University School of Nashville Sixth Grade Math Self-Guided Challenge Curriculum Unit 4 Plane Geometry* This curriculum was written by Joel Bezaire for use at the University School of Nashville, funded by a grant from Quaker Hill in the Summer of 2011. We are making it available for anyone to use. You may not alter, distribute, or disseminate this document except for use in your own personal classroom or home-school/tutoring situation. You may not charge a fee for this document. Do not email this document or share it via USB drive or the Cloud if you wish to share it with someone please direct them to pre-algebra.info so that person can download it for themselves from the source website. All rights reserved by the author and USN (www.usn.org) * This unit uses the computer program Geometer s Sketchpad. Before starting this unit, complete the Geometer s Sketchpad Introduction Supplemental Unit. If you don t want to purchase GSP, you can modify these activities for the free Chrome App named GeoGebra
About This Curriculum What you hold in your hands is a special 6 th grade curriculum, designed by the math teachers at USN to make sure that the best and brightest math students in the grade are being challenged in the way they should be challenged. There are a number of Units in this curriculum. They are very different in the way the activities are arranged: Some use computers, and some don t. Some require a great deal of reading, and some don t. Some might require you to do a great deal of writing, and some won t. Because that is the case you may enjoy some Units more than others, depending on your preferred activities. Because of your particular strengths, you may be invited to do some Units and not invited to do others. What they all have in common is as follows: 1) These are all designed to be self-studies. While you may need to receive some help from a teacher from time to time, if you find yourself having to constantly receive assistance, you might be better served to be with the rest of your 6 th grade class during this unit. 2) This curriculum is designed to be difficult. Since these Units are designed to give a challenge above and beyond the regular 6 th grade curriculum, you should expect to spend some time and effort completing the tasks. Sometimes you may need to set this booklet aside and just think. Sometimes you may need to walk away and work on something else while your brain stews on a difficult problem. All of that is OK. Only when you re convinced that you are stuck should you seek help from a teacher. Of course, the work in this booklet should represent the work of the student, not a parent, sibling, or classmate. 3) This curriculum assumes that you enjoy learning math. We don t put a lot of effort into selling mathematics to you, trying to convince you that mathematics can be fun and worthwhile. We assume that you already think so, and that s why you re tackling this curriculum! 4) Since this is a self-guided curriculum, please let a teacher know if something sparks your interest! There might be a way to spend more time on a topic that you find particularly interesting. You won t know unless you ask Have fun!
Geometry is not true, it is advantageous. Henri Poincare Introduction Way back in 300 BCE, a Greek philosopher named Euclid introduced the concept of points, lines, planes, and all the components that still to this day make up the basis of our understanding of two-dimensional geometry. This Unit of study will study some of those components of Plane (Two-Dimensional) Geometry. This Unit assumes you re already familiar with the basic terminology and properties of points, lines, planes, angles, polygons, as well as area and perimeter formulas for common polygons. We ll actually try and discover some relationships by building shapes, taking measurements, and trying to establish patterns; this process actually has as much to do with algebra as it does geometry. What you ll find over your next 5 or 6 years of studying mathematics is that algebra and geometry are inseparable. The work you do in one area of study inevitably links to the other. We ll begin to study some of those links in this Unit of study. As for the quote at the top: What do you think of it? What does he mean? Keep that quote in the back of your mind as we work through this Unit (and later, Unit 6). See if it makes any more sense to you later on.
1. Relationships in Circles -Terminology You re probably familiar with the basics when it comes to circles: The definition of radius and diameter, how to find the area and circumference, etc. We re going to use Geometer s Sketchpad (GSP) to create a number of circles and examine other relationships within a circle. Before we do this, we need to establish some basic definitions: a) Arcs These arcs have nothing to do with Noah (his was an ark). Arcs can be thought of as partial circumferences. We can measure an arc in degrees, much like an angle. In a circle, there are minor arcs (less than 180 ) and major arcs (greater than 180 but less than 360 ). There are a total of 360 in a circle. Example: In the above picture, we have a circle with a center at point F. There are a number of arcs on this circle, and the measure of two of them are shown. Minor arc HG runs the shortest distance between points H and G and measures around 118. Major arc HIG runs through those three points, and is clearly greater than the minor arc HG. Its measure is shown as approximately 242. Notice that when we add the measures of the two arcs together they total 360. Notice that there are other arc measures that we could have measured. We could have measured minor arc HI and major arc IGH, and their measures would have added up to 360 as well.
b) Tangents Tangents are lines that touch a circle at exactly one point. In the example below, KJ is tangent to Circle L. c) Secants and Chords Secants are lines that touch a circle in exactly two points. Chords are segments of secants contained within the circle. (A diameter is a type of chord, for example). In the example below, OP is a secant to the circle M, while OP is considered a chord of circle M. If we were to draw a segment between points O and N or points P and N, those segments would also be considered chords.
