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Section 2 Drafting Techniques and Skills Chapter 6 Basic Geometric Constructions

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Permission granted to reproduce for educational use only. 3 © Goodheart-Willcox Co., Inc. Objectives Use manual drafting tools and methods to make geometric constructions. Use CAD commands and methods to make geometric constructions. Draw, bisect, and divide lines. Construct, bisect, and transfer angles.

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Permission granted to reproduce for educational use only. 4 © Goodheart-Willcox Co., Inc. Objectives Draw triangles, squares, hexagons, and octagons. Use special techniques to construct regular polygons. Construct circles and arcs.

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Permission granted to reproduce for educational use only. 5 © Goodheart-Willcox Co., Inc. Geometric Constructions Widely used in drafting. Based on principles of plane geometry. Proper instruments required in manual work. Most efficient methods should be used in CAD work.

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Permission granted to reproduce for educational use only. 6 © Goodheart-Willcox Co., Inc. Constructions for Straight Lines Bisecting lines Drawing parallel lines Drawing perpendicular lines Dividing lines

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Permission granted to reproduce for educational use only. 7 © Goodheart-Willcox Co., Inc. Bisect a Line Bisect 1.Strike arcs using endpoints of Line EF as centers. 2.Use equal radii greater than one-half Line EF. 3.Draw Line GH through arc intersections.

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Permission granted to reproduce for educational use only. 8 © Goodheart-Willcox Co., Inc. Bisect a Line 1.Enter Line command and enter Midpoint object snap. 2.Pick near midpoint of Line AB. 3.Draw line in any direction.

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Permission granted to reproduce for educational use only. 9 © Goodheart-Willcox Co., Inc. Draw a Line Through a Point and Parallel to a Given Line 1.Position T-square and triangle to align with Line AB. 2.Slide triangle to align with Point P. 3.Draw Line CD through Point P.

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Permission granted to reproduce for educational use only. 10 © Goodheart-Willcox Co., Inc. Draw a Line Through a Point and Parallel to a Given Line 1.Enter Line command and pick Point C as first point. 2.Enter Parallel object snap and acquire point along Line AB. 3.Draw Line CD along alignment path.

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Permission granted to reproduce for educational use only. 11 © Goodheart-Willcox Co., Inc. Draw a Line Through a Point and Perpendicular to a Line 1.Strike arcs using arbitrary radius and Point P as center. 2.Strike arcs where arcs intersect Line DE using same radius. 3.Draw Line PH.

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Permission granted to reproduce for educational use only. 12 © Goodheart-Willcox Co., Inc. Draw a Line Through a Point and Perpendicular to a Line 1.Enter Line command and specify Point P as first point. 2.Enter Perpendicular object snap and snap to Line AB to draw line.

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Permission granted to reproduce for educational use only. 13 © Goodheart-Willcox Co., Inc. Divide a Line Into a Given Number of Equal Parts 1.Draw vertical line at Point B (Line BC). 2.Place scale so that 11 equal divisions are between Point A and Line BC. 3.Mark points and project them parallel to Line BC on Line AB.

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Permission granted to reproduce for educational use only. 14 © Goodheart-Willcox Co., Inc. Divide a Line Into Proportional Parts 1.Draw vertical line at Point B (Line BC). 2.Place scale so that nine equal divisions are between Point A and Line BC. 3.Mark proportional divisions and project them parallel to Line BC on Line AB.

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Permission granted to reproduce for educational use only. 15 © Goodheart-Willcox Co., Inc. Divide a Line Into a Given Number of Equal Parts 1.Enter Divide command and select Line AB. 2.Specify number of divisions (seven). 3.Change point style to display markers if needed.

