Geometry
Chapter 3 Answers
Angles and Parallel Lines
When explaining how to get from one place to another, you must describe distances and directions carefully. Below, you will use two different methods, bearings and vectors, to explain precisely how to get from one city to another.
Part B and C, Bearings and Vectors
1. In 1997, Linda Finch re-created and completed Amelia Earhart's 1937 around-the-world flight. In this exercise, you will describe the portion of Linda Finch's flight starting in Natal, Brazil and ending in Malaga, Spain.
a. Use bearings to describe the section of Finch's journey between Natal, Brazil and Malaga, Spain. To see a clear map of her route so that you can measure angles with your protractor, look at the Earhart Project Web site.
[Angles will vary but should be close to the following: Natal, Brazil to Dakar, Senegal: bearing of 040; Dakar, Senegal to Las Palmas, Canary Islands: bearing of 002; Las Palmas, Canary Islands to Malaga, Spain: bearing of 045]b. Use vectors to describe the same section of Finch's journey that you described above. You can find distances between any two consecutive cities on Finch's journey by clicking on the map on the first of the two cities that she visited at the Earhart Project Web site.
[Angles will vary but should be the difference between 90° and the corresponding angles in (a). Natal, Brazil to Dakar, Senegal: 1,628 miles at 50°; Dakar, Senegal to Las Palmas, Canary Islands: 801 miles at 88°; Las Palmas, Canary Islands to Malaga, Spain: 792 miles at 45°]c. Now you're going to plan your own trip. Start in your hometown. Decide on two cities in different states that you want to travel to before returning home. Look at the How Far is It? Web site to find distances and bearings between any two cities. Select "See these places on the map" to see the two cities' locations in relation to each other. What city will you start in, and which cities will you visit on your trip?
[Answers will vary. The answers below assume a trip from Reno, NV to Helena, MT to Durango, CO.]d. Use bearings to give directions for the three legs of your trip, beginning and ending in your hometown. Give directions as though you were going to travel in a straight line between each pair of cities.
[Answers will vary. Reno to Helena, bearing of 036.4; Helena to Durango, bearing of 160.2; Durango to Reno: bearing of 287.1]e. Use vectors to describe the legs of your trip.
[Answers will vary. Reno to Helena, 626 miles at 53.6°; Helena to Durango, 677 miles at 289.8°; Durango to Reno: 666 miles at 162.9°]f. What are the differences between the methods of bearings and vectors?
[Answers will vary. Bearings give only the direction between locations while vectors give both direction and distance. Bearings are given clockwise from north while vectors are given counterclockwise from east.]g. Why is it important for someone to understand the method you are using to measure direction if all you give them is an angle measurement?
[An angle measurement is useless as a directional aid unless you know where the fixed side of the angle is and whether the moving side of the angle rotates clockwise or counterclockwise from the fixed side.]
Gears are a crucial component of many mechanical devices. One of these devices that you may be familiar with is the bicycle. The combination of gears on a bicycle determines whether you can easily climb a steep hill or speed along a level road. In this exercise, you will first examine the basic principles of bicycle gears. Next, you will learn how to determine the order of gears on a bicycle from the lowest gear (the one you use to ride up a steep hill) to the highest gear (the one you use on a level or downhill road).
Part C, Making Connections
1. First, let's get acquainted with the bicycle and its gear vocabulary. Either get your own bicycle or go to the Bicycle Web site to see a picture of a bicycle.
a. Sketch a picture of a bike, paying close attention to detail on the front and rear gears.
b. Explain how pedaling a bicycle results in the bicycle's wheels turning.
[As the pedals are turned, the chain which is connected to both the front and rear gears, turns the back wheel. This propels the bicycle.]c. Based on your own bicycle and/or the bicycle you saw in the picture, are the gears in the front or the gears in the back of a bicycle generally larger? [front]
d. Why would this difference in size between the front and the rear gears make pedaling a bicycle more efficient?
[Because the front gears are generally larger, one rotation of the pedals will produce more than one rotation of the rear wheel.]e. If you were going to try to pedal up an extremely steep hill, how do you think you should position the chain on the front chain-rings and back cogs (the series of teeth on the edge of a gearwheel)? Why?
[The chain should be on the smallest chain-ring in the front and largest cog in the back. This way, the rider is rotating the rear wheel as little as possible with one rotation of the pedal.]
