CK-12 Geometry

by CK-12 Foundation

  The slope of the perpendicular line is

  See dashed line in the image below. The equation of the line is

  The second intersection is at

  The distance is .

  The distance is

  Non-Euclidean Geometry

  Learning Objectives

  Understand non-Euclidean geometry concepts.

  Find taxicab distances.

  Identify and understand taxicab circles.

  Identify and understand taxicab midpoints.

  Introduction

  What if we changed the rules of a popular game? For example, what if batters in baseball got five strikes instead of three? How would the game be different? How would it be the same? Up to this point, you have been studying what is called Euclidean geometry. Based on the work of the Greek mathematician Euclid, this type of geometry is based on the assumption that given a line and a point not on the line, there is only one line through that point parallel to the given line (this was one of our postulates). What if we changed that rule? Or what if we changed another rule (such as the ruler postulate)? What would happen? Non-Euclidean geometry is the term used for all other types of geometric study that are based on different rules than the rules Euclid used. It is a large body of work, involving many different types of theories and ideas. One of the most common introductions to non-Euclidean geometry is called taxicab geometry. That will be the principal focus of this lesson. There are many other types of non-Euclidean geometry, such as spherical and hyperbolic geometry that are useful in different contexts.

  Basic Concepts

  In previous lessons, you have learned to find distances on a plane, and that the shortest distance between two points is always along a straight line connecting the two points. This is true when dealing with theoretical situations, but not necessarily when approaching real-life scenarios. Examine the map below.

  Imagine that you wanted to find the distance you would cover if you walked from the corner of and to the corner of and . Using the kind of geometry you have studied until now, you would draw a straight line and calculate its distance.

  But to walk the route shown, you would have to walk through buildings! As that isn’t possible, you will have to walk on streets, working your way over to the other corner. This route will be longer, but since walking through a building is not an option, it is the only choice.

  Our everyday world is not a perfect plane like the coordinate grid, so we have developed language to describe the difference between an ideal world (like the plane) and our real world. For example, the direct line between two points is often referred to by the phrase “as the crow flies,” talking about if you could fly from one point to another regardless of whatever obstacles lay in the path. When referring to the real-world application of walking down different streets, mathematicians refer to taxicab geometry. In other words, taxicab geometry represents the path that a taxi driver would have to take to get from one point to another. This language will help you understand when you should use the theoretical geometry that you have been practicing and when to use taxicab geometry.

  Taxicab Distance

  Now that you understand the basic concepts that separate taxicab geometry from Euclidean geometry, you can apply them to many different types of problems. It may seem daunting to find the correct path when there are many options on a map, but it is interesting to see how their distances relate. Examine the diagram below.

  Each of the drawings above show different paths between points and . Take a moment to calculate the length (in units) of each path.

  Path 3:

  Path 2:

  Path 1:

  Each of those distances is equal, even though the paths are different. The point is that the shortest distance between and is to the right and up. Since , this is consistent with the findings above. What you can learn from this is that it doesn’t matter the order in which you move up or over. As long as you do not backtrack, the length will always be the same.

  Note that the taxicab geometry system looks familiar—in fact it is the same as the coordinate grid system, with the added rule that you can only travel up and down or right and left.

  Example 1

  In the grid below, each vertical and horizontal line represents a street on a map. The streets are evenly spaced.

  June rides her bike from home to school each day along the roads in her town. How far, in feet, does June ride her bike to get to school?

  This is a taxicab geometry question, as June only rides her bike on the streets. Count how many units to the right June travels Now count how may units up June travels Add these two values.

  June travels to get to school. Because the scale shows that is equal to , you can calculate the distance in feet.

  June rides her bike to school.

  Taxicab Circles

  From your previous work in geometry you should already know the definition of a circle—a circle is the set of points equidistant from a center point. Taxicab “circles” look a little different. Imagine selecting a point on a grid and finding every point that was away from it using taxicab geometry. The result is as follows.

  Use logic to work through problems involving taxicab circles. If you work carefully and slowly, you should be able to find the desired answer.

  Example 2

  A passenger in a taxi wants to see how many distinct points she could visit if a cab travels exactly three blocks from where she is standing without turning around. Count the points and draw the taxicab circle with a radius of 3 units.

  Start with a coordinate grid with a point in the middle. Count straight in each direction and mark the points that result.

  Now fill in the other points that involve a combination of moving up, down, and over.

  Count the points to find that there are distinct points on the circle.

