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).