Geometry
Chapter 7 Answers
Similarity
Scale models play huge roles in movies. Spaceships featured in the Star Wars movies have become very popular with model builders. However, at times, the scale is unclear because the size of the actual ship was never known. The actual ships you see on the screen were models themselves! Below you will explore the actual dimensions and model dimensions of an X-wing fighter, the starfighter Luke Skywalker piloted when he destroyed the Death Star in Star Wars: A New Hope.
Part A, Changing the Size of Figures and
Part B, Similar Polygons
1.Go to the Star Wars ships Web site to see statistics for various starships featured in the Star Wars movies.
a. What is the length of the T-65B X-wing fighter?
[12.5 meters]b. If 1 meter = 3.2810 feet, what is the length of the X-wing in feet? In inches?
[41.0125 feet; 492.15 inches]
2. Go to Star Wars Lego Designs Web site to see a model of an X-wing starfighter.
a. Use this model to approximate the ratio of the length of the X-wing to its width.
[About 5:3]b. Based on the information from 1b and 2a, what is the approximate width of the X-wing starfighter in feet? In inches?
[24.6075 feet; 295.29 inches]
3. Now, go to the Star Wars model site to see listings for various Star Wars models. Scroll down to MD-SW-7, the 3-piece Fighter set.
a. How long is the X-wing fighter model?
[8"]b. Assuming that this model is similar to the actual X-wing fighter, use the ratio of length to width you found in 2a to find the width of this X-wing fighter model?
[4.8"]c. What is the similarity ratio of the actual fighter to the model?
[ 492.15/8 = 61.51875/1 ]d. Show that this ratio is the same for the ratios of the lengths and the ratios of the heights.
[Lengths: 492.15/8 = 61.51875/1; Widths: 295.29/4.8 = 61.51875/1 ]e. What will be the relationship between the angles made by the wings in the X-wing model and the actual X-wing fighter?
[The angles will be equal.]
Part C, Areas and Perimeters of Similar Polygons
4. Test your knowledge of surface area.
a. What is the ratio of the surface area of the X-wing fighter to the surface area of the model?
[3784.56:1]b. If it takes 0.75 cups of paint to cover the outside of the model, how many cups of the same paint would it take to cover the actual X-wing fighter?
[2838.42]
Similar triangles occur quite frequently in architecture, even in the pyramids of early Egypt. The Great Pyramid in Ghiza was built somewhere between 2,000 and 3,000 BC and is the most comprehensively surveyed building in the world. Below, you will use your knowledge of similar triangles to calculate dimensions of the Great Pyramid.
Go to the Ghiza Web site to see a picture of the pyramids of Ghiza. The middle one in the row of the three large pyramids is the Great Pyramid. If you look closely, there appears to be a smaller pyramid taken off the top of the Great Pyramid. This smaller missing pyramid is called the capstone. Below is a simplified sketch of the Great Pyramid.
Part A, Similar Triangles
1. Use the Ghiza Web site to answer the following questions.
a. Assuming that the base of the pyramid is parallel to the base of the capstone, how do you know that
b. Go to the Hall of Records Web site to get data about the Great Pyramid. Go to 18:310 to find
the "edge to base" angle. Round this angle off to the nearest degree.
[42°]
Part C, Dilations
2. Study the capstone of the Great Pyramid.
a. Some archaeologists believe that the Great Pyramid is a dilation of its capstone. Look at 4:22 to find the scale factor for this dilation.
b. Look at 11:151 to find the length of one base edge of the Great Pyramid. (Note that "PI" stands for "Pyramid Inch" and is a unit of measure just a bit longer than our current inch.)
[9131 PI]c. Use your answer from 2b to find the length of the corresponding edge of the capstone. Does your answer roughly agree with the information given in 4:11? (Note that one cubit is 25 pyramid inches.)
[161.9 PI; Yes. The side of the capstone is given as about 6.5 cubits which is 162.5 PI.]
d. Go to 0:1 to find the length of, a corner edge.
[8688 PI]
e. Use this to find the length of,a corner edge of the capstone.
[156.8 PI]
Long ago, surveyors used various tools and their knowledge of trigonometry to calculate distances which they couldn't calculate directly. Presently, the tools surveyors use measure distances much more precisely, but trigonometry is still used to calculate distances indirectly.
Part A, Trigonometric Ratios
1. Look at the Uses of Trigonometry Web site to see one simple application of trigonometry to measuring geographical distances.
a. Read through the given explanation. Now, suppose you used the same method to figure out the distance across another river. You set up your survey post directly across from a tree on the riverbank. You head downstream 50 meters, and take a sighting of the tree. Your sighting angle is 62°. Draw a sketch of this situation and calculate the distance across the river.
Answer:
b. In some situations, you can directly measure the distance across a river by stringing a rope across the river. What are some situations in which you would need to perform this measurement indirectly as shown above?
[Sample: The river was too difficult to cross or the river was between cliffs which were difficult to climb.]
Part B, Angles of Elevation and Depression
2. Learn alternate methods of measuring heights.
a. The method above for making an indirect measurement is fairly simple, but requires the surveyor to set up a survey post on one edge of the object to be measured. This is not always possible. However, there is another technique which allows surveyors to measure heights indirectly by setting up two survey posts in convenient spots. Go to the Distant mountain Web site to see an example of the use of this method. Look through the "Hints" and the "Solution" until you feel comfortable with the calculations.
b. Why would it be impossible to use the method in 1a to measure the height of South Sister?
[Because to use this method, you would have to be able to set up a surveying post in the middle of the mountain at its base, and this is impossible.]
3. Geometrical quadrants and measurements.
a. Flemish mathematicians of the sixteenth century used a tool called a geometrical quadrant to make indirect measurements using the method illustrated in 2a. Go to the Figure 19 Web site to see a sketch of a geometrical quadrant in use.
b. Suppose you used a geometric quadrant to measure the height of a mountain. Referring to the notation in the illustration on the internet site given in 3a, you sight the top of the mountain at two positions, points O and F. You also measure the distance between points O and I to be 100 feet.
[837.6 ft]
