CK-12 Geometry Read online

  The drawing takes only seconds to create, but it could help you visualize important information. Remember that there are many different ways to display information. Look at the way a line segment six inches long is displayed below.

  When you approach a problem, think about how you can represent the information in the most useful way. Continue your work on the sample problem by making drawings.

  Let’s return to that example.

  Example 1 (Repeated)

  Ehab drew a rectangle on the chalkboard. was and was . If Ehab draws in the diagonal , what will be its length?

  Think about the different ways in which you could draw the information in this problem. The simplest idea is to draw a labeled rectangle. Be sure to label your drawing with information from the problem. This includes the names of the vertices as well as the side lengths.

  As in most situations that you will encounter, there is more than one correct way to draw this shape. Two more possibilities follow.

  The first example above shows the internal structure of the rectangle, as it is divided into square centimeters. The second example shows the rectangle situated on a coordinate grid. Notice that we rotated the figure by in the second picture. This is fine as long as it was drawn maintaining side lengths. One implication of putting the figure on the coordinate grid is that one square unit on the grid is equivalent to one square centimeter.

  Identifying Your Strategy

  At this point, you have simplified the problem by asking yourself questions about it, and created different representations of the important information. The time has come to establish a formal plan of attack. This is a crucial step in the problem-solving process, as it lays the groundwork for your solution.

  To organize your thoughts, think of your geometric knowledge as a toolbox. Each time you learn a new strategy, technique, or concept, add it to your toolbox. Then, when you need to solve a problem, you can select the appropriate tool to use.

  For now, take a quick look at the representations drawn for the example problem to identify what tools you might need. You can use this section to clearly identify your strategy.

  Example

  Ehab drew a rectangle on the chalkboard. was and was . If Ehab draws in the diagonal , what will be its length?

  In the first representation, there is simply a rectangle with a diagonal. Though there is a way to solve this problem using this diagram, it will not be covered until later in this book. For now, you do not have the tools to solve it.

  The second diagram shows the building blocks that comprise the rectangle. The diagonal cuts through the blocks but presents the same challenges as the first diagram. You do not yet have the tools to solve the problem using this diagram either.

  The third diagram shows a coordinate grid with the rectangle drawn in. The diagonal has two endpoints with specific coordinate pairs. In this chapter, you learned the distance formula to find lengths on a coordinate grid. This is the tool you need to solve the problem.

  Your strategy for this problem is to identify the two endpoints of on the grid as and . Use the distance formula to find the length. The result will be the solution to the problem.

  Making Calculations

  The last step in any problem-solving situation is employing your strategy to find the answer. Be sure that you use the correct values as identified in the relevant information. When you perform calculations, use a pencil and paper to keep track of your work. Many careless mistakes result from mental calculations. Keep track of each step along the way.

  Finally, when you have found the answer, there are two more questions to ask yourself:

  1. Did I provide the information the problem requested?

  Go back to the first stages of the problem. Verify that you answered all parts of the question.

  2. Does my answer make sense?

  Your answer should make sense in the context of the problem. If your number is abnormally large or small in value, check your work.

  Example

  Ehab drew a rectangle on the chalkboard. was and was . If Ehab draws in the diagonal , what will be its length?

  At this point, we have distilled the problem, created multiple representations of the scenario, and identified the desired strategy. It is time to solve the problem.

  The diagram below shows the rectangle on the coordinate grid.

  To find the length of , you must identify its endpoints on the grid. They are and . Use the distance formula and substitute for , for , for , and for .

  is .

  Finally, make sure to ask yourself two more questions to verify your answer.

  1. Did I provide the information the problem requested?

  The problem asked you to identify the length of . That is the information provided with our solution.

  2. Does my answer make sense?

  The value of is slightly larger than or , but that is to be expected in this scenario. It is certainly within reason. A response of or would have been unreasonable.

  Your work on this problem is now complete. The final answer is .

  Lesson Summary

  In this lesson, we explored problem-solving strategies. Specifically, we have learned:

  How to read and understand given problem situations.

  How to use multiple representations to restate problem situations.

  How to identify problem-solving plans.

  How to solve real-world problems using planning strategies.

  These skills are important for any type of problem, whether or not it is about geometry. Practice breaking down different problems in other parts of your life using these techniques. Forming plans and using strategies will help you in a number of different ways.