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Permission granted to reproduce for educational use only. 16 © Goodheart-Willcox Co., Inc. Constructions for Angles Bisecting angles Transferring angles Drawing perpendiculars Laying out angles

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Permission granted to reproduce for educational use only. 17 © Goodheart-Willcox Co., Inc. Angle Symbols and Types of Angles

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Permission granted to reproduce for educational use only. 18 © Goodheart-Willcox Co., Inc. Bisect an Angle 1.Strike Arc D using arbitrary radius. 2.Strike arcs from intersections of arcs and the angle. 3.Draw Line AE where new arcs intersect.

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Permission granted to reproduce for educational use only. 19 © Goodheart-Willcox Co., Inc. Bisect an Angle 1.Enter Line command and pick Point A as first point. 2.Enter Mid Between Points object snap. 3.Pick Point B and then Point C to draw the bisector.

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Permission granted to reproduce for educational use only. 20 © Goodheart-Willcox Co., Inc. Transfer an Angle 1.Strike equal arcs from Points A and A. 2.Draw arc from Point E to intersect Point F. 3.Draw arc with same radius from Point E. 4.Draw Line AC.

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Permission granted to reproduce for educational use only. 21 © Goodheart-Willcox Co., Inc. Transfer an Angle

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Permission granted to reproduce for educational use only. 22 © Goodheart-Willcox Co., Inc. Draw a PerpendicularPerpendicular 1.Lay off three equal segments on Line AB. 2.Strike Arc C with radius of four units and center at Point B. 3.Strike Arc D with radius of five units and center at Point A. 4.Draw perpendicular where arcs intersect.

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Permission granted to reproduce for educational use only. 23 © Goodheart-Willcox Co., Inc. Draw a Perpendicular 1.Enter Line command and activate Perpendicular object snap. 2.Pick near end of given line. 3.Specify second point of line. 4.Ortho mode is quicker if given line is horizontal or vertical.

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Permission granted to reproduce for educational use only. 24 © Goodheart-Willcox Co., Inc. Lay Out a Given Angle 1.Align protractor on Line AB to mark desired angle (35 ). 2.Draw line from vertex indicator (Point C) to Point D.

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Permission granted to reproduce for educational use only. 25 © Goodheart-Willcox Co., Inc. Lay Out a Given Angle 1.Enter Line command and use polar coordinates to draw inclined line at 60 .polar coordinates 2.Specify Point C as first point. as second point. 4.Polar tracking and direct distance entry allow for specific distance.

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Permission granted to reproduce for educational use only. 26 © Goodheart-Willcox Co., Inc. PolygonPolygon and Regular Polygon TypesRegular Polygon

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Permission granted to reproduce for educational use only. 27 © Goodheart-Willcox Co., Inc. Construct a Triangle with Three Sides Given 1.Lay off one side (Side C) and arc with radius equal to one side. 2.Lay off arc with radius equal to remaining side. 3.Draw sides.

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Permission granted to reproduce for educational use only. 28 © Goodheart-Willcox Co., Inc. Construct a Triangle with Three Sides Given 1.Draw Side C. 2.Draw circle with radius equal to Arc A and center at endpoint of line. 3.Draw circle with radius equal to Arc B intersecting first circle. 4.Draw remaining sides.

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Permission granted to reproduce for educational use only. 29 © Goodheart-Willcox Co., Inc. Construct a Triangle with Two Sides and Included Angle Given 1.Lay off Side B. 2.Construct angle at one endpoint. 3.Strike arc with radius equal to Side A. 4.Join endpoints of lines.

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Permission granted to reproduce for educational use only. 30 © Goodheart-Willcox Co., Inc. Construct a Triangle with Two Sides and Included Angle Given 1.Draw Side B. 2.Draw Side A using polar coordinates or tracking. 3.Draw remaining side.

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Permission granted to reproduce for educational use only. 31 © Goodheart-Willcox Co., Inc. Construct a Triangle with Two Angles and Included Side Given 1.Lay off Side AB. 2.Construct Angles A and B at ends of Line AB. 3.Extend lines to their intersection.