2. Relationships in Circles: Inscribed Angles I An angle is said to be inscribed in a circle when its vertex lies on the circle. In the example below, < WVX is inscribed in circle U. We want to examine a particular case of inscribed angles. Use GSP to construct the following: 1) Draw a circle in GSP using the Circle tool. 2) Draw the diameter of the circle using the Line tool. 3) Make points on the circle where the diameter meets the circle using the Point tool. These are the endpoints of the diameter. 4) Construct an inscribed angle in the circle using the Ray tool. Make each ray pass through one of the endpoints of the diameter. Your drawing should look something like this 5) Select the three points on your angle. Use Menu -> Measure to measure the inscribed angle. (Note: You must select the three points so that the vertex is in the middle of your selection. so in the example to the right, I could select the points in the order W, V, X or in the order X, V, W just as long as point V is in the middle because it is the vertex.) 6) Use the Selection tool to drag the vertex of your angle around the circumference of your circle. What do you notice about the measure of your inscribed angle? Complete the following sentence: When an angle inscribed in a circle passes through the endpoints of the diameter,. (What you just did matches what early mathematicians did to find relationships in circles. The difference is they did all of their constructing, drawing, and measuring by hand instead of using a computer program!)
3. Relationships in Circles: Inscribed Angles II 1) Draw a circle in GSP using the Circle tool. 2) Draw an inscribed acute angle somewhere in the circle using the Ray tool. 3) Place points where the rays pass through the circle using the Point tool. Your drawing should look something like this 4) Select the points of the angle, then measure the angle. 5) Select the circle as well the two points where the rays touch the circle. Use Menu -> Measure to measure the Arc Angle. Record the measurements of both the angle and the arc angle below in the chart. 6) Drag the vertex and the points where the rays meet the circle around to make different sized acute angles and arcs. Do this at least 9 more times so you can completely fill out the chart: Measure of Inscribed Acute Angle Measure of Corresponding Arc What is the relationship between an inscribed angle and the arc that it creates? How does this relate to the relationship that you found in the previous exercise, when the inscribed angle met the endpoints of a diameter?
7) Using the same drawing, drag the vertex or the points to make an obtuse angle. What happens to the arc measure? Explain what GSP is telling you, and find the actual arc measure. 8) Given your answer from #7 above, does the relationship you described on the previous page still hold true? If not, explain a new relationship below. 9) Can you give a single, definitive statement that describes the relationship between an inscribed angle and the arc that it creates?
4. Inscribed Angles: Problem Solving Use what you learned in the last two activities to answer the following questions. Do not use GSP to help you on these use the relationships you discovered to answer the questions. a) In the following drawing, what is the measure of major arc GEH? b) In the following drawing, what is the measure of angle I (< LIK)? c) In the following drawing, what is the measure of angle P (< OPM)?
5. Relationships in Circles: Secants I 1) Construct a circle in GSP using the Circle tool. 2) Draw two secant lines that intersect inside the circle using the Line tool. 3) Make points at the 4 places where the secants cross through the circle using the Point tool. Make a point where the secants cross. Your drawing should look something like this: 4) Find the measure of any angle created by the intersection of the secants. 5) Find the measure of the arc created by that angle. This is called the intercepted arc of that angle. 6) Find the measure of the arc opposite the arc you just measured. (That is, neither of the arcs that are directly next to it). This is called the second intercepted arc of that angle. EXAMPLE: If I first decided to find the measure of < TYU in Step 4, the first arc I would measure for Step 5 would be minor arc TU. The next arc I would measure in Step 6 would be minor arc WX. 7) Record all three measures in the chart below. Then drag the intersection point about the interior of the circle and record all three measures at least 9 more times. (Remember to be mindful of what GSP does if a minor arc becomes a major arc while dragging a point around you may need to adjust the measurements GSP is giving you). Angle Measure Intercepted Arc Measure Second Intercepted Arc Measure
8) Keep your drawing but delete your angle and arc measures. Measure the angle adjacent (next to) the angle you had measured previously. Measure both the intercepted arc measure and the second intercepted arc measure for your new angle. Follow the same process as before, recording your data below. Angle Measure Intercepted Arc Measure Second Intercepted Arc Measure Compare your two charts for 7) and 8). Complete the following sentence based on your charts above: When two secants meet in the interior of a circle, any angle created is equal to. Is the above property related at all to what you learned earlier about inscribed angles? If so, how?
