5 Geometry for Drafting Section 5.1 Applied Geometry for Board Drafting Section 5.2 Applied Geometry for CAD Systems

Chapter Objectives • Identify geometric shapes and construc- tions used by drafters. • Construct various geometric shapes. • Solve technical and mathematical prob- lems through geomet- ric constructions using drafting instruments. • Solve technical and mathematical prob- lems through geomet- ric constructions using a CAD system. • Use geometry to reduce or enlarge a drawing or to change its proportions.

Defying Convention It has been said that Zaha Hadid has built a career on defying convention—conventional ideas of architectural space, and of construction. What do you see in the building shown here that defi es convention?

132 Drafting Career

Zaha Hadid, Architect

Architect Zaha Hadid’s designs for the Cincinnati Contemporary Art Center were “like a rollercoaster, a little scary, but exhilarating,” says Center direc- tor Charles Desmarais. Critics said “she was a paper architect, someone who had great respect as a theo- rist and as a thinker about architecture but who hadn't had the opportunity to build.”

“She totally got what we were trying to do,” said Desmarais, “which was to try and bridge that sort of gap between the inside and the outside, between the world and the museum.” She certainly did. Zaha Hadid is the fi rst woman in the world to design a museum and to win the prestigious Pritzker Architec- ture Prize. Academic Skills and Abilities • Math • Computer sciences • Business management skills • Verbal and written communication skills • Organizing and planning skills Career Pathways There is a wealth of opportunities outside the classroom for expanding your drafting knowledge. Learn about annual drafting contests. Even if you do not intend to apply, read about the projects. Find groups such as the Solar-Powered Car Chal- lenge; their ideas will inspire you.

Go to glencoe.com for this book’s OLC to learn more about Zaha Hadid.

133 Zaha Hadid 5.1 Applied Geometry for Board Drafting

Preview In this chapter, you will learn to construct geometric shapes using board drafting techniques. Have you learned geometric terms and formulas in other courses? Content Vocabulary • geometry • vertex • parallel • circumscribe • ellipse • geometric • bisect • polygon • regular construction • perpendicular • inscribe polygon Academic Vocabulary Learning these words while you read this section will also help you in your other subjects and tests. • accurate • methods Graphic Organizer Use a table like the one below to organize the major concepts about the types of geometric constructions.

Bisect Construct Lines Construct Go to glencoe.com for this 1. Arc 1. Triangle 1. book’s OLC for a downloadable 2. 2. 2. version of this graphic organizer. 3. 3. 4. 5. 6.

Academic Standards

English Language Arts NCTE National Council of Teachers of English Students read a wide range of print and nonprint texts to build an understanding of texts (NCTE) NCTM National Council of Teachers of Mathematics Mathematics

Students recognize and use connections among mathematical ideas (NCTM)

134 Chapter 5 Geometry for Drafting Geometry and Geometric Constructions 5 What do you need to be able to understand geometric constructions? B 4 C Geometry is the study of the size and ° 90 2 shape of objects, and of as the relationship 5 = 25 between straight and curved lines in draw- A 4 2 = 16 ing shapes. In ancient times, geometry A222 + B = C was used for measuring land and making 3 3222 + 4 = C accurate right-angle, or 90-degree, corners 9 + 16 = 25 for constructing buildings and other proj- 3 2 = 9 ects. When building the great pyramids, Ancient Egyptians formed right-angle cor- Figure 5-2 ners by using rope with marks or knots at The Pythagorean theorem shown graphically and 3-, 4-, and 5-space sections, and stretching mathematically the rope around carefully placed pegs driven into the ground. (See Figure 5-1.) In the sixth century BCE, the math- This method also works well for triangles ematician Pythagoras studied this method that have the same proportions, such as 6, 8, of forming right angles and proved the and 10 units: theory that the 3-4-5 triangle makes a right angle. This theorem (a2 + b2 = c2) or proof, 62 + 82 = 100 called the Pythagorean theorem, is shown in 36 + 64 = 100 Figure 5-2. 100 = 100

Figure 5-1 Egyptian rope-stretchers used knots divided into 3-4-5 triangles to lay out square corners for buildings.

Section 5.1 Applied Geometry for Board Drafting 135 180° STRAIGHT LINE POINT OF RIGHT (SHORTEST DISTANCE INTERSECTION A ANGLE (90°) A BETWEEN TWO POINTS)

B B INTERSECTING LINES SUPPLEMENTARY ANGLES PARALLEL LINES COMPLEMENTARY ANGLES

HYPOTENUSE

SIDE 60° SIDE

SIDE SIDE ° 60° 60° ALTITUDE 90

EQUILATERAL TRIANGLE BASE SYMBOL FOR BASE ALL SIDES EQUAL LENGTH ISOSCELES TRIANGLE RIGHT ANGLE (90°) TWO SIDES EQUAL LENGTH SCALENE TRIANGLE

SEMI-CIRCLE QUADRANT TANGENT ARC (ONE-QUARTER OF A CIRCLE) CHORD DIAMETER POINT OF TANGENCY

RADIUS SECTOR

TANGENT LINE SEGMENT ANGLE RIGHT ANGLES IN A SEMI-CIRCLE

5 SIDES 6 SIDES

PENTAGON HEXAGON

CONCENTRIC CIRCLES ECCENTRIC CIRCLES

EQUAL OPPOSITE SIDES 7 SIDES 8 SIDES SIDES ARE EQUAL

90° ANGLES 90° ANGLES HEPTAGON OCTAGON SQUARE RECTANGLE

EQUAL SIDES OPPOSITE SIDES ARE EQUAL 9 SIDES 10 SIDES OPPOSITE ANGLES OPPOSITE ANGLES ARE EQUAL ARE EQUAL

RHOMBUS RHOMBOID NONAGON DECAGON

TWO SIDES NO TWO SIDES 12 SIDES ARE PARALLEL ARE PARALLEL

TRAPEZOID TRAPEZIUM DODECAGON

Figure 5-3 Dictionary of drafting geometry

136 Chapter 5 Geometry for Drafting The units may be millimeters, meters, Illustrations made of individual lines and inches, fractions of an inch, or any other unit points drawn in proper relationship to one of measure. Geometric fi gures used in drafting another are known as geometric con- include circles, squares, triangles, hexagons, structions. Drafters, surveyors, engineers, and octagons. Many other shapes are shown architects, scientists, mathematicians, and in Figure 5-3. designers use geometric constructions. To understand geometric constructions, A2 ϩ B2 ϭ C2 you must understand how to describe various 32 ϩ 42 ϭ C2 lines, arcs, and other shapes. This chapter fol- 9 ϩ 16 ϭ 25 lows the identifi cation rules used in geometry.

Geometry Formulas The diameter of a circle is 15″. What is the In addition to solving drafting problems circumference? using geometric constructions, drafters often To fi nd the circumference need to be able to calculate various aspects of of a circle, multiply pi (π) times the diam- geometric constructions. While hundreds of eter of the circle. The approximate decimal these formulas exist, a few are given here as equivalent of pi is 3.1416. examples. What is the area of triangle A where the base is Example: 10″ and the height is 7″? Circumference = πd Circumference = 3.1416 × 2.50 To fi nd the area of any tri- Circumference = 7.85 angle, multiply the base (b) times the height (h) and divide by two. DIAMETER (d) = 2.50 Example: Area = bh/2 Area = 2 × 6/2 Area = 6 square inches

ALTITUDE (h) BASE (b) 2" For help with this math activity, go to the Math Appendix at the back of 6" this book.

Academic Standards Mathematics

Measurement Apply appropriate techniques, tools, and formulas to determine measurements (NCTM)

Section 5.1 Applied Geometry for Board Drafting 137 Lines and arcs are described using their end- B AB points. Therefore, line AB is a line segment LINE AB ARC AB that extends from point A to point B. Arc AB A is an arc whose endpoints are A and B. Angles are described using three points: both end- A points and the vertex, or the point at which B A the two arms of the angle meet. Angle ABC is an angle whose endpoints are A and C and C ANGLE ABC CIRCLE A whose vertex is at point B. Circles are usually specifi ed using their center points, so circle Figure 5-4 A is a circle whose center is at point A. See Identifi cation of lines, angles, arcs, and circles Figure 5-4.

Explain How is the Pythagorean theorem used in geometry?

Bisect a Line, an Arc, or 2. With points A and B as centers and any radius R greater than one-half of AB, an Angle draw arcs to intersect, or cross, line AB Bisect means to divide into two equal parts. as in Figure 5-5B. The radius is the dis- tance from the center of an arc or circle to any point on the arc or circle. The two Bisect a Line or an Arc places where the arcs intersect create Follow these steps to bisect a straight line points C and D. or an arc. 3. Draw line EF through points C and D 1. Draw a line AB and arc AB as shown in (Figure 5-5C). Figure 5-5A.

E A A R A R C C

R R Figure 5-5 D D B B B Bisecting a straight line, F B B E B an arc, and an angle C C

R R

D R D A A A R F ABC

138 Chapter 5 Geometry for Drafting Bisect an Angle 3. With C and D as centers and any radius R more than one- half the radius of arc This construction demonstrates a method 2 CD, draw two arcs to intersect, locating for bisecting a given angle. Refer to Figure 5-6. point E. (Figure 5-6C). 1. Draw given angle AOB (Figure 5-6A). 4. Draw a line through points O and E to 2. With point O as the center and any con- bisect angle AOB (Figure 5-6C). venient radius R1, draw an arc to inter- sect AO and OB to locate points C and D (Figure 5-6B).