2. Now, go to the Gears Web site, and read some basic information about bicycle and other types of gears.
a. The gears on a bicycle are sometimes called chain-rings and cogs. On your sketch above, label the chain-rings and cogs.
[The chain-rings are in the front, and the cogs are in the back.]b. In this article, the author describes how to make a gear chart for a bicycle. What is the purpose of making this gear chart?
[You will know how to go up exactly one gear or down exactly one gear on a bicycle.]c. Read through the author's example for creating a gear chart for a ten-speed bicycle. Note that "gear-inches" are a way to indicate how high or low a gear is. You calculate gear-inches by taking the gear ratio (number of front teeth divided by number of back teeth) and multiplying it by the wheel diameter. Does a large number of gear-inches mean it is easier or harder to pedal the bicycle? [harder]
d. At one point in the article, the author says, "I essentially have only 6 different gears." (He may have made a slight mistake, and meant 7 different gears.) In any case, why does he say this?
[Three pairs of gear combinations have almost exactly the same gear-inches.]e. Now, you get to create a gear chart for your own bike. Use the instructions at this web site to create your gear chart. First, make a chart showing gear-inches for each gear combination. Second, make a chart ranking the gears from 1 to 10 (or the number of gears you have on your bike), where first gear has the fewest gear-inches. Finally, make your gear chart listing the gear, the chain-ring for each gear, the cog for each gear, and a possible use for each gear.
If you do not have a bike available, you can use the following data:
- -3 chain rings with 38, 44 and 50 teeth
- -5 cogs with 14, 18, 22, 26, and 32 teeth
- -wheel diameter: 26 inches
[Note that this bicycle has 15 gear combinations, so the first two tables will have five rows and three columns each.]
Answers for sample data:
Note: Gears 2&3, 5&6, 7&8, and 12&13 are all quite close. This means this bicycle is essentially an 11-speed.
Answer:
Gear Chain-Rings Cog Purpose 1 small 5 (32 tooth) - biggest cog Biggest hills 2 medium 5 (32 tooth) - biggest cog Hills that are fairly steep, not horrible though 3 small 4 (26 tooth) - next to biggest cog Very close to 2nd gear 4 big 5 (32 tooth) - biggest cog Probably won't use because chain will stretch too much 5 medium 4 (26 tooth) - next to biggest cog Pedaling up average hill 6 small 3 (22 tooth) - middle cog Very close to 5th gear 7 big 4 (26 tooth) - next to biggest cog Pedaling up a gentle rise 8 medium 3 (22 tooth) - middle cog Very close to 7th gear 9 small 2 (18 tooth) - next to smallest cog Pulling away fast from a stop 10 big 3 (22 tooth) - middle cog This gear is used the most 11 medium 2 (18 tooth) - next to smallest cog Pedaling on a flat surface, willing to pedal hard 12 small 1 (14 tooth) - smallest cog Pedaling on a flat stretch with a wind coming from behind 13 big 2 (18 tooth) - next to smallest cog Very close to 12th gear 14 medium 1 (14 tooth) - smallest cog Pedaling downhill relatively fast 15 big 1 (14 tooth) - smallest cog Pedaling downhill FAST
3. Now, go to Gear Inch & Shifting Pattern Web site. This site will give you some additional information about a bicycle's gears if you give it the number of teeth on each chain-ring and free-wheel (or cog) and the diameter of the wheel. Enter the data for your bicycle or the sample data given above onto this web page.
a. Which gear combination gives you the shortest distance traveled in a single pedal rotation?
[It is the combination with the smallest number of teeth on the chain-ring and the largest number of teeth on the free-wheel.]b. Which gear combination gives you the longest distance traveled in a single pedal rotation?
[It is the combination with the largest number of teeth on the chain-ring and the smallest number of teeth on the free-wheel.]c. Explain why the lowest gear on a bicycle gives the shortest distance traveled and the highest gear gives the largest distance traveled in a single rotation of the pedal.
[Answers will vary. Students may say that the lowest gear is the easiest and the highest is the hardest. Or, they may say that a smaller number of teeth on the chain-ring will propel the back wheel a shorter distance than a larger chain-ring. Similarly, a larger free-wheel gear will propel the bicycle a shorter distance than smaller free-wheel gear.]
Light interacts with different surfaces in different ways. Some substances do not allow light to pass through them. Some of these, such as mirrors, reflect the light. Other substances do let light through, but bend the light rays. You can see this phenomenon, called refraction, if you put an object in water. Below, we will examine both light reflection and light refraction.