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Permission granted to reproduce for educational use only. 32 © Goodheart-Willcox Co., Inc. Construct a Triangle with Two Angles and Included Side Given 1.Draw Side AB. 2.Draw inclined lines for Angles A and B using polar coordinates or tracking. 3.Enter Trim command and trim lines to their intersection.

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Permission granted to reproduce for educational use only. 33 © Goodheart-Willcox Co., Inc. Construct an Equilateral TriangleEquilateral Triangle 1.Lay off given side (Side AB). 2.Strike two arcs with radii equal to side and centers at Points A and B. 3.Draw remaining sides where arcs meet.

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Permission granted to reproduce for educational use only. 34 © Goodheart-Willcox Co., Inc. Construct an Equilateral Triangle 1.Draw Side AB. 2.Enter Polygon command and specify three sides. 3.Enter Edge option, select endpoint, and specify side length and direction.

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Permission granted to reproduce for educational use only. 35 © Goodheart-Willcox Co., Inc. Construct an Isosceles TriangleIsosceles Triangle If base and side lengths are known, use triangle method for three side lengths. If two angles and one side length are known, use triangle method for two angles and included side length.

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Permission granted to reproduce for educational use only. 36 © Goodheart-Willcox Co., Inc. Construct a Right TriangleRight Triangle 1.If lengths of two sides forming 90 angle are known, construct perpendicular. 2.Join ends to form hypotenuse. hypotenuse

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Permission granted to reproduce for educational use only. 37 © Goodheart-Willcox Co., Inc. Construct a Right Triangle with Hypotenuse and One Side Given 1.Draw hypotenuse (Line AB). 2.Draw semicircle with radius equal to one- half Line AB. 3.Strike Arc AC equal to other given side. 4.Draw Lines AC and BC.

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Permission granted to reproduce for educational use only. 38 © Goodheart-Willcox Co., Inc. Construct a Right Triangle with Hypotenuse and One Side Given 1.Draw hypotenuse (Line AB). 2.Draw circle with midpoint of Line AB as center and radius equal to one-half Line AB. 3.Draw circle with radius equal to other given side. 4.Draw Lines AC and BC.

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Permission granted to reproduce for educational use only. 39 © Goodheart-Willcox Co., Inc. Construct a Square with One Side Given 1.Lay off one side (Line AB). 2.Draw three perpendicular lines (Lines BC, CD, and AD) to correct length.

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Permission granted to reproduce for educational use only. 40 © Goodheart-Willcox Co., Inc. Construct a Square with One Side Given 1.Draw Side AB. 2.Enter Polygon command and specify four sides. 3.Enter Edge option, select endpoint, and specify side length and direction.

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Permission granted to reproduce for educational use only. 41 © Goodheart-Willcox Co., Inc. Construct a Square with Diagonal Given 1.Draw circle with radius equal to half of diagonal. 2.Draw 45 lines across center. 3.Join points where inclined lines intersect circle.

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Permission granted to reproduce for educational use only. 42 © Goodheart-Willcox Co., Inc. Construct a Square with Diagonal Given 1.Enter Polygon command and specify four sides. 2.Pick center point. 3.Enter Inscribed option and specify center radius as one- half the diagonal.

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Permission granted to reproduce for educational use only. 43 © Goodheart-Willcox Co., Inc. Construct a Regular Pentagon with Side Length GivenPentagon 1.Draw Side AB. 2.Construct 108 angle at Point B at distance of Side AB. 3.Draw remaining sides in same manner.

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Permission granted to reproduce for educational use only. 44 © Goodheart-Willcox Co., Inc. Construct a Regular Pentagon with Side Length Given 1.Enter Polygon command and specify five sides. 2.Enter Edge option and select endpoint. 3.Enter side length.

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Permission granted to reproduce for educational use only. 45 © Goodheart-Willcox Co., Inc. Construct an Inscribed Pentagon with Radius Given 1.Draw circle of given radius and bisect radius. 2.Strike Arcs HD, DJ, and DC. 3.Strike Arcs EA and CB. 4.Draw lines between intersection points.