A A A GIVEN ANGLE C C

R2

E O O O R1 R1

R2 D D

B B AOE = EOB B A B C

Figure 5-6 Bisecting an angle

Divide a Line into Any 1. Construct a line of any length at A per- pendicular to line AB, as in Figure 5-7A. Number of Equal Parts Lines are perpendicular when they cross Two methods of dividing a line into equal at 90° angles. parts are described next. Try both methods. 2. Position the scale, placing zero on line AC Can you think of situations in which you would at such an angle that the scale touches need to use one method instead of the other? point B, as in Figure 5-7B. Keeping zero on line AC, adjust the angle of the Divide a Line into Equal Parts scale until any eight equal divisions are This method can be applied to create any included between line AC and point B (in number of equal divisions. In this construc- this case, at 8″). Mark the divisions. tion, you will divide a straight line into eight 3. Draw lines parallel to AC through the equal parts. Refer to Figure 5-7 and follow division marks to intersect line AB Figure these steps: 5-7C. Two lines are parallel when they are always the same distance apart.

B A B A A B

8 7 6 5 4 3 2 1 0 C C C

ABC

Figure 5-7 Dividing a straight line into any number of equal parts

Section 5.1 Applied Geometry for Board Drafting 139 Divide a Line into Five 3. Draw a line connecting point A and the last point on line BC (Figure 5-8C.) Equal Parts Draw lines through each point on BC Follow these instructions to divide a line parallel to this line as shown. into fi ve equal parts. 1. Draw line BC from point B at any conve- nient angle and length (Figure 5-8A). 2. Use dividers or a scale to step off fi ve Summarize Explain how to bisect a line equal spaces on line BC beginning at using board drafting techniques. point B (Figure 5-8B).

ABABAB 1 1 2 2 3 3 4 C 4 C C 5 5

AB C

Figure 5-8 Dividing a straight line into fi ve equal parts

1. Draw line AB and point O (Figure 5-9A). Construct a 2. With O as the center and any conve-

Perpendicular Line nient radius R1, construct an arc inter- Each one of the many procedures to con- secting line AB, locating points C and D struct a line perpendicular to another line (Figure 5-9B). is useful in certain drafting situations. Four 3. With C and D as centers and any radius methods are discussed below. R2 larger than OC, draw arcs intersecting at point E (Figure 5-9C). Method 1 4. Draw a line connecting points E and O to Figure 5-9A shows the given line AB and form the perpendicular line (Figure 5-9C). point O that lies on line AB. Follow these steps to draw a line at point O on line AB so that the two lines are perpendicular.

R2 D D B B B R O O 2 O C R1 C R1

A A A

AB C

Figure 5-9 Constructing a line perpendicular to a given line through a given point on the line (Method 1)

140 Chapter 5 Geometry for Drafting Method 2 passing through line AB to locate point D (Figure 5-10B). Use this method when the given point 3. Construct a line through points D and through which a perpendicular line is drawn C, extending it through the arc to locate lies near one end of the line. point E (Figure 5-10C). 1. Construct given line AB and point O 4. Connect points E and O to form the per- (Figure 5-10A). pendicular line (Figure 5-10C). 2. From any point C above line AB, con- struct an arc using CO as the radius and

E

C C O O B O B B R

D A D A A

AB C

Figure 5-10 Constructing a line perpendicular to a given line through a given point on the line

Method 3 2. Place the T-square and triangle (Figure 5-11B). This construction demonstrates another 3. Slide the triangle along the T-square until way to draw a line perpendicular to a given the edge aligns with point O on line AB line through a given point on the line. Follow (Figure 5-11C). the steps to create a line at O that is perpen- 4. Draw a perpendicular line through point dicular to line AB. Refer to Figure 5-11. O (Figure 5-11C). 1. Construct given line AB and point O (Figure 5-11A).

C B B

O O B

O A A

D A ABC

Figure 5-11 Constructing a line perpendicular to a given line through a given point on the line

Section 5.1 Applied Geometry for Board Drafting 141 Method 4 3. With C and D as centers and CO and DO as radii, draw arcs to intersect, locating Figure 5-12A shows another given line AB point E (Figure 5-12C). and point O that is not on the line. Follow the 4. Connect points O and E to form the per- steps to practice another way to draw a line pendicular line (Figure 5-12C). perpendicular to a given line through a point that is not on the line. 1. Construct given line AB and point O (Figure 5-12A). 2. Construct lines from point O to any two Identify What are perpendicular lines? points on line AB, locating points C and D (Figure 5-12B).

O O O

B B B D D C C A A A

ABCE

Figure 5-12 Constructing a line perpendicular to a given line through a point that is not one the given line

intersecting line AB to locate point C Draw a Parallel Line (Figure 5-13B). The following construction methods create 3. With point C as the center and the same a line that is parallel to another line. Recall radius R , draw an arc through point that lines are parallel when they are always 1 P and line AB to locate point D (Figure the same distance apart. 5-13B). 4. With C as the center and radius R equal Method 1 2 to chord PD, draw an arc to locate point This construction allows you to place a line E. A chord is a straight line between two parallel to a given line. Refer to Figure 5-13. points on a circle (Figure 5-13C). 1. Draw given line AB and point P (Figure 5. Draw a line through points P and E. Line 5-13A). PE is parallel to line AB (Figure 5-13C). 2. With point P as the center and any

convenient radius R1, draw an arc

E

P P P R B R1 B R1 2 B R1 C R1 C A A D A D ABC

Figure 5-13 Using a compass to construct a line parallel to a given line through a given point

142 Chapter 5 Geometry for Drafting Method 2 2. Place the T-square and triangle (Figure 5-14B). The following steps demonstrate another 3. Slide the triangle until the edge aligns way to construct a line parallel to another line with point P (Figure 5-14C). through a given point. Refer to Figure 5-14. 4. Draw a parallel line through point P. See 1. Draw given line AB and point P (Figure (Figure 5-14C). 5-14A).

B B B P P P D A A A C

AB C

Figure 5-14 Using a triangle and T-square to construct a line parallel to a given line through a given point

Method 3 3. Draw a parallel line CD tangent to the arcs. Recall that a line is tangent to an arc Use this method to construct a line parallel to or circle when it touches the arc or circle a given line at a specifi ed distance from the given at one point only (Figure 5-15C). line. Refer to Figure 5-15. Note: See “Construct a Tangent Line to a Circle” later in this chapter for instructions on creating a tangent line. 1. Draw given line AB (Figure 5-15A). Explain What is a chord? 2. Draw two arcs with centers anywhere along line AB. The arcs should have a radius R equal to the specifi ed distance between the two parallel lines (Figure 5-15B).

C

R R D A A A R R B B B A B C

Figure 5-15 Constructing a line parallel to a given line at a specifi ed distance from the given line

Section 5.1 Applied Geometry for Board Drafting 143 BO and AO at C and D and A1O1 at D1. Copy an Angle Refer again to Figure 5-16B. This construction demonstrates a method 4. With D as the center and radius R equal of copying a given angle to a new location 1 2 to chord DC, draw an arc to locate point and orientation. Refer to Figure 5-16. C1 at the intersection of the two arcs 1. Draw given angle AOB (Figure 5-16A). (Figure 5-16C). 2. Draw one side O A in the new position 1 1 5. Draw a line through points O1 and C1 (Figure 5-16B). to complete the angle. Refer again to

3. With O and O1 as centers and any conve- Figure 5-16C.

nient radius R1, construct arcs to intersect

D D O A O A O A GIVEN GIVEN POSITION R2 POSITION GIVEN R1 B POSITION C B CHORD C B GIVEN ANGLE O1 O1 D1 D1 A A1 A1 R1 NEW R2 NEW POSITION POSITION

C1 BC Figure 5-16 Copying an angle

1. Draw base line AB (Figure 5-17A). Construct a Triangle 2. With points A and B as centers and a A triangle is a polygon, or closed fi gure, radius R equal to the length of the two with three sides. The following constructions sides you want, draw intersecting arcs show methods for drawing various types of to locate the third vertex of the triangle triangles. (Figure 5-17B). The other two vertices (plural of vertex) are at the endpoints of Method 1 the base line. This method constructs an isosceles triangle, 3. Draw lines through point A and the ver- which has two sides that are of equal length. tex and through point B and the vertex Refer to Figure 5-17. to complete the triangle (Figure 5-17C).

VERTEX

Figure 5-17 R R BASE Constructing an A B A B A B isosceles triangle ABEQUAL SIDES OF DESIRED LENGTH C

144 Chapter 5 Geometry for Drafting Method 2 draw intersecting arcs to locate the third vertex Refer again to Figure 5-18B. This method constructs an equilateral trian- 3. Draw lines through point A and the gle. An equilateral triangle is one in which all vertex and through point B and the ver- three sides are of equal length and all three tex to complete the triangle (Figure angles are equal. Refer to Figure 5-18. 5-18C). 1. Draw base line AB as in Figure 5-18A. 2. With points A and B as centers and a radius R equal to the length of line AB,

VERTEX

60°

60° 60° BASE ABABAB R AB C

Figure 5-18 Constructing an equilateral triangle

Method 3 2. Draw a line perpendicular to AB at B equal to BC. Note: Construct the perpen- Construct a right triangle using this method dicular line using the method in Figure when you know the length of two sides of the 5-11 or Figure 5-12. triangle. A right triangle is one that has a right 3. Draw a line connecting points A and C (90°) angle at one of its vertices. Given sides to complete the right triangle (Figure AB and BC are shown in Figure 5-19A. 5-19C). 1. Draw side AB in the desired position (Figure 5-19B).

C C

A B B C A B A B

ABC

Figure 5-19 Constructing a right triangle given the lengths of two sides

Section 5.1 Applied Geometry for Board Drafting 145 Method 4 3. With point A as the center and a radius equal to side AB, draw an arc to intersect Use this method to construct a right triangle the semicircle to locate point B (Figure when you know the length of one side and the 5-20C). length of the hypotenuse. See Figure 5-20A 4. Draw line AB and then draw a line to for the given side AB and hypotenuse AC. connect B and C to complete the triangle 1. Draw the hypotenuse AC in the desired (Figure 5-20C). location (Figure 5-20B). 2. Draw a semicircle on AC using ½AC as the radius. Refer again to Figure 5-20B.