6. Relationships in Circles: Secants II 1) Construct a circle in GSP using the Circle tool. 2) Draw two secant lines that intersect outside of the circle using the Line tool. 3) Make points at the 4 places where the secants cross through the circle using the Point tool. Make a point where the secants cross. Your drawing should look something like this: 4) Measure the angle created, the intercepted arc, and the second intercepted arc. Record your data in the chart below. EXAMPLE: In the drawing to the right, I would find the measure of < CAB, minor arc CB, and minor arc DF. 5) Repeat by dragging the vertex around your workspace to change the measure of the angles and the arcs. Repeat this 9 more times until you fill out the entire chart. Angle Measure Intercepted Arc Measure Second Intercepted Arc Measure Complete the sentence based on your chart above: When two secants meet on the outside of a circle, the angle formed is equal to.
7. Problem Solving: Secants Use what you learned in the last two activities to answer the following questions. Do not use GSP to help you on these use the relationships you discovered to answer the questions. a) Use the above drawing to answer What is the measure of < IJG? What is the measure of < IJK? b) In the picture above, what is the measure of minor arc PQ?
8. Relationships in Circles: Secants III Unlike our last relationships this third relationship doesn t have to do with arc measures, so we ll study it separately. 1) Construct a circle in GSP using the Circle tool. 2) Construct two secant lines that intersect inside the circle using the Line tool. 3) Make points at the 4 places where the secants cross through the circle using the Point tool. Make a point where the secants cross. Your drawing should look something like this 4) Select the point where the secants meet and one of the points on the circle. Measure the distance between the points using Menu -> Measure. Record the distance in the chart below. Repeat the process for the other line segments contained in that line. Then repeat the process for the other two line segments that run from the intersection point to the point where the lines meet the circle. EXAMPLE: In the drawing to the right, if you measured the length of segment TY first, you would measure the length of segment YX next. Then you would measure segments YU and WY in some order. 4) Drag the intersection point around the circle 9 more times and record your results for each of the 4 line segments to complete the chart below. Distance between Points Results: First Segment Second Segment Third Segment Fourth Segment
Examine the chart above and see if you can find any connections. In particular, focus on the first two columns (the segments that belong to the same line) and how they relate to the second two columns (the segments that belong to the other line). Do you see any relationships? If you aren t having any luck, round each number to the nearest whole number and see if that helps. Complete the following sentence to describe the relationship you see: When two secants meet in the interior of a circle, the length of segments created. 9. Problem Solving: Secants III Use what you learned in the last activity to answer the following question. Do not use GSP to help you on this use the relationship you discovered to answer the question. In the picture above, what is the length of line segment EB?
10. Visual Demonstration of Pythagorean Theorem Had enough circles for a while? Good. There s plenty more to be done with circles if you like, but let s take a break from that and look at one of the best-known theorems in plane geometry (and algebra ), and that is the Pythagorean Theorem. The Pythagorean Theorem states that in a right triangle, the sum of the squares of the legs equals the square of the hypotenuse. (The legs meet at 90, or are called perpendicular). To use this drawing as an example: a 2 + b 2 = c 2 We re going to use GSP to give a visual demonstration of how The Pythagorean Theorem works. 1) Create a new GSP file and build a right triangle that looks like the one above, using the Segment tool. (Note: There are lots of ways in GSP to ensure that the two legs meet at 90. Spend a little time ensuring that the two legs are in fact perpendicular). 2) We are going to build squares off of each side of the triangle (one way to represent the length of the segment squared is to build squares off of each segment). 3) Let s start by building a square off of the left leg. Use the Selection tool to select the two endpoints and the segment. Go to Menu -> Transform -> Rotate. Click on the top endpoint of the segment, enter 90 into the box, and click Rotate. Your drawing should look like this: 4) Use the Rotate command from the Menu again to build another side of your square. Note that you ll have to pick the right-most endpoint as your point of rotation.
