What information and calculations are needed to determine distance and location?

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1 Think About the Situation The Great Smoky Mountains National Park sits astride the Tennessee - North Carolina border amid the majestic southern climax of the Appalachian Highlands. The most visited of national parks draws more than nine million adventurers and sightseers each year. And for good reason the Smokies are within a day's drive of a third of the U.S. population, and very few places in the East are in their league as an outdoor-recreation destination. The Appalachian Trail meanders across the mountain tops of The Great Smoky Mountains National Park. Conceived in 1921 by Benton McKay and initially cleared and marked in 1923, the Appalachian Trail was completed in 1937 and is a marvelous tribute to the well-meaning individuals such as McKay who overcame many obstacles to create this splendid national treasure. The trail winds for 2,015 miles (actually it varies due to changes in sections of the trail) through parts of 14 states. Its southern terminus is Springer, GA and ends (or begins, depending on your perspective) on Mount Katahdin in Maine. Visitors to the park decided to hike a portion of the trail. They stopped at the first overlook and estimated they had hiked about 2 miles east and 1 mile north. a. Do you think they hiked directly east and directly north? Explain. b. How might they have estimated their travel east and north? c. How far have they hiked? d. How far are they from where they started? Investigation: Distance and Midpoint Formulas Many hikers rely on Global Positioning Systems (GPS) to determine location and distances; however, there are other methods for determining how far you are from a set point. As you work on the following problems, look for answers to this question: What information and calculations are needed to determine distance and location? 1. A coordinate grid can be used to determine how far the visitors are from where they started. a. Represent the distance and direction (2 miles east and 1 mile north) by drawing line segments onto a coordinate grid with the starting point at the origin. Label the coordinate point of the overlook where the visitors stopped. North b. Draw a line segment from the starting point to the overlook. Identify the plane shape that is formed by the line segments drawn on the coordinate grid. c. Find the distance between the starting point and the overlook. d. Show or explain your work. West East 2. In order to calculate the distance, what information was most important and how did you use the information to calculate the distance? South

2 3. The FunFilled Amusement Park is creating a new brochure. They want to include in the brochure distances between some of the most frequently visited attractions. Use the copy of the map and the method from #1 and #2 to find the distances between each pair of attractions. a. Ferris Wheel and Arcade b. Basketball Shot and the Bumper Cars c. Bumper Cars and Water Slide d. Roller Coaster and Arcade e. Concessions and Theater f. Acrobats and Arcade 4. Francine was tired of drawing right triangles and decided to look for a pattern to make finding distances more efficient. She noticed that she used the horizontal distance and the vertical distance between two points for each right triangle she drew. She decided to investigate a way to get the distances without always drawing and counting them. Follow Francine s thought process and help her find a pattern. a. First, she used the coordinates for the Bumper Cars and the Water Slide since they are on the same horizontal line and found the distance. She noticed she could use the coordinates to help her find the distance. What coordinates did she use? (, ) and (, ) Bumper Cars What distance did she find between the two? Water Slide Explain how she used the coordinates, and not the graph to find the distance between the Bumper Cars and Water Slide. b. Francine then decided to try to think about this for ANY two points on the same horizontal line. She wrote down two coordinates (x 1, y 1 ) and (x 2, y 1 ) with x 1 < x 2. Explain how you know that Francine is indicating the points are on the same horizontal line. Write an expression for the distance between the two points. c. Next, she used the coordinates for the Basketball Shot and the Bumper Cars since they are on the same vertical line. Basketball Shot Bumper Cars What coordinates did she use? (, ) and (, ) What distance did she find between the two? Explain how she used the coordinates, and not the graph to find the distance between the Basketball Shot and Water Slide. d. Next, Francine wrote down two coordinates (x 1, y 1 ) and (x 1, y 2 ) with y 1 < y 2. i. Explain how you know Francine s points are on the same vertical line. ii. Write an expression for the distance between the two points.