Figure 5-20 B Constructing a right triangle given the A B SIDE 1 length of one side 2 AC A C ACACand the length of the HYPOTENUSE hypotenuse. ABC

Method 5 3. Connect both ends of line AB with point C to complete the triangle (Figure You can use this method to construct a tri- 5-21C). angle when you know the lengths of all three sides. This method is useful for constructing scalene triangles, which have three different angles and sides of three different lengths. Figure 5-21A shows given triangle sides AB, Compare What is the diff erence between BC, and AC. an isosceles triangle and an equilateral 1. Draw base line AB in the desired location. triangle? 2. Construct arcs from the ends of line AB with radii equal to lines BC and AC to locate point C (Figure 5-21B).

C C

R = AC AB R = BC BC ACABAB A B C

Figure 5-21 Constructing a triangle given the lengths of all three sides

146 Chapter 5 Geometry for Drafting 2. Draw perpendicular bisectors of AB Construct a Circle and BC to intersect at point O (Figure This construction describes a method for 5-22B). creating a circle given three points that lie on 3. Draw the required circle with point O as the circle. Refer to Figure 5-22. the center and radius R = OA = OB = 1. Given points A, B, and C, draw lines AB OC (Figure 5-22C). and BC (Figure 5-22A).

B B B Figure 5-22 A A A Constructing a circle given C O C O C three points that lie on the AB circle

C

2. Draw a line perpendicular to line OA at P Construct Lines Tangent (Figure 5-23B). The perpendicular line to a Circle is the tangent line. The constructions that follow present A A methods of creating lines tangent to a circle.

As you may recall, a line that touches a circle P P at one point only is said to be tangent to the circle. O O Method 1 Use this method to construct a line tangent to a given point on a circle without using a triangle or T-square. Refer to Figure 5-23. A B 1. Given a circle with center point O and tangent point P (Figure 5-23A), draw Figure 5-23 line OA from the center of the circle to Constructing a line tangent to a circle through a extend beyond the circle through point P. given point on the circle (Method 1)

Method 2 Use this method to construct a line tangent O to a given point on a circle using a 30°-60° tri- angle and a T-square. See Figure 5-24. P FIRST POSITION 1. Given a circle with center point O and tan-

gent point P, place a T-square and triangle SECOND POSITION so that you can construct the hypotenuse of the triangle through points P and O. Figure 5-24 2. Hold the T-square, turn the triangle to Constructing a line tangent to a given point on a the second position at point P, and draw circle (Method 2) the tangent line.

Section 5.1 Applied Geometry for Board Drafting 147 Method 3 3. Draw a circle with center A and radius R = AP = AO to locate tangent points T This method creates lines tangent to a circle 1 and T (Figure 5-25B). from a given point outside the circle. See 2 4. Draw lines PT and PT (Figure 5-25C). Figure 5-25. 1 2 These lines are tangent to the circle. 1. Draw a circle with center point O and point P outside the circle (Figure 5-25A). 2. Draw line OP and bisect it to locate point A (Figure 5-25B).

P T1 P T1 P A A

O O O

T2 T2

ABC

Figure 5-25 Constructing a line tangent to a circle from a given point outside the circle (Method 3)

2. Draw a circle with center O and a radius Method 4 1 R, where R = R − R . Refer again to Figure Use this method to construct a line tangent 1 2 5-26A. to the exterior of two circles. Refer to Figure 3. From center point O , draw a tangent O T 5-26. 2 2 to the circle of radius (Figure 5-26B). 1. Draw the two given circles with centers 4. Draw radius O T as shown in Figure O and O and radii R and R (Figure 1 1 2 1 2 5-26B, and extend it to locate point T . 5-26A). 1 5. Draw the needed tangent T1T2 parallel to

TO2 (Figure 5-26C).

T1 T1

T T T2 T2 O O O O O2 O1 2 1 2 1 R R R R2 1

AB C

Figure 5-26 Constructing an exterior common tangent to two circles of unequal radii (Method 4)

148 Chapter 5 Geometry for Drafting 4. Draw radius O T to locate point T (Figure Method 5 1 1 5-27B). Use this method to construct a line tangent to 5. Draw O T parallel to O T. the interior of two circles. Refer to Figure 5-27. 2 2 1 6. Draw the needed tangent T1T2 parallel to 1. Draw the two given circles with centers O1 TO2 (Figure 5-27C). and O2 and radii R1 and R2 (Figure 5-27A).

2. Draw a circle with center O1 and a radius = + R, where R R1 R2. Refer again to Figure 5-27A. Identify What two tools are used in some of 3. From center point O , draw a tangent O T 2 2 the methods described in Section 5.1? to the circle of radius R (Figure 5-27B).

T T T 1 T1

R2 R

O O O 2 1 O2 1 O R1 O2 1 T 2 T2

ABC

Figure 5-27 Constructing an interior common tangent to two circles of unequal radii (Method 5)

1. Given lines AB and CD (Figure 5-28A), Construct Arcs Tangent draw lines parallel to AB and CD at a dis- to Straight Lines and tance R from them on the inside of the angle. The intersection O will be the cen- Other Arcs ter of the arc you need. The following are methods for drawing arcs 2. Draw perpendicular lines from O to AB tangent to other geometric fi gures, such as and CD to locate the points of tangency straight lines and other arcs. T (Figure 5-28B). Construct an Arc Tangent to 3. With O as the center and radius R, draw the needed arc (Figure 5-28C). Two Straight Lines The technique is shown for two lines at an acute angle, an obtuse angle, and a right angle. Refer to Figure 5-28.

Section 5.1 Applied Geometry for Board Drafting 149 A D A D A D R R O O O T T R

B B T B T C C C ACUTE ANGLE

D

D D R R O O R O T T A B C A T B C A T B C

OBTUSE ANGLE

D CT D CT D A A A R R T T O O O R

B B B RIGHT ANGLE AB C

Figure 5-28 Constructing an arc tangent to two straight lines at an acute angle, an obtuse angle, and a right angle

= Construct an Arc Tangent to 2. Draw an arc with center O1 and radius R + R, where R is the radius of the Two Given Arcs 1 desired tangent arc. See Figure 5-29B. Refer to Figure 5-29 for the steps in con- The intersection O is the center of the structing this arc. tangent arc. 1. Draw two arcs having radii R and R 1 2 3. Draw lines O1O and O2O to locate tan- (Figure 5-29A). The radii R and R may 1 2 gent points T1 and T2 (Figure 5-29C). be equal or unequal. 4. With point O as the center and radius R, draw the tangent arc needed. RADIUS OF TANGENT ARC

O1 O2 O1 O2 O1 O2 R1 R2

T1 T2 R AB C R+R1 R+R2

O O Figure 5-29 Constructing an arc tangent to two given arcs

150 Chapter 5 Geometry for Drafting 4. Draw a line from O perpendicular to AB Construct an Arc Tangent to a 1 Line and an Arc to locate tangent point T. 5. Draw a line from O to O1 to locate tan- Use this method to construct an arc tan- gent point T on CD (Figure 5-30C). gent to a line and an arc, given the line, the 1 6. With point O1 as the center and radius R, arc, and the radius R of the desired tangent draw the tangent arc. arc. Refer to Figure 5-30. 1. Draw given line AB and arc CD as shown

in Figure 5-30A. Recall What three types of angles do you 2. Draw a line parallel to line AB, at dis- create constructing an arc tangent to two lines? tance R, toward arc CD. (Figure 5-30B). + 3. Use radius R1 R to locate point O1. Refer again to Figure 5-30B.

T T B A B A B O A O C O C T1 R + R1 R R1 1 D D D O1 O1

A A A RADIUS R

O1 O1 T T R1 B B B R + R1 T1 R1 C C C O O D D O D

AB C

Figure 5-30 Constructing an arc tangent to line and an arc

3. Complete the square by drawing perpen- Construct a Square dicular lines at each end of line AB to A square is a rectangle with all four sides intersect the diagonals. Draw the last line equal. You can construct a square in several from the intersection of the diagonal and ways. The method you choose depends on the the vertical lines. Draw the lines in the other geometry in the drawing. order shown by the numbered arrows. Construct a Square When the Length of One Side Is Known 3 Use this method to construct a square when you know the length of a side. Refer to Figure 5-31 Figure 5-31. Constructing a 1 2 1. Given the length of the side AB, draw square given the line AB. length of a side 2. Construct 45° diagonals from the ends of line AB. Refer again to Figure 5-31. AB

Section 5.1 Applied Geometry for Board Drafting 151 Construct a Square Inscribed in A B a Circle A square or other polygon is inscribed in a circle when its four corners are tangent to O the circle. Refer to Figure 5-32.

1. Draw the given circle with center point O. D C 2. Draw 45° diagonals through the center point O to locate points A, B, C, and D. Refer again to Figure 5-32. Figure 5-32 3. Connect points A and B, B and C, C and Constructing a square inscribed within a circle D, and D and A to complete the square.

Construct a Square Circumscribed in a Circle A square or other polygon is circumscribed O about a circle when the square fully encloses the circle and the circle is tangent to the square on all four sides. Refer to Figure 5-33. 1. Draw the given circle with center point O. 2. Draw 45° diagonals through the center Figure 5-33 point O. Constructing a square circumscribed about a 3. Draw sides tangent to the circle, inter- circle secting at the 45° diagonals, to complete the square.

