GSP Application
Dr. Kimberly Vincent
THE SHORTEST PATH BETWEEN OBJECTS
Due Monday December 15th
You can do the following activity with patty paper or Geometer’s Sketchpad (GSP). If using patty paper then record
measurements in a table on the board so the data from all students can be used to make observations. If using GSP each
student can generate many examples using the dynamic feature. Today you will use Geometer’s Sketchpad.
ACTIVITY 1: The Burning Tent Problem—The Guess
When on his annual camping trip along the St. Joe River, John, took a hike up Marble Creek. Rounding the last bend in
the trail on his return to the campsite, he could see up ahead that his tent was on fire. A quick glance around told him
that he forgot to fill his bucket and take it back to the campsite. Instead John left his water bucket hanging on a branch
along the trail he was hiking. John must run to the river fill his bucket and run to the burning tent to put out the fire.
What’s the shortest path he can take?
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Construct a picture with a line segment AB representing the river.
Anywhere along one side of the river place any point labeled J, representing where John is standing.
Anywhere along the same side of the river, as John, place any point labeled T, representing the burning tent.
Burning Tent
John
River
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Think about the path John will run. He must first run to the river then to the burning tent.
Mark the point along the river that you think would make the shortest distance for John to run.
Label this point R.
Questions to Consider. Write your answers in a text box, using size 10 or 12 font. I will not read anything smaller.
Put your name in a text box. Print your sketch after answer Q1, Q2, and Q3.
Q1. What measurements could we make to be more accurate? Make the measurements. Using the calculator cut and
past the measurements and add them together. You should see the length of the shortest path in your list of
measurements on the screen. Then move the point until you think the distance is shortest.
Now, measure JRA and TRB . Using the dynamic features of GSP move things around and observer what
happens to the two angles. Q2 Make a conjecture about the angels when the path is shortest. Write your conjecture and
how you can verify it in a text box.
Q3. Move John and observe what happens. Move the burning tent and observe what happens. Move the river and
observe what happens. Make a conjecture on their relationship based on the data and your observations.
GSP Application
Dr. Kimberly Vincent
Activity 2: The Burning Tent Problem --The Mathematical Approach.
 First construct the image of T by reflecting T in the line AB. Label the image of T as T’.
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Now construct the line JT’ with a dotted line. Label the intersection of JT ' and AB as E.
Measure the total distance from John to this intersection point, to the tent (that is measure the distance from J
to E and add that to the distance from E to T).
Measure the angles JEA and TEB . Record the measurements.
Questions to Consider. Write your answers in a text box, using size 10 or 12 font. I will not read anything smaller.
Put your name in a text box. Print out your final sketch after answer Q4, Q5.
Q4. What is the relationship of the angles?
Q5. Make a conjecture as to what you think point E is?
Measure the path John takes to E to the tent. How was your first guess? Adapt your conjecture if need be.
GSP Application
Dr. Kimberly Vincent
The Shortest Mirror Problem
What’s the shortest mirror you’d need on the wall in order to see your full reflection from your toes to the top of your
head?
Activity 3: Constructing Paths Light Travels
Draw You and the Mirror
 Open a new sketch and construct two parallel line segments to represent you and a mirror.
 Label the endpoints of one line segment with “Top of Head” and “Toes”.
 Place a point below the top of your head and label it “Eye”.
Top of Head
Eye
Mirror
Feet
Draw One Beam of Light by following the steps below:
 Reflect the point representing the top of your head across the line segment representing the Mirror.
 Construct the line segment from the image to your eyes.
 Label the intersection of this line segment with the Mirror as T.
 Construct the path a beam of light will take from the top of your head to your eyes.
Draw A Second Beam of Light by following the steps below:
 Reflect the point representing your toes across the line segment representing the Mirror.
 Construct the line segment from the image of your toes to your eyes.
 Label the intersection of this line segment with the Mirror as B.
 Construct the path a beam of light will take from your toes to your eyes.
Questions to Consider. Write your answers in a text box, using size 10 or 12 font. I will not read anything smaller.
Put your name in a text box. Print out your final sketch after answer Q5.
Q5. What will you see between T and B when you look in the mirror? Move points and lines and make observations to
see if the relationships you see observe always holds. Adjust the description of the relationship so it always holds.
GSP Application
Dr. Kimberly Vincent
Measuring
 Measure the distance between T and B. Record it on the table on the instructors projected computer file.
 Measure and record the distance between the top of your head and your toes (in your picture—not for real).
Questions to Consider. Write your answers in a text box, using size 10 or 12 font. I will not read anything smaller.
Put your name in a text box. Print out your final sketch after answer Q6, Q7.
Q6. What relationship is there between a person’s height and the distance between T and B?
Q7 Generalization
What is the shortest mirror a person n feet tall will need in order to see their full reflection?