September 11
Assignment 1
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Assignment 1 |
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1) page 17, Ex. 1.3, 1.4, 1.6, 1.7, and 1.8
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2) Using Sketchpad, find the flaw in problem 1.5 on page 17
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3) Using Sketchpad, find the best locations for each survivor, and explain your findings.
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Maria loves to surf and discovers the the surfing is outstanding on all three of the island's coasts. She crafts a surfboard from a fallen tree and spends her days surfing.
On the other hand, Paul sorely misses civilization and spends his time going to a different corner of the island each day to check for passing ships.
Each survivor wants to build a shelter that best suits their respective needs. They have no interest in living in the same place. However, if it turns out to be advantageous to live in the same area, neither is against that idea. María wants her shelter to be closest to the three beaches as she visits them with equal frequency. Paul wants his shelter to be situated so that he can create the shortest possible paths to the three corners of the island. In other words, the sum of the distances to the sides of the triangles must be minimized for María; and the sum of the distances to the vertices must be minimized for Paul. Where should each of them live?
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1) Construct an equilateral triangle ABC.
3) Relabel D as Paul, if you wish.
4) Construct E anywhere inside the triangle.
7) Hide the perpendicular lines. Relabel E as María, if you wish
9) Calculate DA + DB + DC
To do this calculation, select Calculate under the Measure Menu, and a little calculator appears. You can then single click on each separate measurement as a quantity to be used in the calculator.
11) Calculate EF + EG + EH
12) Move the points D and E (Paul and María) around inside the triangle to find the solutions.
Your image will be a
variation on this image.ttt
The triangle DEF (in green below) is called the Pedal Triangle of triangle ABC with respect to P. As P is moved about, the pedal triangle will have different shapes. Answer the questions below about the Pedal Triangle.
HINT: This construction works best if all three sides of triangle ABC are extended and the perpendiculars are drawn to the extended sides.
1) Select the interior of the Pedal Triangle EFG and Measure the Area.
2) Drag point P around to make the area of triangle EFG change.
i) How large can you make it?ii) How small can you make it?
iii) Is it ever equal to 0?
iv) Is there a position where the Pedal Triangle appears to be equilateral? Is there more than one position?
v) Construct the circumcircle of triangle ABC. Drag point P to the center of the circumcircle? What happened?
vi) Drag point P to any point on the circumcircle. What happened?
Write down your conclusions!
THE PEDAL TRIANGLE
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