2. Draw line BC and extend it to make line Construct a Pentagon CD equal to AC. Refer to Figure 5-34A A pentagon is a fi ve-sided polygon. When its for steps 1 and 2. fi ve sides are exactly the same length and all 3. With radius AD and points A and B as of its angles are equal, it is called a regular centers, draw intersecting arcs to locate polygon. The following methods demon- point O (Figure 5-34B). strate the construction of regular pentagons. 4. With the same radius and O as the cen- Regular Pentagon When the ter, draw a circle. 5. Step off AB as a chord to locate points E, Length of One Side Is Known F, and G. Connect the points to complete To use this method, refer to Figure 5-34. the pentagon (Figure 5-34C). 1. Given line AB, construct a perpendicular line AC equal to one-half of the length of AB. F Figure 5-34 G E Constructing a D D O O regular pentagon C C given the length of one side

AB AB A B ABC

152 Chapter 5 Geometry for Drafting Inscribe a Pentagon within a 4. With C as center and radius CE, draw an arc to locate point F. Circle 5. Draw chord CF. This chord is one side of Refer to Figure 5-35 for this method. the pentagon. 1. Draw the given circle with diameter AB 6. Step off chord CF around the circle and radius OC (Figure 5-35A). The to locate points G, H, and J. Draw diameter of a circle is the distance across the chords to complete the pentagon the circle through its center point. (The (Figure 5-35C). symbol for diameter is Ø.) 2. Bisect radius OB to locate point D (Figure 5-35B).

3. With D as center and radius DC, draw an Explain What is a regular polygon? arc to locate point E.

C C C

R

F F G O E O D O ABABAB R

J H

ABC

Figure 5-35 Inscribing a regular pentagon within a circle

Construct a Hexagon 2. With the T-square and 30°-60° triangle, A hexagon is a six-sided polygon. The fol- draw the tangents in the order shown in lowing methods demonstrate construction for Figure 5-36. regular hexagons, which have six sides of equal length, six internal angles of equal size, and six external angles of equal size. 1

Construct a Regular Hexagon 5 4 When the Distance across the DISTANCE ACROSS FLATS Flats Is Known 3 6 This method constructs a regular hexagon when you know the distance across the 2 fl ats, or sides. The distance across the fl ats is the distance from the midpoint of one side through the center point to the midpoint Figure 5-36 of the opposite side of the polygon. Refer to Constructing a regular hexagon given the Figure 5-36. distance across the fl ats 1. Given the distance across the fl ats of a regular hexagon, draw centerlines and a circle with a diameter equal to the dis- tance across the fl ats.

Section 5.1 Applied Geometry for Board Drafting 153 Construct a Regular Hexagon 3. Connect the points to complete the When the Distance across the hexagon. Corners Is Known C D Method 1 Use this method to construct a regular hexagon when you know the distance across A B the corners. The distance across the corners is the distance from one vertex through the center point to the opposite vertex. Refer to F E Figure 5-37. DISTANCE ACROSS CORNERS 1. Given the distance AB across the corners, draw a circle with AB as the diameter. Figure 5-37 2. With A and B as centers and the same Constructing a regular hexagon given the radius, draw arcs to intersect the circle at distance across the corners (Method 1) points C, D, E, and F.

Method 2 5

This construction demonstrates another 1 3 method of constructing a regular hexagon 60° 30° given the distance across the corners. Refer to A B 30° Figure 5-38. 4 1. Given the distance AB across the corners, 60° 2 draw lines from points A and B at 30° to line 6 AB. The lines can be any convenient length. 2. With the T-square and 30°-60° triangle, Figure 5-38 draw the sides of the hexagon in the Constructing a regular hexagon given the order shown. distance across the corners (Method 2)

Explain What is a fl at of a hexagon?

2. With the T-square and 45° triangle, draw Construct an Octagon lines tangent to the circle in the order An octagon is an eight-sided polygon. The shown to complete the octagon. following methods demonstrate the construc- 1 tion of regular octagons. 57 Construct an Octagon Circumscribed about a Circle 3 4 Refer to Figure 5-39 as you follow the steps 8 6 in constructing an octagon circumscribed about a circle. 2 1. Given the distance across the fl ats, draw centerlines and a circle with a Figure 5-39 diameter equal to the distance across Constructing a regular octagon circumscribed the fl ats. about a circle given the distance across the fl ats

154 Chapter 5 Geometry for Drafting Construct an Octagon Inscribed C G E within a Circle Refer to Figure 5-40 as you follow the 45° steps for constructing an octagon inscribed A B within a circle.

1. Given the distance across the corners, F H draw centerlines AB and CD and a circle D with a diameter equal to the distance across the corners. 2. With the T-square and 45° triangle, draw Figure 5-40 diagonals EF and GH. 3. Connect the points to complete the Inscribing a regular octagon within a circle given octagon. the distance across the corners of the octagon

Construct an Octagon Inscribed CD within a Square Refer to Figure 5-41 as you follow the steps to construct an octagon inscribed within O a square. 1. Given the distance across the flats, con- struct a square having sides equal to AB. AB 2. Draw diagonals AD and BC with their Figure 5-41 intersection at O. With A, B, C, and D as centers and radius R = AO, draw arcs to Inscribing a regular octagon within a square given the distance across the fl ats intersect the sides of the square. 3. Connect the points to complete the octagon.

Describe How many sides does an octagon have?

Construct an Ellipse (Figure 5-42A). The major axis AB and minor axis CD are given. They intersect at O. An ellipse is a regular oval. It is sym- = metrical around two axes that form a right 1. With C as center and radius R AO, angle. The shorter axis is the minor axis, and draw an arc to locate points F1 and F2 the longer one is the major axis. This sec- (Figure 5-42A). tion demonstrates methods for drawing an 2. Place pins at points F1, C, and F2 (Figure ellipse. 5-42B). 3. Tie a string around the three pins and Pin-and-String Method to remove pin C. Construct an Ellipse 4. Put the point of a pencil in the loop and draw the ellipse. Keep the string tight This illustration demonstrates the use of the when moving the pencil (Figure 5-42C). pin-and-string method of drawing a large ellipse

Section 5.1 Applied Geometry for Board Drafting 155 C C C

O O O A B A B A B F 1 R = AO F2 F1 F2

D D D AB C Figure 5-42 Constructing an ellipse by the pin-and-string method

Trammel Method to Construct 2. On the trammel, move point O along minor axis CD and point D along major an Ellipse axis AB and mark points at A (Figure This method demonstrates the use of the 5-43B). trammel to draw an ellipse. A trammel is a 3. Use a French curve or irregular curve to piece of paper or plastic on which specifi c connect the points to draw the ellipse distances have been marked off. Figure (Figure 5-43C). 5-43A shows the major axis AB and minor axis CD, intersecting at O. 1. Cut a strip of paper or plastic to use as a trammel. Mark off distances AO and OD on the trammel (Figure 5-43A).

C C

a FIRST POSITION A O O o B A d B

a o o D D d D SECOND POSITION AB C a

Figure 5-43 Constructing an ellipse by the trammel method

Use of Major and Minor Axes to 1. Lay off OF and OG, each equal to AB – Construct an Ellipse CD Refer again to Figure 5-44A. 2. Lay off OJ and OH, each equal to three- This method constructs an approximate fourths of OF. ellipse by using its major and minor axes. This 3. Draw and extend lines GJ, GH, FJ, and method works when the minor axis is at least FH (Figure 5-44B). two-thirds the size of the major axis. Figure 5-44A shows the major axis AB and minor axis CD, intersecting at O.

156 Chapter 5 Geometry for Drafting 4. Draw arcs with centers F and G and radii FD and GC to the points of tangency

(Figure 5-44C). Identify What tool is used with the trammel 5. Draw arcs with centers J and H and method? radii JA and HB to complete the ellipse. The points of tangency are marked T in (Figure 5-44C).

C C C T T F F F

J H H J H J A B A B A B O O O

G G T G T

D D D ABC

Figure 5-44 Constructing an approximate ellipse when the minor axis is at least two-thirds the size of the major axis

Reduce or Enlarge a Drawing The following techniques reduce or enlarge an existing drawing.

ENLARGED SIZE E Reduce or Enlarge a Square or F

Rectangular Drawing ORIGINAL SIZE B A If a drawing is square or rectangular, use a diagonal line method to reduce or enlarge it. REDUCED SIZE Refer to Figure 5-45. 1. Draw a diagonal through corners D and B. D 2. Measure the width or height you need G C along DC or DA (example: DG). DIAGONAL 3. Draw a perpendicular line from that point (G) to the diagonal. Figure 5-45 4. Draw a line perpendicular to DE intersect- Reducing or enlarging a square or rectangular area ing at point F.

Section 5.1 Applied Geometry for Board Drafting 157 Reduce or Enlarge a Drawing 8 A That Is Not Square or Rectangular 7 Use this method to reduce or enlarge a 6 drawing that is not square or rectangular. 5

Refer to Figure 5-46. 8 4 7 1. Draw a grid larger or smaller than the one 6 3 shown at B. The size of the grid depends 5 4 2 on the amount of enlargement or reduc- 3 2 1 tion needed. 1 0 0 2. Use dots to mark key points on the sec- 1234567 12 3 456 7 ond grid corresponding to points on the BC original drawing at A. 3. Connect the points and darken the lines Figure 5-46 to complete the new drawing. Reducing or enlarging a drawing of a sailboat

Multiply pi (π) times the diameter of Section 5.1 Assessment the circle. The approximate decimal After You Read equivalent of pi is 3.1416. Circumference = πd Self-Check Circumference = 3.1416 × 2.50″ 1. List various geometric shapes and con- Circumference = 7.85″ structions used by drafters. 2. Describe one method for constructing Drafting Practice a geometric shape. 5. Draw the gasket shown in Figure 5-47. 3. Explain how to solve technical and Before beginning, determine an appropriate mathematical problems through geometric scale and sheet size. Do not dimension. constructions using drafting instruments. Ø64 R48 Academic Integration 2x Ø24 Mathematics R24 4. Calculate Circumference Calculate the circumference of a circle with a dia- meter of 2.50 inches. Calculating Circumference 178 METRIC To fi nd the circumference of a circle:

DIAMETER (d) = 2.50 Figure 5-47

Go to glencoe.com for this book’s OLC for help with this drafting practice.