  Taxicab Midpoints

  Much like finding taxicab distances, you can also identify taxicab midpoints. However, unlike traditional midpoints, there may be more than one midpoint between two points in taxicab geometry. To find a taxicab midpoint, trace a path between the given paths along the roads, axes, or lines. Then, divide the distance by and count that many units along your path. This results in identifying a taxicab midpoint. As you will see, there may be more than one midpoint between any two points.

  Example 3

  Find the taxicab midpoints between and in the diagram below.

  Start by finding the taxicab distance . You will have to travel to the right and up. Add these two values to find the distance.

  The taxicab distance, is Use the diagram and identify how many points are away from and . These will be the taxicab midpoints.

  There are three taxicab midpoints in this scenario, shown in the diagram above.

  Now we have seen two major differences between taxicab geometry and Euclidean geometry. Based on a new definition of the “distance between two points” in taxicab geometry, the look of a circle has changed, and one of the fundamental postulates about the midpoint has changed.

  These short examples illustrate how one small change in the rules results in different rules for many parts of the taxicab geometry system.

  Other non-Euclidean geometries apply to other situations, such as navigating on the globe, or finding the shape of surfaces on bubbles. All of these different systems of geometry follow postulates and definitions, but by changing a few key rules (either the postulates or definitions) the entire system changes. For example, in Taxicab geometry we see that by changing the definition of "the distance between two points" we also changed the meaning of midpoint.

  Lesson Summary

  In this lesson, we explored one example of non-Euclidean geometry. Specifically, we have learned:

  Where non-Euclidean geometry concepts come from.

  How to find taxicab distances.

  How to identify and understand taxicab circles.

  How to identify and understand taxicab midpoints.

  These will help you solve many diffe
rent types of problems. Always be on the lookout for new and interesting ways to apply concepts of non-Euclidean geometry to mathematical situations.

  Points To Consider

  Now that you understand lines and angles, you are going to learn about triangles and their special relationships.

  Review Questions

  Solve each problem.

  What is the taxicab distance between points and in the diagram below?

  Draw a taxicab circle on the diagram below with a radius of

  What is the taxicab distance between and in the diagram below?

  Draw one of the taxicab midpoints between and on the diagram above.

  How many taxicab midpoints will there be between and ? How do you know you have found them all?

  What is the taxicab distance between points and in a coordinate grid?

  Is there a taxicab midpoint between and ? Why?

  What are the coordinates of one of the taxicab midpoints between and (10,1)?

  How many taxicab midpoints will there be between points and on a coordinate grid? What are their coordinates?

  If you know that two points have a taxicab distance of between them, do you have enough information to tell how many taxicab midpoints there will be between those two points? Why or why not?

  What are some similarities and differences between taxicab geometry and Euclidean geometry?

  Review Answers

  See below:

  One possible answer:

  Any of the following points are correct. There are a total of three midpoints, and by systematic checking there are no more.

  No, since the distance between the two points is odd, there are no midpoints as they happen in the “middle of a block”—or they are not at a point in the coordinate grid with integer coordinates.

  Any of the following coordinate pairs is correct: and

  There are four midpoints between and The midpoints are at and

  No, there may be one midpoint if the two points have the same or coordinate, or as many as

  Answers will vary, but some major ideas: In taxicab geometry all distances are integers, while in Euclidean geometry distances can be rational and real values. In Euclidean geometry there is only one midpoint of a segment, but in taxicab geometry there may be multiple midpoints for a segment. Both types of geometry use “lines” between points but in the case of taxicab geometry, lines must be vertical or horizontal (along the grid). One other interesting difference is that taxicab circles appear to be squares in Euclidean geometry

  Chapter 4: Congruent Triangles

  Triangle Sums

  Learning Objectives

  Identify interior and exterior angles in a triangle.

  Understand and apply the Triangle Sum Theorem.

  Utilize the complementary relationship of acute angles in a right triangle.

  Identify the relationship of the exterior angles in a triangle.

  Introduction

  In the first chapter of this course, you developed an understanding of basic geometric principles. The rest of this course explores specific ideas, techniques, and rules that will help you be a successful problem solver. If you ever want to review the basic problem solving in geometry return to Chapter 1. This chapter explores triangles in more depth. In this lesson, you’ll explore some of their basic components.