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Permission granted to reproduce for educational use only. 46 © Goodheart-Willcox Co., Inc. Construct an Inscribed Pentagon with Radius Given 1.Enter Polygon command and specify five sides. 2.Pick center point. 3.Enter Inscribed option and specify given radius.

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Permission granted to reproduce for educational use only. 47 © Goodheart-Willcox Co., Inc. Construct a Hexagon with Distance across Flats GivenHexagon 1.Draw circle equal in diameter to given distance. 2.Draw lines tangent to circle. 3.Draw remaining sides.

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Permission granted to reproduce for educational use only. 48 © Goodheart-Willcox Co., Inc. Construct a Hexagon with Distance across Flats Given 1.Enter Polygon command and specify six sides. 2.Pick center point. 3.Enter Circumscribed option and specify radius as half the given distance.

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Permission granted to reproduce for educational use only. 49 © Goodheart-Willcox Co., Inc. Construct a Hexagon with Distance across Corners Given 1.Draw circle equal in diameter to given distance. 2.Draw 60 lines across center. 3.Join intersection points.

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Permission granted to reproduce for educational use only. 50 © Goodheart-Willcox Co., Inc. Construct a Hexagon with Distance across Corners Given 1.Enter Polygon command and specify six sides. 2.Pick center point. 3.Enter Inscribed option and specify radius as half the given distance.

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Permission granted to reproduce for educational use only. 51 © Goodheart-Willcox Co., Inc. Construct an Octagon with Distance across Flats GivenOctagon 1.Draw circle equal in diameter to given distance. 2.Draw sides tangent to circle.

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Permission granted to reproduce for educational use only. 52 © Goodheart-Willcox Co., Inc. Construct an Octagon with Distance across Corners Given 1.Draw circle equal in diameter to given distance. 2.Draw 45 lines across center. 3.Join intersection points.

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Permission granted to reproduce for educational use only. 53 © Goodheart-Willcox Co., Inc. Construct an Octagon 1.Enter P olygon command and specify eight sides. 2.Pick center point. 3.Use Circumscribed option if distance across flats is known. 4.Use Inscribed option if distance across corners is known.

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Permission granted to reproduce for educational use only. 54 © Goodheart-Willcox Co., Inc. Construct an Inscribed Regular Polygon with Circle Diameter Given

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Permission granted to reproduce for educational use only. 55 © Goodheart-Willcox Co., Inc. Construct a Regular Polygon with Side Length Given

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Permission granted to reproduce for educational use only. 56 © Goodheart-Willcox Co., Inc. Transfer and Rotate a Plane Figure

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Permission granted to reproduce for educational use only. 57 © Goodheart-Willcox Co., Inc. Transfer and Rotate a Plane Figure 1.Draw Line AB in desired orientation. 2.Enter Rotate command. Use Copy, Reference, and Points options to copy and rotate polygon. 3.Relocate polygon. Erase Line AB drawn in Step 1.

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Permission granted to reproduce for educational use only. 58 © Goodheart-Willcox Co., Inc. Duplicate a Plane Figure with Irregular Curves

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Permission granted to reproduce for educational use only. 59 © Goodheart-Willcox Co., Inc. Duplicate a Plane Figure with Irregular Curves 1.Enter Copy command and select entire object or individual entities. 2.Select base point. 3.Specify second point. 4.Different selection methods can be used to simplify operation if selecting individual objects.

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Permission granted to reproduce for educational use only. 60 © Goodheart-Willcox Co., Inc. Enlarge or Reduce a Plane Figure Using Grid Squares

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Permission granted to reproduce for educational use only. 61 © Goodheart-Willcox Co., Inc. Enlarge or Reduce an Object 1.Enter Scale command and select object. 2.Select base point. 3.Enter scale factor. 4.Use Copy option if duplicating object.

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Permission granted to reproduce for educational use only. 62 © Goodheart-Willcox Co., Inc. Circle and Arc ConstructionsArc Circles have center, diameter, and radius. diameterradius Irregular curve constructions common in drafting.