158 Chapter 5 Geometry for Drafting 5.2 Applied Geometry for CAD Systems

Preview In this section, you will learn to construct geomteric shapes using CAD techniques. Content Vocabulary • object snap • ogee curve Academic Vocabulary Learning these words while you read this section will also help you in your other subjects and tests. • intervals • specify Graphic Organizer Use a diagram like the one below to organize the CAD commands discussed in the section.

Go to glencoe.com for this Inscribe a polygon book’s OLC for a downloadable version of this graphic organizer.

POLYGON

Academic Standards

English Language Arts NCTE National Council of Teachers of English Students employ a wide rage of strategies as they write and use diff erent writing process elements NCTM National Council appropriately to communicate with diff erent audiences for a variety of purposes (NCTE) of Teachers of Use information resources to gather information and create and communicate knowledge (NCTE) Mathematics

Mathematics

Geometry Specify location and describe spatial relationships using coordinate geometry and other representational systems (NCTM)

Section 5.2 Applied Geometry for CAD Systems 159 specify an object snap, type the fi rst three let- Using Geometry with ters of its name. CAD Systems What do object snaps allow a drafter to do?

The techniques for creating geometry in Explain How do you specify an object snap? AutoCAD and other CAD programs differ sig- nifi cantly from board drafting techniques. With CAD programs, the software creates the Bisect or Divide a Line, geometry, but you must understand the geo- metric principles before you can direct the an Arc, or an Angle software to create the geometry to achieve What actions do the LINE, TRIM, and the correct result. DIVIDE commands perform? This section consists of a series of exam- ple exercises in which you will use CAD Lines and arcs are usually bisected to fi nd a techniques to create the same geometry beginning point for a new line or arc. described in the fi rst section of this chapter. You can also use the same techniques Bisect a Line or an Arc described in that section. However, drafters In AutoCAD, the point that lies at the who use CAD systems usually take advan- exact middle of a line or arc is known as the tage of the streamlined methods when the midpoint. Because AutoCAD has a Midpoint software offers them. By working through object snap, bisecting a line or arc—fi nding its these constructions, you will begin to under- midpoint—is simply part of the construction stand how to draw the basic geometry in of the new line or arc. AutoCAD. 1. Draw a line and an arc (Figure 5-48A). To work through the constructions, open 2. Enter the LINE command, but do not a new drawing in AutoCAD. Use the tem- enter a fi rst point. Instead, type MID (for plate specifi ed by your instructor, or start midpoint) and press Enter. a new drawing using AutoCAD’s default 3. At the “of” prompt, select the line you acad.dwt template. Your instructor will advise drew in step 1. Depending on the version you on how many constructions to include in of AutoCAD you are using, you may see each drawing fi le. Be sure to save your work a yellow triangle appear at the midpoint frequently. of the line. In any case, the fi rst point of the new line you are creating begins at Object Snaps the exact midpoint of the original line, AutoCAD has a set of features known as shown as point C in Figure 5-48B. object snaps that allow you to “snap” auto- D matically to important points on any Auto- A A CAD object. Object snaps you will use in this C section include: SNAP TO MIDPOINT • Midpoint • Intersection B B • Nearest • Quadrant B B Endpoint Perpendicular • • C • Center • Tangent Specifying the Intersection object snap, for example, allows you to snap to the intersec- A A D tion of two existing lines or arcs. This can be AB useful if you have used two arcs to locate the beginning of a new line. Object snaps have Figure 5-48 many other uses, too, as you will see as you Bisecting a line or arc in AutoCAD work through the following constructions. To

160 Chapter 5 Geometry for Drafting 4. Pick another point anywhere in the sor to draw a radius similar to the one in drawing area and press Enter to end the Figure Figure 5-49B. LINE command. 3. Enter the TRIM command and press 5. Repeat steps 2 through 4, but this time Enter to select all of the objects on the select the arc in step 3. This results in a line screen automatically. Then pick any that starts at point C and bisects the arc point on the circle outside angle AOB. (Figure 5-48B). This procedure trims away all of the cir- cle except for an arc that extends from Bisect an Angle one arm of angle AOB to the other. See The CAD method for bisecting an angle is Figure 5-49C. very similar to the board drafting method. 4. Enter the LINE command. Use the Inter- Refer to Figure 5-49. section object snap to place the fi rst point of the line at point O. Then use the 1. Use the LINE command to draw two con- Midpoint object snap to place the second nected line segments to create angle AOB point of the line at the exact midpoint of (Figure 5-49A). the arc. Refer again to Figure 5-49C. This 2. Enter the CIRCLE command and specify line bisects angle AOB. point O as its center point. Use the cur-

A A A

C C

O O O

D D

B B B

A B C

Figure 5-49 Bisecting an angle in AutoCAD

Divide a Line into Eight Equal point style from the dialog box that appears. Parts See Figure 5-50B. AutoCAD includes a DIVIDE command that divides lines, arcs, and other geometry AB into equal parts. The following procedure divides a line into eight equal parts. Refer to Figure 5-50. AB 1. Draw a line of any length as in Figure 5-50A. Figure 5-50 2. Enter the DIVIDE command. Dividing a line into equal parts in AutoCAD 3. When prompted for the number of seg- ments, type 8 and press Enter. Markers appear at equal intervals along the line to divide it into eight parts. If you cannot see these markers, you will need to change the point style. To do so, enter DDP- Explain Why might a line or arc need to be TYPE at the keyboard and select a different bisected?

Section 5.2 Applied Geometry for CAD Systems 161 3. Before specifying the second point of the line, type PER to enter the Perpendicular object snap. Then pick a point on line AB and press Enter. The resulting line is per- Presetting Object pendicular to line AB. Snaps Construct Lines Parallel to a If you know that you will be using Given Line certain object snaps frequently for a To create parallel lines in AutoCAD, use the particular drawing, you can set OFFSET command. Refer to Figure 5-52. AutoCAD to use them automatically, without having to specify them each 1. Draw the line AB. time you use them. Object snaps that 2. Enter the OFFSET command and enter have been preset in this way are known an offset distance of 1. This will place the as running object snaps. To set running second line 1 unit away from line AB. object snaps, enter the OSNAP com- 3. When prompted to select the object to mand. A dialog box appears. Pick the offset, pick line AB. Object Snap tab of the dialog box to 4. When prompted for the side to offset, see the available object snaps. Pick the pick a point anywhere above line AB. The check boxes next to the object snaps parallel line CD appears. you want to run automatically and pick OK to close the dialog box. Notice that the OFFSET command is still active. You can offset as many lines or arcs as you want without reentering the command. This can save time when you are working on a technical drawing. Construct Lines with a 5. Press Enter to end the command. D

CAD System B B Most CAD systems can construct a full vari- C ety of lines. A A

Construct a Perpendicular Line Figure 5-52 Follow these steps to create a line perpen- Creating a line parallel to a given line using the dicular to a given line. Refer to Figure 5-51. OFFSET command 1. Draw given line AB. 2. Reenter the LINE command and pick point O as the fi rst point of the new line. Construct a Polygon AutoCAD provides a POLYGON command to create regular polygons with 3 to 1,024

USE THE sides. Equilateral triangles and squares are O PERPENDICULAR examples of regular polygons that have three OBJECT SNAP FOR SECOND POINT and four sides, respectively. The constructions

B in this section use the following geometry: • square, or four-sided polygon A • pentagon, or fi ve-sided polygon • hexagon, or six-sided polygon Figure 5-51 Creating a line perpendicular to a given line Create a Square through a point that does not lie on the given line Use this method to construct a polygon, in this case a square, when you know the length

162 Chapter 5 Geometry for Drafting of one of its sides. It can be very useful when the center of the polygon to be. Instead of pick- you need to construct a polygon that shares ing a point on the circle to defi ne the radius, a line with other geometrical shapes in the enter a numerical value at the keyboard. drawing. Circumscribe a Hexagon about 1. Enter the POLYGON command, and specify 4 as the number of sides. Press E a Circle (Edge) and pick a point on the screen. This method circumscribes a hexagon 2. Either pick another point on the screen about a circle with a known center point and for the second endpoint of the edge or radius. Refer to Figure 5-54. use polar coordinates to specify where 1. Create the circle. the endpoint should be. If you use polar 2. Enter the POLYGON command and coordinates, the length of the line you specify 6 sides. specify becomes the length of one side 3. Use the Center object snap to select the of the square. The square appears on the center of the circle as the center point of screen. the hexagon. 4. Enter C (Circumscribed) to circumscribe Inscribe a Pentagon in a Circle the polygon about the circle. When Use this method to inscribe a pentagon in prompted for the radius of the circle, use a circle with a known center point and radius. the Nearest object snap to snap to a point Refer to Figure 5-53. on the circle. The hexagon appears inside 1. Create the given circle. the circle, with the point you picked on 2. Enter the POLYGON command and spec- the circle as one of the vertices. ify 5 sides. 3. Use the Center object snap to select the center of the circle as the center point of the pentagon. 4. Enter I (Inscribed) to inscribe the poly- gon in the circle. When prompted for the radius of the circle, use the Nearest object snap to snap to a point on the circle. The pentagon appears inside the circle with the point you picked using the Nearest object snap as one of the vertices. Figure 5-54 Using the POLYGON command to circumscribe a hexagon about a circle

You can use this method to “circumscribe” a polygon about a circle even if the circle does not exist. Follow the preceding four steps, but for the center point, pick a point where you Figure 5-53 want the center of the polygon to be. Instead Using the POLYGON command to inscribe a of picking a point on the circle to defi ne the regular pentagon within a circle radius, enter a numerical value at the keyboard.