  Points to Consider

  This chapter focused on the basic postulates of geometry and the most common vocabulary and notations used throughout geometry. The following chapters focus on the skills of logic, reasoning, and proof. Review the material in this chapter whenever necessary to maintain your understanding of the basic geometric principles. They will be necessary as you continue in your studies.

  Review Questions

  Suppose one line is drawn in a plane. How many regions of the plane are created?

  Suppose two lines intersect in a plane. How many regions is the plane divided into? Draw a diagram of your answer.

  Now suppose three coplanar lines intersect at the same point in a plane. How many regions is the plane divided into? Draw a diagram of your answer.

  Make a table for the case of and coplanar lines intersecting at one point.

  Generalize your answer for number 4. If coplanar lines intersect at one point, the plane is divided into __________ regions.

  Bindi lives twelve miles south of Cindy. Mari lives five miles east of Bindi. What is the distance between Cindy's house and Mari’s house? Model this problem by drawing it on a coordinate grid. Let Bindi’s house bet at the origin, . Use the labels for Bindi’s house, for Mari’s house, and for Cindy’s house.

  What are the coordinates of Cindy's and Mari’s house?

  Use the distance formula to find the distance between

  Suppose a camper is standing north of a river that runs east-west in a perfectly straight line (we have to make some assumptions for geometric modeling!). Her tent is north of the river, but downstream. See the diagram below).

  The camper sees that her tent has caught fire! Luckily she is carrying a bucket so she can get water from the river to douse the flames. The camper will run from her current position to the river, pick up a bucket of water, and then run to her tent to douse the flames (see the blue line in the diagram). But how far along the river should she run (distance in the diagram) to pick up the bucket of water if she wants to minimize the total distance she runs? Solve this by any means you see fit—use a scale model, the distance formula, or some other geometric method.

  Does it make sense for the camper in problem 7 to want to minimize the total distance she runs? Make an argument for or against this assu
mption. (Note that in real-life problem solving finding the “best” answer is not always simple!).

  Review Answers

  See the table below

  Number of Coplanar Lines Intersecting at One Point Number of Regions Plane is Divided Into

  Every number in the right-hand column is two times the number in the left-hand column, so the general statement is: “If coplanar lines intersect at one point, the plane is divided into regions.”

  Cindy’s House: ; Mari’s house:

  One way to solve this is to use a scale model and a ruler. Let . Then you can draw a picture and measure the distance the camper has to run for various locations of the point where she gets water. Be careful using the scale!

  Now make a table for all measurements to find the best, shortest total distance.

  Distance to Water Distance from Water to Tent Total Distance

  It looks like the best place to stop is between and . Based on other methods (which you will learn in calculus and some you will learn later in geometry), we can prove that the best distance is when she runs downstream to pick up the bucket of water.

  Answers will vary. One argument for why it is not best to minimize total distance is that she may run slower with the full bucket of water, so she should take the distance she must run with a full bucket into account.

  Chapter 2: Reasoning and Proof

  Inductive Reasoning

  Learning Objectives

  Recognize visual patterns and number patterns.

  Extend and generalize patterns.

  Write a counterexample to a pattern rule.

  Introduction

  You learned about some of the basic building blocks of geometry in Chapter 1. Some of these are points, lines, planes, rays, and angles. In this section we will begin to study ways we can reason about these building blocks.

  One method of reasoning is called inductive reasoning. This means drawing conclusions based on examples.

  Visual Patterns

  Some people say that mathematics is the study of patterns. Let’s look at some visual patterns. These are patterns made up of shapes.

  Example 1

  A dot pattern is shown below.

  A. How many dots would there be in the bottom row of a fourth pattern?

  There will be dots. There is one more dot in the bottom row of each figure than in the previous figure. Also, the number of dots in the bottom row is the same as the figure number.

  B. What would the total number of dots be in the bottom row if there were patterns?

  There would be a total of dots. The rows would contain and dots.

  The total number of dots is .

  Example 2

  Next we have a pattern of squares and triangles.

  A. How many triangles would be in a tenth illustration?

  There will be triangles. There are squares, with a triangle above and below each square. There is also a triangle on each end of the figure. That makes triangles in all.

  B. One of the figures would contain triangles. How many squares would be in that figure?

  There will be squares. Take off one triangle from each end. This leaves triangles. Half of these triangles, or triangles, are above and triangles are below the squares. This means there are squares.