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Permission granted to reproduce for educational use only. 63 © Goodheart-Willcox Co., Inc. Construct a Circle Through Three Given Points 1.Draw and bisect Lines AB and BC. 2.The center of circle is intersection of bisectors. 3.The radius equals distance from intersection to any point.

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Permission granted to reproduce for educational use only. 64 © Goodheart-Willcox Co., Inc. Locate the Center of a Circle or Arc 1.Draw two nonparallel chords. 2.Bisect each chord. 3.Center of circle or arc is intersection of bisectors.

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Permission granted to reproduce for educational use only. 65 © Goodheart-Willcox Co., Inc. Construct a Circle Through Three Points 1.Enter Circle command. 2.Enter Three Points option. 3.Select three points individually. 4.To locate center point of arc or circle, use Center object snap.

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Permission granted to reproduce for educational use only. 66 © Goodheart-Willcox Co., Inc. Construct a Circle Within a Square 1.Draw 45 diagonals to locate center. 2.Locate midpoint of one side of square (Point E). 3.The radius equals distance from center to Point E.

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Permission granted to reproduce for educational use only. 67 © Goodheart-Willcox Co., Inc. Construct a Circle Within a Square 1.Enter Circle command. 2.Enter Two Points option. 3.Select midpoints of Sides AB and CD using Midpoint object snap.

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Permission granted to reproduce for educational use only. 68 © Goodheart-Willcox Co., Inc. Construct a Line Tangent to a Circle or ArcTangent 1.Align 45 triangle and straightedge with center of circle and point of tangency (Point P). 2.Slide triangle until other edge contacts Point P. 3.Draw line.

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Permission granted to reproduce for educational use only. 69 © Goodheart-Willcox Co., Inc. Construct a Line Tangent to a Circle or Arc

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Permission granted to reproduce for educational use only. 70 © Goodheart-Willcox Co., Inc. Construct a Circle or Arc Tangent to a Line at a Given Point

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Permission granted to reproduce for educational use only. 71 © Goodheart-Willcox Co., Inc. Construct a Circle or Arc Tangent to a Line at a Given Point

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Permission granted to reproduce for educational use only. 72 © Goodheart-Willcox Co., Inc. Construct a Circle or Arc Through a Point and Tangent to a Line

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Permission granted to reproduce for educational use only. 73 © Goodheart-Willcox Co., Inc. Construct a Circle or Arc Through a Point and Tangent to a Line

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Permission granted to reproduce for educational use only. 74 © Goodheart-Willcox Co., Inc. Construct a Circle or Arc Through a Point and Tangent to a Circle

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Permission granted to reproduce for educational use only. 75 © Goodheart-Willcox Co., Inc. Construct a Circle or Arc Through a Point and Tangent to a Circle

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Permission granted to reproduce for educational use only. 76 © Goodheart-Willcox Co., Inc. Construct a Circle or Arc Tangent to Two Circles

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Permission granted to reproduce for educational use only. 77 © Goodheart-Willcox Co., Inc. Construct a Circle or Arc Tangent to Two Circles 1.Enter Circle command and enter Tangent, Tangent, Radius option. 2.Pick near one circle to specify first tangent point. 3.Pick near second circle to specify second tangent point. 4.Enter given radius.

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Permission granted to reproduce for educational use only. 78 © Goodheart-Willcox Co., Inc. Construct a Circle or Arc Tangent to a Line and a Circle

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Permission granted to reproduce for educational use only. 79 © Goodheart-Willcox Co., Inc. Construct a Circle or Arc Tangent to a Line and a Circle 1.Enter Circle command and Tangent, Tangent, Radius option. 2.Pick near line to specify first tangent point. 3.Pick near circle to specify second tangent point. 4.Enter given radius. 5.Use Trim command to create arc segments.