You can use this method to “inscribe” a Construct an Ellipse polygon in a circle even if the circle does not Of the two axes of an ellipse, the shorter exist. Follow the preceding four steps, but for axis is the minor axis, and the longer one the center point, pick a point where you want is the major axis. In AutoCAD, the ELLIPSE

Section 5.2 Applied Geometry for CAD Systems 163 command allows you to create ellipses FIRST AXIS (MAJOR) THIRD of any size by defining the axes. Refer to POINT

Figure 5-55. FIRST POINT SECOND 1. Enter the ELLIPSE command and pick a POINT point anywhere in the drawing area as the fi rst endpoint of the fi rst axis. 2. Pick another point as the second end- SECOND AXIS point of the fi rst axis. (MINOR) 3. As the ellipse begins to appear on the Figure 5-55 screen, select a third point to specify the Using the ELLIPSE command. other axis. Notice that you do not have to specify two points for the second axis. When you specify the third point, AutoCAD calculates the last point automatically, so that the second axis is at right angle to the fi rst.

Note that you can control the orientation Copy an Angle of the angle by entering a numerical value This construction demonstrates a method for the angle of rotation instead of using the of copying a given angle to a new location cursor. and orientation. Refer to Figure 5-56. 1. Draw the angle AOB (Figure 5-56A). 2. Enter the COPY command and use a window to select both arms of the angle. To do this, pick a point below and to the AB C right of the angle, and then pick another A A A

point above and to the left of the angle. O O ORIGINAL O ORIGINAL The selected lines become dashed to show that they are selected. Press Enter to proceed to the next prompt. B B B P 3. For the point of displacement, pick C point O. P 4. When asked for the second point of dis- COPY ROTATED placement, pick another point anywhere COPY C D on the screen. An exact copy of angle D AOB appears (Figure 5-56B). Press Enter to end the COPY command. Figure 5-56 5. To change the orientation of the sec- Copying and changing the orientation of an angle ond angle, enter the ROTATE command, in AutoCAD. select both legs of the second angle, and press Enter. 6. Specify a point anywhere on the angle as the base point. This is the point about

which the angle will rotate. Contrast How can you control the orientation 7. Move the cursor to reposition the angle of an angle other than by using the cursor? at a new orientation (Figure 5-56C).

164 Chapter 5 Geometry for Drafting 3. Create a second circle with the same Construct a Triangle radius, placing its center point at point B This type of polygon can be created using Refer again to Figure 5-57B. the LINE command as described in the fol- 4. Enter the LINE command and enter END lowing methods. Note: The POLYGON com- to use the Endpoint object snap to place mand is usually used to create an equilateral the fi rst point of the line at point A. triangle. Use the Intersection object snap to place the second point of the line at the upper Construct an Isosceles Triangle intersection of the two circles. Then use The following method is for constructing the Endpoint object snap for point B. See an isosceles triangle. Refer to Figure 5-57. Figure 5-57C. 1. Draw the given base line AB (Figure 5. Erase the two circles. The remaining tri- 5-57A). angle is an isosceles triangle. 2. Create a circle with its center point at point A and a radius equal to the length of the sides you want. See Figure 5-57B.

USE INTERSECTION VERTEX OBJECT SNAP

R R BASE A B A B A B A

B C

Figure 5-57 Constructing an isosceles triangle using AutoCAD

Construct a Right Triangle 3. Use the Endpoint object snap to place Construct a right triangle using this method the third point at point A, completing when you know the length of two sides of the the right triangle. triangle. In this construction, sides AB and BC C are given. Side AB is 2.50 units long, and side BC is 3.25 units long. Refer to Figure 5-58.

1. Draw side AB using the LINE command USE ENDPOINT 2 OBJECT SNAP and polar coordinates: @2.50<0. Leave 3 the LINE command active. @3.25<90 B 2. Specify the coordinates for side BC: A @2.50<0 @3.25<90. This creates line BC perpendic- 1 ular, to side AB. Leave the LINE command Figure 5-58 active. Constructing a right triangle in AutoCAD given the length of two sides

Section 5.2 Applied Geometry for CAD Systems 165 Construct Tangents You already know several methods for creating a circle: • specify a center point and a radius • substitute the diameter for the radius by Object Tracking pressing the D key before entering the The process for extending a numerical value line that is described in step 4 of • specify two points on the diameter of the Construct a Tangent Line is known as object tracking. If this does not seem circle to work for you, enter the OSNAP • specify three points on the diameter of the command, go to the Object Snap tab, circle and make sure the Object Snap Track- AutoCAD also allows you to create a circle ing On box is checked. If this option is that is tangent to two other objects in AutoCAD not available in your version of Auto- by specifying the tangent objects and a value CAD, you can achieve the same eff ect for the radius of the circle. As you may recall, a by using the EXTEND command. line is tangent to a circle if the line touches the circle at one point only. Refer to Figure 5-59. 1. Before you can use this option, you must have at least two lines in the drawing to specify as tangents. Use the LINE com- (5.00,7.50) mand to create the two lines. Use coor- (6.00,7.00) dinate values to place the endpoints of the lines at the coordinates shown in the illustration. 2. Enter the CIRCLE command. Enter T at (2.50,5.50) the keyboard to select the tan tan radius (6.50,5.00) (Ttr) option. 3. At the appropriate prompts, pick any- Figure 5-59 where on the two lines as the two tan- Creating a circle tangent to two other objects gents. Specify a radius of 1.00. The circle given the radius of the circle. appears as in Figure 5-59.

Construct a Tangent Line same general direction. AutoCAD displays Because AutoCAD has a Tangent object an “Extension” message that shows the snap, creating tangent lines is fairly easy. Refer length and angle of the extended line. to Figure 5-60. P TANGENT 1. Draw a circle anywhere in the drawing POINT area using the cursor to specify any radius. 2. Enter the LINE command. Pick any point outside the circle as the fi rst point of the line. 3. Enter the Tangent object snap and move the cursor near the circle. Select a point on the circle. The line automatically Figure 5-60 snaps to the tangent point on the circle. Using the Tangent object snap to create a line 4. To extend the line beyond the tangent tangent to a circle. point, keep moving the cursor in the

166 Chapter 5 Geometry for Drafting 2. Use the OFFSET command to offset both lines 1 unit to the inside.

Summarize How do you create a circle that 3. Enter the ARC command. At the prompt, is tangent to two other objects in AutoCAD? enter C (Center), and use the Intersec- tion object snap to snap to the intersec- tion of the two lines you offset in step 2. Construct a Tangent Arc 4. Use the Perpendicular object snap to AutoCAD has an ARC command that gives place the ends of the arc perpendicular CAD users great fl exibility in creating arcs. to lines AB and CD (Figure 5-61B). However, sometimes the best solution is to use the tan tan radius option of the CIRCLE com- mand, trimming away the unneeded parts of A D A D the circle. This section illustrates a few of the 1 1 ways to create arcs in AutoCAD. O O Construct an Arc Tangent to B B Two Lines C C The procedure for constructing an arc tan- AB gent to two lines in AutoCAD is similar to the board drafting procedure. In CAD, the pro- Figure 5-61 cedure is the same whether the angle is an Using AutoCAD to create an arc tangent to two acute, obtuse, or right angle. Therefore, only lines an acute angle is shown in Figure 5-61. 1. Draw given lines AB and CD (Figure 5-61A).

Construct an Arc Tangent to C

Two Given Arcs TANGENT This method uses the CIRCLE command POINT 2 TANGENT B D to construct an arc tangent to two given arcs. POINT 1 Refer to Figure 5-62. 1. Enter the ARC command and follow the prompts to enter the start point, second point, and endpoint of arcs AB and CD. The A radii of the arcs may be equal or unequal. BREAK 2. Enter the CIRCLE command. At the POINTS prompt, enter T (tan tan radius). Select points on the given arcs near the tangent Figure 5-62 locations. Note that you have only to Using the CIRCLE command to construct an arc pick a point somewhere near the tangent tangent to two given arcs point. AutoCAD calculates the exact tan- gents for you. 3. Specify a radius of 1.50 to make the tan- gent circle appear. 4. Enter the BREAK command, and pick two points on the circle to break the arc out of the circle. Use the ERASE com- Determine How does the CAD procedure mand to erase the unwanted portion of for constructing an arc tangent to two lines the circle. The remaining arc is tangent diff er for acute, obtuse, and right angles? to the two given arcs.

Section 5.2 Applied Geometry for CAD Systems 167 4. Create two circles. For the fi rst, use the Construct an Ogee Curve intersection of the vertical line from An ogee curve is a reverse curve that looks point B and the lower perpendicular as something like an S. The CAD procedure for the center point. For the radius, enter drawing an ogee curve is similar to the board the Endpoint object snap and snap to drafting procedure. Refer to Figure 5-63. point E. For the second circle, use the 1. With Ortho on, draw lines AB and CD intersection of the vertical line from (Figure 5-63A). Then turn Ortho off point C and the upper perpendicular as and use the Endpoint object snap with the center point. For the radius, use the the LINE command to draw line BC. Endpoint object snap to snap to point 2. Enter the BREAK command. This com- E. It does not matter if the circles extend mand is used to “break” a single line, arc, off the screen. (Figure 5-63C). circle, or other geometry into two distinct 5. Notice that the two circles are tangent to objects. At the prompt, enter F (First), and each other at point E. One circle is also use the Nearest object snap to pick a point tangent to line AB, and the other is tan- E on line BC through which the curve is gent to line CD. To fi nish the ogee curve, to pass. Refer again to Figure 5-63. Line BC enter the TRIM command, press Enter to becomes two lines: BE and EC. select all of the objects, and trim away 3. Construct perpendiculars at the mid- the unwanted parts of the circles. Erase points of lines BE and EC. The length of lines BE, EC, and the vertical and perpen- the perpendicular lines does not matter. dicular lines. See Figure Figure 5-63D Erase any circles or arcs used for con- for the fi nished curve. struction before continuing to step 4 (Figure 5-63B).