  To check: With squares, there is a triangle above and below each square, making squares. Add one triangle for each end and we have triangles in all.

  C. How can we find the number of triangles if we know the figure number?

  Let be the figure number. This is also the number of squares. is the number of triangles above and below the squares. Add for the triangles on the ends.

  If the figure number is , then there are triangles in all.

  Example 3

  Now look at a pattern of points and line segments.

  For two points, there is one line segment with those points as endpoints.

  For three noncollinear points (points that do not lie on a single line), there are three line segments with those points as endpoints.

  A. For four points, no three points being collinear, how many line segments with those points as endpoints are there?

  . The segments are shown below.

  B. For five points, no three points being collinear, how many line segments with those points as endpoints are there?

  . When we add a 5th point, there is a new segment from that point to each of the other four points. We can draw the four new dashed segments shown below. Together with the six segments for the four points in part A, this makes segments.

  Number Patterns

  You are already familiar with many number patterns. Here are a few examples.

  Example 4 – Positive Even Numbers

  The positive even numbers form the pattern

  What is the positive even number?

  The answer is . Each positive even number is more than the preceding one. You could start with , then add , , to get the number. But there is an easier way, using more advanced mathematical thinking. Notice that the even number is , the even number is , and so on. So the even number is

  Example 5 – Odd Numbers

  Odd numbers form the pattern

  A. What is the odd number?

  The answer is . We can start with and add . . Or, we notice that each odd number is less than the corresponding even number. The even number is (example 4), so the odd number is .

  B. What is the odd number?

  . The even number is (example 4), so the odd number is

  Example 6 – Square Numbers

  Square numbers form the pattern

  These are called square numbers because

  A. What is the square number?

  The answer is . The square number is .

  B. The square number is . What is the value of ?

  The answer is . The square number is .

  Conjectures and Counterexamples

  A conjecture is an “educated guess” that is often based on examples in a pattern. Examples suggest a relationship, which can be stated as a possible rule, or conjecture, for the pattern.

  Numerous examples may make you strongly believe the conjecture. However, no number of examples can prove the conjecture. It is always possible that the next example would show that the conjecture does not work.

  Example 7

  Here’s an algebraic equation.

  Let’s evaluate this expression for some values of .

  These results can be put into a table.

  After looking at the table, we might make this conjecture:

  The value of is for any whole number value of .

  However, if we try other values of , such as , we have

  Obviously, our conjecture is wrong. For this conjecture, is called a counterexample, meaning that this value makes the conjecture false. (Of course, it was a pretty poor conjecture to begin with!)

  Example 8

  Ramona studied positive even numbers. She broke some positive even numbers down as follows:

  What conjecture might be suggested by Ramona’s results?

  Ramona made this conjecture:

  “Every positive even number is the sum of two different positive odd numbers.”

  Is Ramona’s conjecture correct? Can you find a counterexample to the conjecture?

  The conjecture is not correct. A counterexample is . The only way to make a sum of two odd numbers that is equal to is: , which is not the sum of different odd numbers.

  Example 9

  Artur is making figures for a graphic art project. He drew polygons and some of their diagonals.

  Based on these examples, Artur made this conjecture:

  If a convex polygon has sides, then there are diagonals from any given vertex of the polygon.

  Is Artur’s conjecture correct? Can you find a counterexample to the conjecture?

  The conjecture appears to be correct. If Artur draws other polygons, in every case he will be able to draw diagonals if the polygon has sides.

  Notice that we have not prov
ed Artur’s conjecture. Many examples have (almost) convinced us that it is true.

  Lesson Summary

  In this lesson you worked with visual and number patterns. You extended patterns to beyond the given items and used rules for patterns. You also learned to make conjectures and to test them by looking for counterexamples, which is how inductive reasoning works.

  Points to Consider

  Inductive reasoning about patterns is a natural way to study new material. But we saw that there is a serious limitation to inductive reasoning: No matter how many examples we have, examples alone do not prove anything. To prove relationships, we will learn to use deductive reasoning, also known as logic.

  Review Questions

  How many dots would there be in the fourth pattern of each figure below?

  What is the next number in the following number pattern?

  What is the tenth number in this number pattern?

  The table below shows a number pattern.

  What is the value of when ?

  What is the value of when ?

  Is a value of in this pattern? Explain your answer.

  Give a counterexample for each of the following statements.

  If is a whole number, then

  Every prime number is an odd number.

  If and , then .