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Permission granted to reproduce for educational use only. 80 © Goodheart-Willcox Co., Inc. Construct a Circle or Arc Tangent to Two Parallel Lines

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Permission granted to reproduce for educational use only. 81 © Goodheart-Willcox Co., Inc. Construct an Arc Tangent to Two Parallel Lines

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Permission granted to reproduce for educational use only. 82 © Goodheart-Willcox Co., Inc. Construct a Circle or Arc Tangent to Nonparallel Lines

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Permission granted to reproduce for educational use only. 83 © Goodheart-Willcox Co., Inc. Construct an Arc Tangent to Nonparallel Lines

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Permission granted to reproduce for educational use only. 84 © Goodheart-Willcox Co., Inc. Connect Parallel Lines with Reversing Arcs of Equal Radii

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Permission granted to reproduce for educational use only. 85 © Goodheart-Willcox Co., Inc. Connect Parallel Lines with Reversing Arcs of Equal Radii

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Permission granted to reproduce for educational use only. 86 © Goodheart-Willcox Co., Inc. Connect Parallel Lines with Arcs of Unequal Radii

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Permission granted to reproduce for educational use only. 87 © Goodheart-Willcox Co., Inc. Connect Parallel Lines with Arcs of Unequal Radii

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Permission granted to reproduce for educational use only. 88 © Goodheart-Willcox Co., Inc. Lay off the Length of the Circumference of a Circle 1.Draw Line AB equal to three times circle diameter. 2.Strike Arc C equal to circle radius. 3.Draw Line DE perpendicular to centerline. 4.Line EB is rectified length.

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Permission granted to reproduce for educational use only. 89 © Goodheart-Willcox Co., Inc. Calculate and Lay off the Circumference of a Circle Multiply circle diameter by pi ( ). Lay off length on straight line. For CAD applications, enter List command and pick circle to display value.

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Permission granted to reproduce for educational use only. 90 © Goodheart-Willcox Co., Inc. Lay off the Length of an Arc 1.Draw Chord AB. Draw Line CA equal to one- half Line AB. 2.Draw Line AD. 3.Draw Arc BD with radius equal to Line BC. 4.Line AD is arc length. 5.Use List command in CAD to obtain length.

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Permission granted to reproduce for educational use only. 91 © Goodheart-Willcox Co., Inc. Lay off a Given Length on an Arc 1.Divide tangent line of specific length (Line AB) into four parts. 2.Strike Arc DB from Point C with radius of CB. 3.Arc AD is equal in length to Line AB.

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Permission granted to reproduce for educational use only. 92 © Goodheart-Willcox Co., Inc. Lay off a Given Length on an Arc 1.Divide tangent line of specific length (Line AB) into four parts. 2.Draw circle from Point C with radius of CB. 3.Arc AD is equal in length to Line AB. 4.Use Arc command for arc segments if chord length is known.chord length

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Permission granted to reproduce for educational use only. 93 © Goodheart-Willcox Co., Inc. Review Questions 1.Which of the following is used to bisect a line? A. Dividers B. Scale C. Template D. Compass

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Permission granted to reproduce for educational use only. 94 © Goodheart-Willcox Co., Inc. Review Questions 2.Which of the following is used to draw a line through a point parallel to a given line? A. Dividers B. Scale C. Triangle and T-square D. Protractor C. Triangle and T-square

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Permission granted to reproduce for educational use only. 95 © Goodheart-Willcox Co., Inc. Review Questions 3.The _____ command can be used to draw parallel lines. A. Spline B. Offset C. Divide D. Arc B. Offset

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Permission granted to reproduce for educational use only. 96 © Goodheart-Willcox Co., Inc. Review Questions 4.The _____ method is used to divide a line into equal parts. A. vertical line B. construction C. compass D. 3,4,5 A. vertical line

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Permission granted to reproduce for educational use only. 97 © Goodheart-Willcox Co., Inc. Review Questions 5.A(n) _____ angle is an angle more than 90 . obtuse

