The Burning Tent Problem
The first thing we need to do in this problem is construct a sketch. First I created my river, which was a line. Next I constructed a point and labeled it C for the camper. I then constructed another point on the same side of the river as the camper and labeled it T for the tent. Then I constructed a point on the river. I drew segments from the camper to the point and from the tent to the point. This represent some path that the camper would take from her position to the river to her burning tent. The goal of the problem is to find the shortest distance from the camper to the river to her tent. To do this I measured the length of the path from the camper to the river and from the river to the tent. In the algebra tab of GeoGebra, I found the sum of these two distances. I observed that as I drug the point on the river the length of the paths changed and so did the distance. As I moved the point on the river to the left the distance decreased and then began to increase again and the same occurred when I moved the point to the right. In my mind I thought of these distanced as a parabola. I moved the point until the distance was the lowest value but there was a range where these values were the same. So I decided to take my measurement out another decimals place, but again there was a range of this length. I did this until there was only point with the smallest distance. I had to take my decimals out 5 places. Once I found the shortest path I measured the angles formed from the lines. The angle at which the camper runs to the river is the incoming angle and the angle at which the camper runs from the river to the tent is the outgoing angle. I found the measure of these angles are saw that there were almost equal. They were only hundredths apart. From this I said that they were equal. So from this we can say that when we have the shortest path the angles formed are equal.
Next I wanted to see if there was a way of finding the distance without a guess and check method. I know from previous knowledge that the shortest distance between two points is a straight line. So I started to think how can I have a straight line between these two points that still has the camper running to the river for water. From my previous knowledge of reflections I thought that I could try reflecting the path from the river to the tent. When I did this I observed that this formed a straight line.
With this knowledge from above, I wanted to find the shortest path without guess and check. So I constructed the sketch where I had the river and the camper and the tent on the same side of the river. With the idea that the shortest distance between two points lies along a straight line, I decided to reflect the tent across the river. This was labeled T'. Once I did this I constructed a line segment connecting the camper and the reflected tent. This distance would be the shortest distance from the camper to the tent. However the path is not a realistic path for this scenario. Next I constructed the intersection of the path from the camper to the reflected tent and the river. This is the point that the camper runs to the river to get water. Next I constructed a path from this point on the river to the tent. This should be our shortest distance. We can check this by comparing the sum of the distance of the path from the camper to the river and then from the river to the tent to the path from the camper to the reflected tent. We can see from adding the segments that they are equal. Also we need to check to see that the angles are equal. From the angle measure tool we can see that the incoming and outgoing angles are of equal measure. They are equal because they are vertical angles. and are vertical angles by the definition of vertical angles. are equal because they are formed by reflected segments making .