Geometry
3D Perspective and Spacial Visualization
Anamorphic 3-D Drawing
1 and 2 Point Perspective Labs
Heights & Areas of Triangles with Trig
SOUTH MOUNTAIN
tan20 = H/X+82 tan25 = H/X H=(X+82)tan20 H=xtan25 (x+82)tan20=xtan25 xtan20+82tan20 = xtan25 82tan20=xtan25-xtan20 82tan20=x(tan25-tan20) x = 82tan20/(tan25-tan20) x = 291.63 tan20 = H/291.63+82 (373.63)(tan20) = H=136 |
WEST ROCKSPIRE
H/X+45=tan15 H/X=tan20 H=(X+45)tan15 H=xtan20 (x+45)tan15=xtan20 xtan15+45tan15=xtan20 45tan15=xtan20-xtan15 45tan15=x(tan20-tan15) X = 45tan15/(tan20-tan15) X=125.57 tan15 = (H/125.57)+5 H=50.7 |
EAST MOUNTAIN
tan10=H/(x+90) tan15=H/X H=(x+90)tan10 H=xtan15 xtan10+90tan10=xtan15 90tan10=xtan15-xtan10 90tan15=x(tan15-tan10) x = 90tan10/(tan15-tan10) x = 173.2 tan15=H/173.2 (173.2)(tan15) = H H = 46.4 |
HexaflexagonWhen creating my hexaflexagon, I had to be mindful of all of the aspects of rotational symmetry and line of reflection symmetry that create hexaflexagon. Each diamond in the shape is created off of the others with rotational symmetry. Each triangle is reflected across the line of reflection symmetry that goes through the middle of the diamond. This created the second triangle that makes up the second half of the diamond. As a result, when I colored the hexaflexagon I made sure that all of the diamond shapes were made up of the same pattern. The pattern on the side shown in the picture is probably my favorite because I like how it makes another triangle with the three circles on each diamond. If I could redo this project, I would try harder to make each diamond of each side exactly the same so that the effect would be more pronounced. From this activity, I learned that my artistic ideas will be executed better if I take more time to refine them in my head. That way I can fix any mistakes in my head instead of having to do it while in the process of creating it.
Two Rivers Geogebra LabSCENARIO:
There is a sewage treatment plant at the point where two rivers meet. You want to build a house near the two rivers (upstream from the sewage plant, naturally), but you want the house to be at least 5 miles from the sewage plant. You visit each of the rivers to go fishing about the same number of times but being lazy, you want to minimize the amount of walking you do. You want the sum of the distances from your house to the two rivers to be minimal, that is, the smallest distance. The big question is this. Where should you build your house? We will use the sketch to model the scenario and to answer the question. The first picture to the left DOES NOT meet the requirements because the house is located within the five mile radius from the sewage plant, and is 1.38 miles from one river while 2.19 miles from the other. Since the segment from the house to the sewage plant DOES NOT bisect the angle of the two rivers from the sewage plant, the house is located closer to the West River than the East River. The second picture to the left DOES meet the requirements because the house is located on the outside of the five mile radius from the sewage plant, and is 2 miles from both rivers. This way the lazy inhabitant of the house will not have to walk a longer distance to either river, and the house is a reasonable distance from the sewage plant. Since the segment from the house to the sewage plant DOES bisect the angle of the two rivers from the sewage plant, the house is located exactly between the West River and the East River. Snail Trail Graffiti LabIn this lab, I started off by making a circle, and then dividing it into 6 sections by rotating points on the circle and making segments with those points. After that I made a random point in one section, and then by reflecting it across into other sections I made 5 more. Afterwards, I hid the segments and some points in order to clear up space. When I was finished, I was able to move the turquoise point (which affected the other colored points) around to create art! This design demonstrated reflection symmetry and rotation symmetry through the rotation of the segments and points and the reflection of points. In the process of creating this design, I learned what kind of designs look appealing to the eye, and how reflection across 3 or more sections creates a 360 degree design.
The Burning Tent LabSCENARIO:
A camper out for a hike is returning to her campsite. The shortest distance between her and her campsite is along a straight line, but as she approaches her campsite, she sees that her tent is on fire! She must run to the river to fill her canteen, and then run to her tent to put out the fire. What is the shortest path she can take? In this exploration you will investigate the minimal two-part path that goes from a point to a line and then to another point. The first picture to the left IS acceptable because it shows the shortest distance that the camper can take to run to the river and then to her campsite. If she filled her canteen at a spot on the river farther to the left or farther to the right, then one of the parts of her path will be unnecessarily long and it will take her longer to put out her fire. The minimal distance of the path is 8.796. The way I reached this conclusion is I reflected the point Tent Fire across the line of reflection that is the river to get Tent Fire prime. The minimal two-part path runs through the point Tent Fire prime because it illustrates the most direct path. That path is the shortest distance possible from the Camper to the River and from the River to the Tent Fire while still meeting in the middle. This way the Camper will be able to save her tent quickly and efficiently. The second picture to the left IS NOT acceptable. It adds unnecessary distance and therefore it would take longer to put out the burning tent. It shows that the camper walked a longer distance to the river and a shorter distance from the river to the Tent Fire. However, the distance from the Camper to the River has been lengthened more drastically than the distance from the River to the fire has been shortened. Therefore, the two-part path is overall longer. Evidence of this theory is given when you look at the numbers: the first path is 8.796 while the second path is 8.88. |