BREAK POINT C D C D

E E

A B A B

A B

C D C D

E E

A B A B C D

Figure 5-63 Creating an ogee curve in AutoCAD.

168 Chapter 5 Geometry for Drafting 3. The base point is the point around which Reduce or Enlarge a the scaling will occur. Use the Center Drawing object snap to select the center of the cir- To change the size of objects in an Auto- cles for the base point. CAD drawing, you can use the SCALE com- 4. Enter a scale factor of .75 to scale the cir- mand. Note that this process is different from cles to 75% of their original size (Figure using the ZOOM command to make objects 5-64B). You can check their size by using on the screen appear larger or smaller. It is also the grid, remembering that the dots on different from choosing a standard scale in the grid are spaced at intervals of .50. paper space to scale a drawing for printing. When you use the SCALE command, you change the actual dimensions of the objects you see on the screen. You can scale all of the objects in the drawing at once or scale only those objects that you select. This construction demonstrates the effect of scaling objects in AutoCAD. Refer to Figure 5-64. AB 1. Set the snap and grid to .50. Use the Figure 5-64 snap, grid, and coordinate display to create two concentric circles (both with Using the SCALE command to enlarge or reduce the same center point). Make the radius the physical size of a drawing in AutoCAD. of one circle 2.00 units, and make the radius of the second circle 1.00 unit Notice that you must enter a decimal frac- (Figure 5-64A). tion. The number 1 stands for 100%, or full 2. Enter the SCALE command. Pick both size. If you enter 75, the circles will enlarge to circles to scale, and press Enter. 75 times their original size.

Section 5.2 Assessment Academic Integration After You Read English Language Arts 3. Read the following content vocabulary and technical terms from this chapter. Self-Check Organize the terms using one of them as 1. Describe how technical and mathemat- the heading under which the others are ical problems related to geometric con- listed as examples. structions can be solved using CAD. isosceles equilateral 2. Explain how to reduce or enlarge the triangle scalene physical size (dimensions) of a drawing using CAD. Drafting Practice Repeat the board drafting practice in Section 5.1, this time using CAD techniques.

Go to glencoe.com for this book’s OLC for help with this drafting practice.

Section 5.2 Applied Geometry for CAD Systems 169 5 Review and Assessment Chapter Summary Section 5.1 Section 5.2 • Geometry is the study of the size and • Using CAD object snaps for geometric shape of objects and their relationship to constructions greatly increase the effi - each other. ciency of the drawing process and reduces • Drafters, surveyors, engineers, architects, the time involved in preparing accurate, scientists, mathematicians, and design- high-quality drawings. ers use geometric constructions to show • In CAD, many commands are available proper relationships between individual for drawing basic geometric shapes. lines and points. Examples include CIRCLE, POLYGON, • Geometric shapes discussed in this ARC, and ELLIPSE. chapter include lines, triangles, squares, circles, arcs, angles, pentagons, hexagons, polygons. • The most important principles of drafting include accuracy. Work that is not accurate may give designers wrong information.

Review Content Vocabulary and Academic Vocabulary 1. Use each of these content and academic vocabulary words in a sentence or drawing. Content Vocabulary • polygon (p. 144) Academic Vocabulary • geometry (p. 135) • inscribe (p. 152) • accurate (p. 135) • geometric construction (p. 137) • circumscribe (p. 152) • methods (p. 139) • vertex (p. 138) • regular polygon (p. 152) • intervals (p. 161) • bisect (p. 138) • ellipse (p. 155) • specify (p. 163) • perpendicular (p. 139) • object snap (p. 160) • parallel (p. 139) • ogee curve (p. 168)

Review Key Concepts 2. List geometric shapes that drafters use. 3. Demonstrate how to construct various geometric shapes accurately. 4. Describe how technical and mathematical problems related to geometric constructions can be solved using board-based drafting. 5. Describe how technical and mathematical problems related to geometric constructions can be solved in a computer environment. 6. List the steps involved in using geometry to enlarge or to change a drawing’s proportions.

170 Chapter 5 Geometry for Drafting Engineering Prep 7. What do Engineers Do? Multiple Choice Question Webster’s Dictionary defi nes engineering For Directions Choose the letter as “the application of science and mathemat- of the best answer. Write the ics by which the properties of matter and the letter for the answer on a sources of energy in nature are made useful to separate piece of paper. people.” According to the National Academy 10. Which of the following is an example of Engineering (NAE), there are more than of a polygon? two million practicing engineers in the United A. Circle States. What are all these engineers doing? In B. Angle what ways are engineers making things use- C. Triangle ful for people? Using the Internet or library, D. Parallel Line research a type of engineering, such as archi- tectural or biomedical engineering. Then write a one-page paper, summarizing what makes TEST-TAKING TIP the fi eld of engineering important and name In a multiple-choice test, the answers one major innovator working in the fi eld. should be specifi c and precise. Read the question fi rst, then read all the answer choices before you choose. Eliminate answers that you know are incorrect. 8. Productivity and Accountability You and a classmate have been assigned a project that represents a signifi cant part of your grade. You are both to participate equally Win in completing it. The two of you agree to the Competitive parts for which each will be responsible. You fi nished your work, but your partner did not. Events How do you handle this situation? Prepare a 11. Technical Math bulleted list to show your options, to use as a Organizations such as SkillsUSA offer a basis for a class discussion. variety of architectural, career, and draft- ing competitions. Completing activities such as the one below will help you pre- Mathematics pare for these events. 9. Calculate Area Activity Complete the STEM Math- Determine the area of a triangle with a base ematics exercise on this page. Then team of 6 inches and a height of 2 inches. with a partner and check each other’s work, going over any concepts that Calculating Area might be unclear.

To fi nd the area of a triangle, multiply the base (b) times the height (h) and divide by Go to glencoe.com for this book’s two. Area = bh/2. OLC for more information about competitive events. ALTITUDE (h) BASE (b) 2"

6"

Review and Assessment 171 5 Problems

Drafting Problems The problems in this chapter can be per- formed using board drafting or CAD tech- niques. The problems are presented in order of diffi culty, from least to most diffi cult. Problems 1 through 18 are designed for work- ing four problems on an A-size sheet, laid out as shown in Figure 5-65. Draw each problem three times the size shoown. If you are using board drafting, use dividers to pick up the Figure 5-65 dimensions from the problems, and step off each measurement three times. If you are using a CAD system, use a scale to measure the dimen- 9. Draw a circle with a 3″ diameter Figure sions, and create the geometry in the CAD sys- 5-66I. Inscribe a square in the circle. tem at three times the measured size.

10. Draw a circle with a 3″ diameter Figure 1. Draw and bisect line AB Figure 5-66A. 5-66I. Inscribe a regular pentagon in the circle. 2. Draw line AB Figure 5-66B. Construct a

perpendicular at point P. 11. Draw a circle with a 3″ diameter Figure

5-66I. Circumscribe a regular hexagon 3. Draw line AB Figure 5-66C. Divide line about the circle.

AB into fi ve equal parts. 12. Draw a circle with a 3″ diameter Figure

4. Draw line AB Figure 5-66D. Construct 5-66I. Circumscribe a regular octagon

line CD through point P so that CD is par- about the circle. allel to AB and equal in length to line AB. 13. Draw a circle with a 3″ diameter Figure

5. Draw angle ABC Figure 5-66E. Bisect 5-66J. Construct a tangent line through

angle ABC. point P.

6. Draw angle ABC Figure 5-66F. Copy 14. Locate points A, B, and C on the drawing

the angle in a new location, beginning sheet Figure 5-66K. Construct a circle through these three points. with line A1B1.

7. Draw base line AB Figure 5-66G. Con- 15. Draw the two lines shown in Figure

struct an isosceles triangle using base line 5-66L. Construct an arc having a radius AB and sides equal to line CD. R tangent to the two lines.

8. Draw base line AB Figure 5-66H. Con- 16. Draw the two arcs shown in Figure

struct a triangle on base AB with sides 5-66M. Construct an arc having a radius equal to BC and AC. R tangent to the fi rst two arcs.

172 Chapter 5 Geometry for Drafting 17. Draw a 3.00″ square Figure 5-66N. Construct a regular octagon within the square.

18. Construct an ellipse that has a 4.00″ major axis and a 2.50″ minor axis Figure 5-66O.

A

P B AB

30° AB C

A B1 A

A1 P C

B B B A DE F

D C C

A BASE B C A BASE B B A GH I

R P B A

C

JK L

MAJOR AXIS

R MN O

Figure 5-66

Problems 173 5 Problems

Problems 19 through 24: These problems provide additional practice in geometric con- 21. Draw the adjustable fork shown in structions. They are designed to be drawn Figure 5-69. Use the following dimen- one per drawing sheet. Before beginning each sions: A = 220 mm; B = 80 mm; C = drawing, determine an approximate scale and 40 mm; D = 27 mm; E = 64 mm; F = sheet size. Do not add dimensions to your 20 mm; G = 8 mm; H = 10 mm. drawing. G F

H 19. Draw the handwheel shown in Figure E 5-67. Use the following dimensions: C D A = Ø7.00″; B = Ø6.12″; C = Ø5.50″; D = R1.25″; E = Ø2.00″; F = Ø1.00″; G B (keyway) = .20″ wide × .10″ deep; H = A Ø.38″; I = R.38″; J = R.20″; K = 1.00″. METRIC A Figure 5-69 B H I C G D F E 22. Draw the rod support shown in Figure

5-70. K J

Figure 5-67 R3.50 5.50 Ø1.06 3 HOLES R1.00

20. Draw the combination wrench shown in

Figure 5-68. Use the following dimen- sions: square: 1.00″; octagon: 1.38″ across 2.75 fl ats; isosceles triangle: 2.75″ base, 2.00″ 7.62 sides; pentagon: inscribed within Ø1.38Љ ″ circle; hexagon: 1.25 across fl ats. If you Figure 5-70 are using board drafting techniques, do not erase construction lines.