5) Now select the 4 vertices of the square you created. Go to Menu -> Construct -> Quadrilateral Interior. It will create the interior of the square you just created. Your drawing should look like this: 6) Build a square off of the second (bottom) leg, extending down below the triangle. Use the Menu -> Transform -> Rotate feature again, but this time you ll have to use the rotate feature three times to complete your square. Select the 4 vertices, and construct the quadrilateral interior. Your drawing should look like this: Note: You can go to Menu -> Display -> Color to change the color of your polygon interiors. This would be a good idea, because we ll need to tell them apart later on. 7) We have constructed the two squares built off of the legs. We now need to construct the square off of the hypotenuse. If the Pythagorean Theorem is true, we should be able to show that the areas of the Green and Yellow squares add up to the area of the new square we are about to create. 8) Create the square off of the hypotenuse using the Menu -> Transform - > Rotate feature. Rotate the square down into the yellow square, not above the triangle. Select the 4 vertices, and construct the quadrilateral interior. Your drawing should look like this:
9) Now, the brown square we just created should be equal in area to the green and yellow squares. That would be a visual example of the Pythagorean Theorem at work. Notice that much of the brown square is already overlapped by the green and yellow squares. All that remains to be shown is this Do these areas fit inside of these areas?
10) Print out your drawing. Cut out the entire shape, then cut out the three shapes shown in the first drawing on the previous page. Arrange them so that they cover up the remaining brown square. Explain how they can be arranged to fit inside the two other areas of the brown square:. 11) a) Explain in your own words how this activity demonstrates the Pythagorean Theorem: b) If you worked on Unit 1 of this challenge program, explain why this activity does not prove the Pythagorean Theorem:
11. Problem Solving: Fold To The Max Take a regular sheet of blank printer paper. Fold the top left corner down so that it touches the bottom of the paper. How far away from the lower left corner does it need to be in order to maximize the area of the right triangle created in the lower left corner? (In other words, what s the biggest right triangle that can be created in the bottom left of the paper by folding the top left corner down?) Explain how you solved the problem below, and explain how you know your answer is correct:
12. Problem Solving: Bridges of Konigsburg In the city of Konigsburg, Prussia (now part of Russia), there was a long-standing riddle about the bridges in the city. A river ran through the city, creating two islands. These islands and both sides of the river were connected by seven (7) bridges, as shown here: The riddle went something like this: Can a person find a route through the city so that he crosses each of the seven bridges exactly one time? The islands could not be reached or left by any other means besides the bridges. Try to solve the riddle of the Bridges of Konigsburg. On the next page is the solution, but spend some time trying to solve it yourself before looking.
The Solution: In 1735 a man by the name of Leonhard Euler solved the riddle of the Bridges of Konigsburg. His solution to this seemingly mundane riddle actually launched a branch of mathematics called topology, a topic that you could study in university to this day. What Euler did is changed the four landmasses to points (or nodes), and made the bridges paths that connected them, like this: Each node represents one of the landmasses, and each black line represents a bridge. Euler classified each node as either even or odd, depending on the number of paths that were attached to it. So, on the drawing above, the top node is odd (3 paths), the left node is odd (5 paths), the right node is odd (3 paths), and the bottom node is odd (3 paths). But then, Euler noted that in order to arrive at and leave a node successfully, you had to have an even number of paths (since arriving/leaving happen in pairs). The most number of odd nodes you could have is two: One to start at, and one to finish at. Every other node needs to be even, so that you can arrive to it and leave from it. Therefore, since the Bridges of Konigsburg problem contains four odd nodes, it s impossible.
Problem Solving: Topology For each drawing below, determine if it s possible to visit each node while traveling every path exactly one time. If it isn t possible, explain why. If it is possible, state which node(s) you must begin or end at (if applicable). a) b) c) In topology, a successful path is called an Euler Path, after the man who first answered the Bridges of Konigsburg problem.