3 5. Draw the horizontal and vertical segments for the line segment, d, given on the diagram to the right. a. Write expressions to represent the length of the horizontal segment and the length of the vertical segment. y (x 2, y 2 ) Horizontal: (x 1, y 1 ) Vertical: b. Recall the Pythagorean Theorem and use it to help write an expression for the length of the given line segment, d. c. Use the expression to find the distances between each pair of points. i.(3, -2) and (5, -1) ii.(2, -1) and (-4, 3) iii.(-1, -3) and (4,1) iv. (0.5, 2.1) and (4, 2.1) d. Compare with others and resolve any differences. Summarize the Mathematics In this investigation, you developed methods for calculating the distance, or length of a line segment, on a coordinate plane. i. Using (x 1, y 1 ) and (x 2, y 2 ), write the formula that can be used for calculating distance. ii. Explain in words how the formula determines the distance. iii. If you have calculated the distance of a line segment on the coordinate plane, using the formula, and you are uncertain that you did it correctly, what other way can you calculate the distance to check your work?

4 Investigation: Distance and Midpoint Formulas (Continued) Travis and Janelle want to take a trip to the mall to do some holiday shopping. However, they have a very limited amount of time to do so. Once at the mall, they will need to split up and shop at the stores separately to get everything on their shopping list quickly. Janelle printed a layout of the mall to map out their route. The black rectangles, on the map, represent the entrances(s) to the stores, while the stars represent the entrance doors to the mall. We will use the map to help Travis and Janelle determine their meeting locations to check in after shopping at some of the stores. 6. Travis decides to head to the cell phone store first, while Janelle goes to purchase some shoes. They agree to meet back at the midpoint of their locations. At what coordinate would they meet? How did you determine this point? How could you use the coordinates from the cell phone store and shoe store to calculate the midpoint location? Cell Phone Store Shoe Store Meeting Location (Midpoint) (, ) (, ) (, ) 7. If Travis started at the West Entrance of the Food Court and Janelle started at the restrooms, and they wanted to meet halfway between each other (their midpoint) to check in, where would they meet? At what coordinate would they meet? How did you determine this point? How could you use the coordinates from the Food Court s west entrance and the restrooms to calculate the midpoint? Food Court West Entrance Restroom Meeting Location (Midpoint) (, ) (, ) (, ) 8. They decide to check in after Janelle goes to the Party Store and Travis goes to the Movie Store, where will their midpoint meeting location be? At what coordinate would they meet? How did you determine this point? How could you use the coordinates from the Party Store and the Movie Store to calculate the midpoint? Party Store Movie Store Meeting Location (Midpoint) (, ) (, ) (, ) 9. How could you find the midpoint of ANY two points (x 1, y 1 ) and (x 2, y 2 ) without graphing? Using these two points, what expression would represent finding the midpoint of any horizontal distance? Using these two points, what expression would represent finding the midpoint of a vertical distance? Write the expressions in their appropriate place below for calculating a midpoint coordinate: Midpoint: (, )

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6 10. If Travis and Janelle check in at the location (17, 8.5) and you know Travis left the Music Store, where was Janelle coming from? 11. Above, you were given the midpoint (meeting location), and Travis s beginning location, which could be called an endpoint. How did you use these two pieces of information to find Janelle s original location, or the other endpoint? 12. How could you the coordinates to find an endpoint when you know the midpoint and one of the endpoints? 13. Complete the table below by filling in the missing information. For some, you are given the endpoints that determine a line segment and must find the midpoint, the point on the segment that is the same distance from each endpoint. For others, you must find the missing endpoint. Endpoint 1 Endpoint 2 Midpoint (-3,2) (1,2) (-4,1) (-3, 1) (3,3) (3, 0.5) (1,-3) (1,2) (3,2) (1,1) (-3,-1) (2,3) (4,-2) (0.5, 0.5) Summarize the Mathematics You just developed efficient ways to calculate the midpoint of a segment on a coordinate plane i. What is the formula for calculating the midpoint? ii. iii. Explain in words how the formula determines the midpoint. Explain how to find a missing endpoint when given the midpoint and the other endpoint. Check Your Understanding On a coordinate grid graph the points A(0, 1), B(2,5), and C( 4, 1) a) Find the coordinates of the midpoints of AB and AC. Call these midpoints D and E, respectively and plot them on the graph. b) Find and compare the lengths of DE and BC.