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Transcript
Amanda Lutje Math 329 September 21, 2015 Homework 4 – Spheres and Triangle Exploration Part 1 1) a. The shortest path between Anchorage and Saint Petersburg is calculated in 1b, which is if you head directly west on a great circle between these cities. This path was found to be 13343.391 kilometers. We can find the distance heading directly east by subtracting this from the circumference of the great circle connecting these cities. If the radius is the 6371 km, then the circumference is 40030.17359 by formula. So we get 40030.17359 km 13343.391 km = 26686.78259 km. b. 2) Assuming the borders of Ideal City are as such, the different colors represent the different districts. I calculated the distances between Roosevelt and Franklin, and Jefferson and Franklin. Using the distance I halved it to get the midpoint, which was the radius for the circles. I made boundaries cutting through where these circles meet to determine where they are separated. They are not perfect squares, but I used the properties of circles to visualize the areas that are actually closest to each school. Part 2 1) Equilateral Triangle: Three-sided polygon. All sides are the same length, and angles are equal to 60 degrees. Isosceles Triangle: Three-sided polygon. At least two sides and two angles are equal. Scalene Triangle: Three-sided polygon where no sides or angles are equal. Acute Triangle: Three-sided polygon where at least one angle is less than 90 degrees, all angles greater than 0 degrees. Right Triangle: Three-sided polygon where only one angle equal to 90 degrees. Obtuse Triangle: Three-sided polygon where only one angle is greater than 90 degrees, but less than 180. 2) On the sphere you can have a triangle with three right angles. But you can also have an acute and obtuse triangle. So the sum of the interior angles in a triangle on a sphere can be up to 300 degrees considering these triangles. 3) A sphere can be cut up into 8 different right-angles triangles, using perpendicular great circles to separate them. They satisfy the definition of a triangle in that the 3 points are connected by a line, but the angles are greater than the Euclidean space would allow. 4) Above is what the “two-sided polygon” would look like. In my definition this would not qualify as a triangle, because I explicitly define triangles as having three straight sides. I would classify this as an irregular polygon though, because it is a closed figure with vertices. 5) The Venn diagram I made would almost work for triangles on the sphere. You would have to allow for equilateral triangles to be right triangles. But the rest should still hold.