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For Dummies is a registered trademark of Wiley Publishing, Inc. in the United States and other countries. Used here by license. Studying geometry is sort of a Dr. Jekyll-and-Mr. Hyde thing. You have the ordinary, everyday geometry of shapes (the Dr. Jekyll part) and the strange world of geometry proofs (the Mr. Hyde part). Every day, you see various shapes all around you (triangles, rectangles, boxes, circles, balls, and so on), and you’re probably already familiar with some of their properties: area, perimeter, and volume, for example. In this book, you discover much more about these basic properties and then explore more-advanced geometric ideas about shapes. Geometry proofs are an entirely different sort of animal. They involve shapes, but instead of doing something straightforward like calculating the area of a shape, you have to come up with an airtight mathematical argument that proves something about a shape.
This process requires not only mathematical skills but verbal skills and logical deduction skills as well, and for this reason, proofs trip up many, many students. If you’re one of these people and have already started singing the geometry-proof blues, you might even describe proofs — like Mr. Hyde — as monstrous. But I’m confident that, with the help of this book, you’ll have no trouble taming them. This chapter is your gateway into the sensational, spectacular, and superduper (but sometimes somewhat stupefying) subject of this book: geometry. If you’re tempted to ask, “Why should I care about geometry?” this article will give you the answer.
Studying Shapes Have you ever reflected on the fact that you’re literally surrounded by shapes? Look around. The rays of the sun are — what else? — rays. The book in your hands has a shape, every table and chair has a shape, every wall has an area, and every container has a shape and a volume; most picture frames are rectangles, CDs and DVDs are circles, soup cans are cylinders, and so on and so on. Can you think of any solid thing that doesn’t have a shape? This section gives you a brief introduction to these one-, two-, and three-dimensional shapes that are all-pervading, omnipresent, and ubiquitous — not to mention all around you.
One-dimensional shapes There aren’t many shapes you can make if you’re limited to one dimension. You’ve got your lines, your segments, and your rays. That’s about it. But it doesn’t follow that having only one dimension makes these things unimportant — not by any stretch. Without these one-dimensional objects, there’d be no two-dimensional shapes; and without 2-D shapes, you can’t have 3-D shapes. Think about it: 2-D squares are made up of four 1-D segments, and 3-D cubes are made up of six 2-D squares. And it’d be very difficult to do much mathematics without the simple 1-D number line or without the more sophisticated 2-D coordinate system, which needs 1-D lines for its x- and y-axes.
Two-dimensional shapes As you probably know, two-dimensional shapes are flat things like triangles, circles, squares, rectangles, and pentagons. The two most common characteristics you study about 2-D shapes are their area and perimeter. These geometric concepts come up in countless situations in the real world. You use 2-D geometry, for example, when figuring the acreage of a plot of land, the number of square feet in a home, the size and shape of cloth needed when making curtains or clothing, the length of a running track, the dimensions of a picture frame, and so on.
The formulas for calculating the area and perimeter of 2-D shapes are covered in Parts III through V. I devote many chapters in this book to triangles and quadrilaterals (shapes with four sides); I give less space to shapes that have more sides, like pentagons and hexagons. Shapes of any number of straight sides, called polygons, have more-advanced features such as diagonals, apothems, and exterior angles. You may be familiar with some shapes that have curved sides, such as circles, ellipses, and parabolas. The circle is the only curved 2-D shape covered in this book. In Part V, you investigate all sorts of interesting circle properties involving diameters, radii, chords, tangent lines, and so on.
Three-dimensional shapes You work with prisms (a box is one example), cylinders, pyramids, cones, and spheres. The two major characteristics of these 3-D shapes are their surface area and volume. Three-dimensional concepts like volume and surface area come up frequently in the real world; examples include the volume of water in a fish tank or backyard pool. The amount of wrapping paper you need to wrap a gift box depends on its surface area. And if you wanted to calculate the surface area and volume of the Great Pyramid in Egypt — you’ve been dying to do this, right? — you couldn’t do it without 3-D geometry. Here are a couple of ideas about how the three dimensions are interrelated.
Two-dimensional shapes are enclosed by their sides, which are 1-D segments; 3-D shapes are enclosed by their faces, which are 2-D polygons. And here’s a nifty real-world example of the relationship between 2-D area and 3-D volume: A gallon of paint (a 3-D volume quantity) can cover a certain number of square feet of area on a wall (a 2-D area quantity). (Well, okay, I have to admit it — I’m playing a bit fast and loose with my dimensions here. The paint on the wall is actually a 3-D shape. There’s the length and width of the wall, and the third dimension is the thickness of the layer of paint. If you multiply these three dimensions together, you get the volume of the paint.)