Part C, Making Connections
1. Go to the Light and Optic site to see a basic discussion of light reflection.
a. According to this article, what is the law of reflection, or Fresnel's Law?
[The angle of incidence equals the angle of reflection.]b. Sketch Figure 1.
Answers for "b", "c" and "d":
c. Suppose that the measure of angle i is 51°. Write the measures of all the other angles in your sketch above.
d. Label points on each of the line segments in your sketch using letters of your choice. Label the point where the light hits the surface point O.
e. Give all pairs of complementary angles in your sketch.
Answer:
f. Give all linear pairs in your sketch.
Answer:
2. Now, go to the Reflections Web site to see a discussion of reflections from curved surfaces.
a. Suppose you stood in front of a convex mirror. What would your image look like?
[It would look much larger than you actually are.]b. Suppose you stood very close to a concave mirror. What would your image look like? [Normal, but smaller than actual size.]
c. When you stand far away from a concave mirror, why is your image flipped upside down?
[The light rays from your toes will be reflected to the top of the image and the light rays from your head will be reflected to the bottom of the image.]d. What is the focus of a concave mirror?
[It's the point where the mirror tends to focus light rays.]
3. Next, we'll look at the phenomenon called refraction. Read the article at the Light, Reflection, and Refraction Web site.
a. According to this article, what is refraction?
[When light is bent at the interface between two substances.]b. What do you think causes refraction?
[The beams of light changing speed between two substances.]
4. Look at the Total internal reflection Web site. Read the article and study the picture at the very end.
a. What is happening to the beam of light?
[Part of it is being reflected and part is being refracted.]b. Sketch this picture. Be sure to include the original beam of light, the reflected beam, the refracted beam and the surface of the water.
Answer for "b" and "c":
c. Let's suppose that the acute angle between the original beam and the water surface measures 46°, and the acute angle between the refracted beam and the water surface measures 7°. Find the measures of all the other angles in your sketch, and write them in your sketch.
Parallel lines play an important role in many forms of architecture. Below, you will examine four different architectural structures and the part that parallel lines and transversals play in each.
Part A, Transversals and Angles
1. Take a look at some of these interesting architectural examples.
a. Look at the Art Gallery Installation Web site. What are some of the architectural references in this installation?
[The artist starts with a floor plan, adds actual 2x4 framing for walls, and then places photographs on the gallery wall of the frame for the house.]b. Describe some examples of parallel lines and transversals in this installation.
[Answers may vary. The pictures of the framed house have wall studs which are parallel and cross bars which form transversals. Similarly, the 2x4's which are actually put up in the installation are parallel, and the pieces of wood which are placed diagonally suggest transversals.]c. Now, we'll look at the architecture of houses.
c. Now, we'll look at the Look at the Palmer House in Jefferson County, Florida at the Historic Places Web site. Give examples of parallel lines and their corresponding transversals.
[Answers may vary. The horizontal lines on the house are parallel. The diagonals which compose the different parts of the roof all suggest transversals.]d. Below is a simple sketch of the roof of the house. Suppose the architect decided that the roof over the entryway should make a 40° angle with the horizontal and that the roofs for the small dormer windows should be parallel to the entryway roof. Based on your knowledge of angles associated with parallel lines and their transversals, write in the measures of all the other angles in the sketch below.
Answer:
e. Describe examples of parallel lines and their corresponding transversals in the building on the San Juan de Aspalaga Web site in Jefferson County, Florida.
[Answers may vary. All of the vertical boards on the side of the house are parallel. The sections of the roof act as transversals.]f. Below is a simple sketch of the front of the building. Suppose that the architect decided that the angle at the very top of the roof should be 110°. Based on your knowledge of angles associated with parallel lines and their transversals, write in the measures of all the other angles in the sketch.
Answer:
g. Why are many houses' roofs slanted rather than horizontal?
[Answers may vary. So that rain and snow drain off easily.]h. Now, we'll look at parallels and transversals in the Eiffel Tower Web site in Paris. Study the design of the tower directly above the arch. Sketch a portion of this section which illustrates the use of parallels and transversals.
Answer:
i. Why do you think the architect put diagonal transversals in this section of the tower?
[Answers may vary. To strengthen the tower or for aesthetic reasons]j. Why did the architect place diagonal transversals slanting in two directions rather than just one?
[Answers may vary. If transversals slanted in only one direction, the structure would collapse on itself.]
