The main objective of POW 3 is to find the correct location to place a patrol station between highways, so that the station will be equidistance from the highways, but it has to be the shortest possibly path from the station to highway. So what we are really doing is finding where to plot a point, which has to be at an equal distance and shortest possible path from a number of lines, which are intersecting. I have four situations, having two, three, four, and five highways; I have to find where I can place the patrol station so that it is the shortest path and equidistance from all the highways. From completing this problem of the week, I will know where and how to plot a certain object so that it becomes an equal distance from having two or more lines intersecting each other.
My plan for solving this problem is, begin first using two highways (lines) and discovering all the different ways the two highways interest and working from there to see where I can place the station so that it is equidistance and the shortest path from the two highways. I plan to follow these same steps when dealing with three, four, and five and more highways. After finding where I can put the patrol station in all four cases I plan to place them in categories. I will do this by looking at the drawings of each example I am going to draw, then placing them into categories.
Process:
The only difficulties I have, was finding the shortest possible path, and making sure that it is equal distance from all the highways. What helped solve this minor problems were the homework assignments we did that told us how to find the distance between two points and finding the center point, for example, homework 6: the distance formula, homework 7: a snack in the middle, homework 8: Proving with Distance- Part 1.
These homework assignments taught me how to find the mid point between tow points, and how to find the distance between two points. Another huge assignment that helped had to be, homework 13: Another kind of Bisector, which showed how angles also have bisectors as well as lines. This came helpful when dealing with where highways intersect, finding when dividing the angle at where they intersect, if it is still equidistance.
Other basic geometric skills that helped where, forming polygons with the highways, when dealing with the cases of three or more lines, knowing how to find their angles, for example when dealing with three highways, you can make either an isosceles or equilateral triangle, knowing that in an isosceles triangle there are two angles of equal measurement, as well as in a equilateral triangle all three angles are congruent. Another method that helped was the one about having the circle inscribed or circumscribed in the polygon, this would help in the way that it would be much easier to find the center point and equal distance between the highways.
Solutions:
When having two highways there are many ways the lines can intersect and many places where to station the patrol station. From wherever the two highways intersect, you find where the angle bisector is, meaning you can place that station along any point of the bisector path. Also if the two highways intersect at a right angle, you can use perpendicular bisector to find where to put the station, and after placing that point, it should create a perfect square.
In the case of three highways, there are also many ways in which the highways can intersect. You can have all three highways forming an equilateral or isosceles triangle, or having two lines parallel and having the third cross them as if it were a transversal. These are just a few of all the possible intersections.
From these examples is why the answer to, “Is it possible to place the station the same distance from all three highways,” is yes! If the lines intersect at one point then the station would have to be where they all intersect. If they intersect with two points, then as I said you can have two of the lines being parallel and the third crossing them. In the case of having three intersection points, it can form an equilateral triangle. In all three cases the patrol station will be equidistance from the highways.
When dealing with more then three highways, possible arrangements of highways that allow you to place the patrol station equidistance to all of them, is making regular polygons, for example, in the case of four highways, you can make a perfect square, or with five highways you can make a pentagon, or a hexagon with six sides. All you have to do is find the center point in each of the polygons and place the station there and it will be equidistance form all the highways. You can do this by having a circle circumscribed on the polygon.
Evaluation:
The geometric ideas and skills I gained from completing “POW 3: On Patrol” are, well the idea of the angle bisector was developed from the homework, but I was able to further more experiment and become accustom to it through this problem of the week. I also found out that there are many possible ways to place a point between lines, but it might not always be the shortest.
Simple real world applications are for example cutting a pizza or pie, if you want equal congruent pieces, then you have to find the center point, and making a cut to the crust, which would be equidistance from all the points on the crust to the center. Or it could be as hard as where to build the bathroom so that it is equidistance from the bedrooms. These are just a few ways of how POW 3 can be used in real life situations.
While completing POW 3, I think I did a very good job of keeping my research and experiments organized, while checking sure that I followed all the instructions and requirements in the write-up and rubric. What I did well was that of going step by step. First I started with the two highways and then I moved on to three, four and so on. I actually did very well in not making assumption, the only assumption I did do was that all the highways were equal in length. To minimize my errors, what I did was draw all the examples, when I thought of something.
Also I made sure that I was following all the requirements from both the write-up and the rubric. Regarding my logical thinking skills, I would have to say they have improved from the beginning of the school year up to know March 6, 2005 at 8:03pm, LOL No honestly a good part of my logical thinking skills have improved. This thanks to all the harsh homework assignments we have done. And of course from all the problems of the week we have done, which help us further more, extend our thoughts and understandings of the main unit problem. In no way do I think anything has kept me from making progress in this geometry class.