Historical Highlights The study of geometry has impacted architecture, engineering, astronomy, physics, medicine, and warfare, among other fields, in countless ways for well over 5,000 years. I doubt anyone will ever be able to put a date on the discovery of the simple formula for the area of a rectangle (Area = length Å width), but it likely predates writing and goes back to some of the earliest farmers.
Some of the first known writings from Mesopotamia (in about 3500 B.C.) deal with the area of fields and property. And I’d bet that even pre-Mesopotamian farmers knew that if one farmer planted an area three times as long and twice as wide as another farmer, then the bigger plot would be 3 Å 2, or 6 times as large as the smaller one. The architects of the pyramids at Giza (built around 2500 B.C.) knew how to construct right angles using a 3-4-5 triangle. Right angles are necessary for the corners of the pyramid’s square base, among other things. And of course, you’ve probably heard of Pythagoras (circa 570–500 B.C.) and the famous right-triangle theorem named after him. Archimedes (287–212 B.C.) used geometry to invent the pulley. He developed a system of compound pulleys that could lift an entire warship filled with men.
The Chinese knew how to calculate the area and volume of many different geometric shapes and how to construct a right triangle by 100 B.C. In more recent times, Galileo Galilei (1564–1642) discovered the equation for the motion of a projectile and designed and built the best telescope of his day. Johannes Kepler (1571–1630) measured the area of sections of the elliptical orbits of the planets as they orbit the sun. René Descartes (1596–1650) is credited with inventing coordinate geometry, the basis for most mathematical graphing (see Chapter 18).
Isaac Newton (1642–1727) used geometrical methods in his Principia Mathematica, the famous book in which he set out the principle of universal gravitation. Closer to home, Ben Franklin (1706–1790) used geometry to study meteorology and ocean currents. George Washington (1732–1799) used trigonometry (the advanced study of triangles) while working as a surveyor before he became a soldier. Last but certainly not least, Albert Einstein discovered one of the most bizarre geometry rules of all: that gravity warps the universe. One consequence of this is that if you were to draw a giant triangle around the sun, the sum of its angles would actually be a little larger than 180°. This contradicts the 180° rule for triangles, which works until you get to an astronomical scale. The list of highlights goes on and on.
Geometric Proofs Geometry proofs are an oddity in the mathematical landscape, and just about the only place you find geometry proofs is in a geometry course. If you’re in a course right now and you’re wondering what’s the point of studying something you’ll never use again, I get back to that in a minute in the section “When Am I Ever Going to Use This?” For now, I just want to give you a very brief description of what a geometry proof is.
A geometry proof — like any mathematical proof — is an argument that begins with known facts, proceeds from there through a series of logical deductions, and ends with the thing you’re trying to prove. Mathematicians have been writing proofs — in geometry and all other areas of math — for over 2,000 years. (See the sidebar about Euclid and the history of geometry proofs.) The main job of a present-day mathematician is proving things by writing formal proofs. This is how the field of mathematics progresses: As more and more ideas are proved, the body of mathematical knowledge grows. Proofs have always played, and still play, a significant role in mathematics. And that’s one of the reasons you’re studying them. Part II delves into all the details on proofs, but in the sections that follow, I get you started in the right direction.
Everyday Example of Proofs You probably never realized it, but sometimes when you think through a situation in your day-to-day life, you use the same type of deductive logic that’s used in geometry proofs. Although the topics are different, the basic nature of the argument is the same. Here’s an example of real-life logic. Say you’re at a party at Sandra’s place. You have a crush on Sandra, but she’s been dating Johnny for a few months. You look around at the partygoers and notice Johnny talking with Judy, and a little later you see them step outside for a few minutes. When they come back inside, Judy’s wearing Johnny’s ring. You weren’t born yesterday, so you put two and two together and realize that Sandra’s relationship with Johnny is in trouble and, in fact, may end any minute.