  Interior and Exterior Angles

  Any closed structure has an inside and an outside. In geometry we use the words interior and exterior for the inside and outside of a figure. An interior designer is someone who furnishes or arranges objects inside a house or office. An external skeleton (or exo-skeleton) is on the outside of the body. So the prefix “ex” means outside and exterior refers to the outside of a figure.

  The terms interior and exterior help when you need to identify the different angles in triangles. The three angles inside the triangles are called interior angles. On the outside, exterior angles are the angles formed by extending the sides of the triangle. The exterior angle is the angle formed by one side of the triangle and the extension of the other.

  Note: In triangles and other polygons there are TWO sets of exterior angles, one “going” clockwise, and the other “going” counterclockwise. The following diagram should help.

  But, if you look at one vertex of the triangle, you will see that the interior angle and an exterior angle form a linear pair. Based on the Linear Pair Postulate, we can conclude that interior and exterior angles at the same vertex will always be supplementary. This tells us that the two exterior angles at the same vertex are congruent.

  Example 1

  What is in the triangle below?

  The question asks for . The exterior angle at vertex measures Since interior and exterior angles sum to you can set up an equation.

  Thus, .

  Triangle Sum Theorem

  Probably the single most valuable piece of information regarding triangles is the Triangle Sum Theorem.

  Triangle Sum Theorem

  The sum of the measures of the interior angles in a triangle is

  Regardless of whether the triangle is right, obtuse, acute, scalene, isosceles, or equilateral, the interior angles will always add up to Examine each of the triangles shown below.

  Notice that each of the triangles has an angle that sums to

  You can also use the triangle sum theorem to find a missing angle in a triangle. Set the sum of the angles equal to and solve for the missing value.

  Example 2

  What is in the triangle below?

  Set up an equation where the three angle measures sum to Then, solve for .

  Now that you have seen an example of the triangle sum theorem at work, you may wonder, why it is true. The answer is actually surprising: The measures of the angles in a triangle add to because of the Parallel line Postulate. Here is a proof of the triangle sum theorem.

  Given: as in the diagram below,

  Prove: that the measures of the three angles add to or in symbols, that .

  Statement Reason

  1. Given in the diagram

  1. Given

  2. Through point , draw the line parallel to . We will call it

  2. Parallel Postulate

  3.

  3. Alternate interior Angles Theorem

  4.

  4. Alternate interior Angles Theorem

  5.

  5. Angle Addition postulate

  6.

  6. Linear Pair Postulate

  7.

  7. Substitution (also known as “transitive property of equality”)

  8.

  8. Substitution (Combining steps 3, 4, and 7).

  And that proves that the sum of the measures of the angles in ANY triangle is

  Acute Angles in a Right Triangle

  Expanding on the triangle sum theorem, you can find more specific relationships. Think about the implications of the triangle sum theorem on right triangles. In any right triangle, by definition, one of the angles is a right angle—it will always measure This means that the sum of the other two angles will always be resulting in a total sum of

  Therefore the two acute angles in a right triangle will always be complementary and as one of the angles gets larger, the other will get smaller so that their sum is .

  Recall that a right angle is shown in diagrams by using a small square marking in the angle, as shown below.

  So, when you know that a triangle is right, and you have the measure of one acute angle, you can easily find the other.

  Example 3

  What is the measure of the missing angle in the triangle below?

  Since the triangle above is a right triangle, the two acute angles must be complementary. Their sum will be We will represent the missing angle with the variable and write an equation.

  Now we can use inverse operations to isolate the variable, and then we will have the measure of the missing angle.

  The measure of the missing angle is

  Exterior Angles in a Triangle

  O
ne of the most important lessons you have learned thus far was the triangle sum theorem, stating that the sum of the measure of the interior angles in any triangle will be equal to You know, however, that there are two types of angles formed by triangles: interior and exterior. It may be that there is a similar theorem that identifies the sum of the exterior angles in a triangle.

  Recall that the exterior and interior angles around a single vertex sum to as shown below.

  Imagine an equilateral triangle and the exterior angles it forms. Since each interior angle measures each exterior angle will measure

  What is the sum of these three angles? Add them to find out.

  The sum of these three angles is In fact, the sum of the exterior angles in any triangle will always be equal to You can use this information just as you did the triangle sum theorem to find missing angles and measurements.

  Example 4

  What is the value of in the triangle below?

  You can set up an equation relating the three exterior angles to Remember that does not represent an exterior angle, so do not use that variable. Solve for the value of the exterior angle. Let's call the measure of the exterior angle .