draw a circle tangent to two circles I GivenTwo circles and the radius of the third circle = R + C2 + C1
Draw a circle tangent to two circles I GivenTwo circles and the radius of the third circle = R R + R2 center of the arc R + R1 C R R2 R1 + C2 + C1 Repeat
When circle tangent to other circle Tangent point R1 R2 C1 C2 The center of two circles and tangent point must lie on the same straight line !!!
Draw a circle tangent to two circles II Given + C2 + C1 C+ Example
Draw a circle tangent to two circles II GivenTwo circles and the radius of the third circle = R + + C2 C1
Draw a circle tangent to two circles II GivenTwo circles and the radius of the third circle = R R R2 R1 + + C2 C1 R R2 C R R1 Repeat
Draw a circle tangent to two circles III GivenTwo circles and the radius of the third circle = R R2 R1 + C2 C1 + R R1 R + R2 C
Tangents- CW 1 85 85
R=18+22=40 R=44+22=66
Construct a Hexagon given distance Across Flats (Circumscribed) Given distance across the flats of a hexagon, draw centerlines and a circle with a diameter equal to the distance across flats With parallel edge and 30 60 triangle, draw the tangents
Construct a Hexagon given distance Across Corners (Inscribed) Given distance AB across the corners, draw a circle with AB as the diameter C D With A and B as centers and the same radius, draw arcs to intersect the circle at points C, D, E, and F A B Connect the points to complete the hexagon F E
Construct an Octagon Across Flats (Circumscribed) Given the distance across the flats, draw centerlines and a circle with a diameter equal to the distance across flats 1 5 7 With a parallel edge and 45 triangle, draw lines tangent to the circle in the order shown to complete the octagon 3 4 8 6 2
Construct an Octagon Across Corners (Inscribed) Given the distance across the corners, draw centerlines AB and CD and a circle with a diameter equal to the distance across corners C G E B A With the T-square and 45 triangle, draw diagonals EF and GH H F D Connect the points to complete the octagon
General Method to Draw any Polygon Note: See Page (29) of Your Textbook
Draw an Approximate Ellipse GivenMajor and minor axes
Draw an approximate ellipse GivenMajor and minor axes J E C F A B G O K D H Repeat
Note:(Ellipse Drawing) Also see Figure(3.24), page(43) from textbook