You glance over in Sandra’s direction and see her leaving the room with tears in her eyes. When she comes back, you figure it might not be a bad idea to go over and talk with her. (By the way, this story about a party gone bad is based on Lesley Gore’s No. 1 hit from the ’60s, “It’s My Party.” The sequel song, also a hit, “Judy’s Turn to Cry,” relates how Sandra got back at Judy. Check out the lyrics online.) Now, granted, this party scenario might not seem like it involves a deductive argument. Deductive arguments tend to contain many steps or a chain of logic like, “If A, then B; and if B, then C; if C, then D; and so on.” The party fiasco might not seem like this at all because you’d probably see it as a single incident. You see Judy come inside wearing Johnny’s ring, you glance at Sandra and see that she’s upset, and the whole scenario is clear to you in an instant. It’s all obvious — no logical deduction seems necessary.
Turning everyday logic into a proof Imagine that you had to explain your entire thought process about the party situation to someone with absolutely no knowledge of how people usually behave. For instance, imagine that you had to explain your thinking to a hypothetical Martian who knows nothing about our Earth ways. In this case, you would need to walk him through your reasoning step by step. Here’s how your argument might go. Note that each statement comes with the reasoning in parentheses:
1. Sandra and Johnny are going out (this is a given fact).
2. Johnny and Judy go outside for a few minutes (also given).
3. When Judy returns, she has a new ring on her finger (a third given).
4. Therefore, she’s wearing Johnny’s ring (much more probable than, say,
that she found a ring on the ground outside).
5. Therefore, she’s going out with Johnny (because when a boy gives a girl
his ring, it means they’re going out).
6. Therefore, Sandra and Johnny will break up soon (because a girl will not
continue to go out with a guy who’s just given another girl his ring).
7. Therefore, Sandra will soon be available (because that’s what happens after someone breaks up).
8. Therefore, I should go over and talk with her (duh). This eight-step argument shows you that there really is a chain of logical deductions going on beneath the surface, even though in real life your reasoning and conclusions about Sandra would come to you in an instant. And the argument gives you a little taste for the type of step-by-step reasoning you use in geometry proofs. You see your first geometry proof in the next section.
Sampling a simple geometrical proof Geometry proofs are like the party argument in the preceding section, only with a lot less drama. They follow the same type of series of intermediate conclusions that lead to the final conclusion: Beginning with some given facts, say A and B, you go on to say therefore, C; then therefore, D; then therefore, E; and so on till you get to your final conclusion.
For this proof, you’re told that segment PS is congruent to (the same length as) segment WZ, that PQ is congruent to WX, and that QR is congruent to XY . (By the way, instead of saying is congruent to all the time, you can just use the symbol E to mean the same thing.) You have to prove that RS E YZ. Now, you may be thinking, “That’s obvious — if PS is the same length as WZ and both segments contain these equal short pieces and the equal medium pieces, then the longer third pieces have to be equal as well.” And of course, you’d be right. But that’s not how the proof game is played. You have to spell out every little step in your thinking so your argument doesn’t have any gaps. Here’s the whole chain of logical deductions:
1. PS E WZ (this is given).
2. PQ E WX and QR E XY (these facts are also given).
3. Therefore, PR E WY (because if you add equal things to equal things, you get equal
totals).
4. Therefore, RS E YZ (because if you start with equal segments, the whole segments PS and WZ, and take away equal parts of them, PR and WY , the parts that are left must be equal). In formal proofs, you write your statements (like PR E WY from Step 3) in one column, and your justifications for those statements in another column.
Hate proofs? Blame Euclid. Euclid (circa 385–275 B.C.) is usually credited with getting the ball rolling on geometry proofs. (If you’re having trouble with proofs, now you know who to blame!) His approach was to begin with a few undefined terms such as point and line and then to build from there, carefully defining other terms like segment and angle. He also realized that he’d need to begin with some unproved principles (called postulates) that he’d just have to assume were true. He started with ten postulates, such as “a straight line segment can be drawn by connecting any two points” and “two things that each equal a third thing are equal to one another.”
After setting down the undefined terms, the definitions, and the postulates, his real work began. Using these three categories of things, he proved his first theorem (a proven geometric principle), which was the side-angleside method of proving triangles congruent (see Chapter 9). And then he proved another theorem and another and so on. Once a theorem had been proved, it could then be used (along with the undefined terms, definitions, and postulates) to prove other theorems. If you’re working on proofs in a standard high school geometry course, you’re walking in the footsteps of Euclid, one of the giants in the history of mathematics — lucky you!
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For Dummies is a registered trademark of Wiley Publishing, Inc. in the United States and other countries. Used here by license.