R1.25

2.50 3.50 3.00 3.00 12.00

Figure 5-68 174 Chapter 5 Geometry for Drafting 23. Draw the adjustable table support shown 24. Draw the tilt scale shown in Figure 5-72.

in Figure 5-71. Use the following dimensions: AB = 44 mm; AX = 66 mm; AC = 140 mm; AD =

R1.00 184 mm; AE = 216 mm; AF = 222 mm; AG = 236 mm; H = R24 mm; I = R16 mm; 60? J = R5 mm; K = Ø12 mm. R.38

20 30 G R7.00 1 4 70 60 0 F R1.62 0 50 45 90 E J ° R.75 50 D R1.00 15? 45° C ° R.38 45 Ø.75 4.00 Figure 5-72 8.00 B

I K Figure 5-71 A H X METRIC Design Problems Design problems have been prepared to challenge individual students or teams of students. In these problems, you will apply skills learned mostly in this chapter but also in other chapters throughout the text. The problems are designed to be completed using board drafting, CAD, or a combination of the two. Be creative and have fun! Challenge Your Creativity 2. Design an octagon-shaped jewelry box

1. Design an educational toy to help tod- with a hinged lid. The overall size should dlers develop manual dexterity, spatial not exceed 160 mm across the corners relationships, and color association. The of the octagon by 90 mm high. Material: toy should be similar to Figure 5-73, but optional. Do not dimension. expanded to include at least six geomet- ric shapes of different colors. Material: 1″ Teamwork thick pine. 3. Design and draw a cover for your 8.50″ ×

11.00″ or 11.003 × 17.00″ set of technical drawings. Use various geometric shapes in the design. Geometric shapes, such as cir- cles, squares, hexagons, octagons, ellipses, 2 2 etc., can be used to enhance the design. Use colors where desired. Use block letters to add information on the cover, such as your name, the school name, the course title, the instructor’s name, and the year.

Figure 5-73

Problems 175 UNIT 1 Hands-On Math Project

Create a Logo for Your Own Business

Your Project Assignment TIP! A logo will be used to promote your Explore the opportunities to become an business and give it an identity. It should entrepreneur by working as a freelance draft- be eye-catching, simple, and speak to the sperson from your home. Create a logo for your needs of your potential customers. business. • Prepare three fi nished drawings of your logo Use what you have learned in Chapters 1–5 at three different sizes. to create a plan for starting your own freelance business. Your challenge is to: Applied Skills • Identify opportunities for employment as • List and categorize the opportunities you an independent draftsperson by researching uncover in your research. Include contact local classifi ed ads and regional and national information for each company, information online job search sites. about qualifi cations and requirements, type • Choose a focus for your home-based business of company, and the nature of the work. based on your interests and abilities. Will • Outline the educational requirements, and your clients be manufacturing companies, identify schools or programs where you could engineering fi rms, or architects? Are you obtain the necessary training. stronger at board drafting, or computer-aided • Write a paragraph or two about your interests drafting? and abilities. Discuss why you chose the focus • Explain the educational requirements for the for your business that you did. kind of work you have chosen. • List the steps, materials, and tools you used to • Create fi nished drawings for a business create the drawings for your logo. Explain the logo to use on stationery, business cards procedure you used to reduce or enlarge your brochures, etc. initial drawing to create three versions. The Math Behind the Project The primary math skills you will use to com- plete this project are geometry modeling, algebra, Math Standards and measurement. To get you started, remember these key concepts, and follow this example: Geometry Use visualization, spatial reasoning, and Geometry—Ratio, Proportion, geometric modeling to solve problems (NCTM) and Scale Problem Solving Solve problems that arise in To understand how to reduce or enlarge the mathematics and other contexts (NCTM) size of a drawing, think about the terms ratio, proportion, and scale. A ratio is a comparison NCTM National Council of Teachers of Mathematics of two numbers. For example, a rectangle has a

176 Chapter 5 Geometry for Drafting length of 2 inches and width of 3 inches. The unknown quantity. For example, consider this ratio of length to width is 2/3. problem: When two ratios are equal, they form a pro- What is the length of a rectangle 6 inches portion. One way to determine whether two wide that is proportional to another rectangle ratios form a proportion is to check their cross 2 inches wide and 3 inches long? products. For example, to fi nd out if a 2/3 rect- Use l to represent the length of the enlarged angle is proportional to a 6/8 rectangle, multiply rectangle drawn to scale. the numerator of each ratio by the denominator __ 2 = __ 6 2l = 6 × 3 2l = 18 l = 9 of the other. If the resulting products are equal, 3 l the fi gures are proportional. The length should be 9 inches. __ 2 ? __ 6 2 × 8 ? 6 × 3 16 ≠ 18 To determine the scale factor of the enlarged 3 8 drawing, write a ratio comparing similar sides, Since the products are not equal, the rectan- and reduce. For example, 6/2 or 9/3. In both gles are not proportional. In other words, they cases, the ratios reduce to 3. When a fi gure is are not drawn to scale. enlarged, the scale factor is greater than one. When two similar fi gures are identical in size, Solving Proportion Problems the scale factor is equal to one. When a fi gure is To use proportions to solve problems, set reduced, the scale factor is less than one. up two ratios using a letter symbol for the

Designers of Famous Logos In the United States and around the world, famil- iar corporate logos dot the landscape. You can spot your favorite fast food restaurant or gas station from far away because their powerful logos are easy to rec- ognize and prominently displayed. What makes these symbols so eff ective? Who designed them? One of the most infl uential logo designers of the twentieth century is Milton Glaser. He designed the famous “I Love New York” logo. He also designed the “bullet” you see on DC Comics. A good logo catches the eye. It may also say something about the product or service off ered, or make the observer curious. Research Activity Find out more about Milton Gla- ser and the things he has designed. What characteris- tics do his logos and other objects have in common? Also research the principles behind good logo design. Write a one-page summary of your fi ndings. Bonus! Incorporate the principles of good logo design into your creation.

Unit 1 Hands-On Math Project 177 Car Culture/Corbis UNIT 1 Hands-On Math Project

Project Steps: enlarge or reduce it so you have three ver- sions: one should be sized for use on a busi- Design Your Future! ness card, one for use on stationery, and one STEP 1 Research for use on a Web site. TEAMWORK Collaborate: Ask a classmate to review the design of your logo before you Explain the type of drafting work you are best • continue. Ask for feedback on the technical suited for and most interested in pursuing. aspects of your drawing as well as the overall • Look for job opportunities in local classifi ed concept. ads and on the Internet. Make phone calls to these companies to fi nd out more about STEP 4 available opportunities. Present • Phone other similar fi rms in your area and ask if they ever hire outside fi rms to handle Prepare a presentation combining your any of their drafting needs. research with your completed drawings using the checklist below. • Find out more about logo design and think about what you want to communicate with your logo. Presentation Checklist TIP! Write a script, and practice your phone Did you remember to… inquiry skills before you call prospective clients. ✓state your objectives for your business? ✓ STEP 2 Plan describe the services you will off er? ✓show and discuss your logo? • Defi ne and write out your overall goal for ✓explain the process you used to create the this project. logo and what you hope to achieve with it? ✓ • Gather the appropriate supplies and tools for describe the services you will off er? board drafting. ✓show and discuss your logo? • Set up to prepare your drawing fi le with ✓explain the process you used to create the AutoCAD. logo and what you hope to achieve with it? ✓show your preliminary sketches and explain Refer to the Math Concepts on the how you created your logo? previous page, or go to glencoe.com ✓ for this book’s OLC for more informa- demonstrate the basic sketch or CAD drawing? tion on the math concepts used in ✓show the three versions of your logo and this project. explain how they could be used? ✓review the drafting principles involved in completing your logo? STEP 3 Apply ✓explain any problems you encountered and how you overcame them? • Make several preliminary sketches of ideas ✓turn in your research and planning notes to you have for your logo. your teacher? • Complete one version of your logo, then

178 Chapter 5 Geometry for Drafting STEP 5 Build Your Portfolio

The purpose of a portfolio is to showcase your education and examples of your work and accomplishments. The purpose of a portfolio is to showcase • Organize your drawings in a manner that will your education and examples of your show your ideas well. work and accomplishments. A typical portfolio might include the following: • Attach a written introduction and a descrip- tion of your design. • Career summary and goals • Résumé STEP 6 • List of accomplishments Evaluate Your Technical • Education and certifi cations Skills • Samples of your work • Job or Job-shadowing experience Assess yourself before and after your Getting Started presentation. To prepare the written components for 1. Is your research thorough? your portfolio, you will need access to a 2. Did you plan your steps carefully? computer with Microsoft Word, Pages, 3. Did you organize your visuals so that they or other word processing application. showcase your ideas? Use this software to create the written 4. Is your presentation creative and effective? components of your portfolio. 5. During your presentation, do you make eye contact and speak clearly enough? 1. Career summary and goals: Prepare a brief summary of your specifi c career goals. Describe the Rubrics Go to glencoe.com to this industry or job that interests you. book’s OLC for a printable evaluation rubric and Academic Assessment. 2. Résumé: If you have not already done so, use the information from Chapter 1 to prepare your résumé. Include in your résumé a list of accomplishments, education, and certifi cations you hold. 3. Samples of your work: Now that you have completed your business planning and design project for this unit, include your drawings as sam- ples of your work in your portfolio. Save Your Work In the following Units, you will add more elements to your portfolio. Keep items you want to save for your portfolio in a special folder as you progress through this class.

Unit 1 Hands-On Math Project 179 Image Source Black/Alamy