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Permission granted to reproduce for educational use only. 98 © Goodheart-Willcox Co., Inc. Review Questions 6.Which of the following is used to measure and lay out angles? A. Dividers B. Scale C. Protractor D. Template C. Protractor

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Permission granted to reproduce for educational use only. 99 © Goodheart-Willcox Co., Inc. Review Questions 7.A(n) _____ triangle has all sides and all angles equal. equilateral

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Permission granted to reproduce for educational use only. 100 © Goodheart-Willcox Co., Inc. Review Questions 8.The _____ command is used to transfer an angle. Rotate

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Permission granted to reproduce for educational use only. 101 © Goodheart-Willcox Co., Inc. Review Questions 9.A(n) _____ is a regular polygon with six equal sides. hexagon

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Permission granted to reproduce for educational use only. 102 © Goodheart-Willcox Co., Inc. Review Questions 10.Which of the following is not specified when drawing objects with the Polygon command? A. Number of sides B. Major axis C. Center point D. Side length B. Major axis

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Permission granted to reproduce for educational use only. 103 © Goodheart-Willcox Co., Inc. Review Questions 11.When using the Polygon command to construct an octagon, use the _____ option if the distance across the corners is known. Inscribed

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Permission granted to reproduce for educational use only. 104 © Goodheart-Willcox Co., Inc. Review Questions 12.A(n) _____ is a line or curve that touches the surface of a circle or an arc at only one point. tangent

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Permission granted to reproduce for educational use only. 105 © Goodheart-Willcox Co., Inc. Review Questions 13.Which of the following options is not used when drawing circles with the Circle command? A. Three Points B. Two Points C. Tangent, Tangent, Radius D. Edge

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Permission granted to reproduce for educational use only. 106 © Goodheart-Willcox Co., Inc. Review Questions 14.Laying off the circumference of a circle is also referred to as locating the _____ length. rectified

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Permission granted to reproduce for educational use only. 107 © Goodheart-Willcox Co., Inc. Review Questions 15.The _____ length is the length of a line that connects the endpoints of an arc. chord

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Permission granted to reproduce for educational use only. 108 © Goodheart-Willcox Co., Inc. Glossary Arc A part of a circle. Bisect To divide something into two equal parts. Chord length The length of a line that connects the endpoints of an arc.

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Permission granted to reproduce for educational use only. 109 © Goodheart-Willcox Co., Inc. Glossary Diameter The length of a straight line passing through the center of a circle and terminating at the circumference on each end. Equilateral triangle A triangle with three equal sides and three equal angles (three 60° angles).

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Permission granted to reproduce for educational use only. 110 © Goodheart-Willcox Co., Inc. Glossary Hexagon A polygon having six sides. Hypotenuse The side directly opposite the 90° angle of a right triangle. Isosceles triangle A triangle with two equal sides and two equal angles.

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Permission granted to reproduce for educational use only. 111 © Goodheart-Willcox Co., Inc. Glossary Octagon A polygon having eight sides. Pentagon A polygon having five sides. Perpendicular A line or plane drawn at a right angle (90 angle) to a given line or plane.

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Permission granted to reproduce for educational use only. 112 © Goodheart-Willcox Co., Inc. Glossary Polar coordinates Point locations measured by entering a given distance and angle from a previous point. Polygon A plane geometric figure with three or more sides. Radius The straight-line distance from the center of a circle or arc to its circumference.

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Permission granted to reproduce for educational use only. 113 © Goodheart-Willcox Co., Inc. Glossary Rectified length The length of a curved surface laid out on a straight line. Regular polygon An object with sides of equal length and included angles.

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Permission granted to reproduce for educational use only. 114 © Goodheart-Willcox Co., Inc. Glossary Right triangle A triangle with one 90 angle. Tangent A line or curve drawn to the surface of an arc or circle so that it contacts the arc or circle at only one point. A line drawn from the center of the circle or arc to the point of tangency is perpendicular to the tangent object.

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