The Center of a Triangle: An Exploration . . . . . . . . . . . . . . name(s) _______________________

1)What is the center of a triangle? Discuss some possible answers to this question before you go any further. There are not necessarily any "right" answers to this question. You may give an answer or answers, pose questions related to the problem, sketch ideas, or whatever you feel is appropriate:

2) Why would you want to find the center of a triangle? Consider the following situations.

a) You are a city planner. The three towns of Kai, Pua and Lani have pooled their funds and want to build a rec center. The 3 towns are sketched below. Where would you put the rec center so as to be fair to all 3 towns? Sketch (not on the computer) and explain your solution.

b) You are a sculptor and have just completed a large metal mobile. You want to hang this artwork in the State Capitol so that it will be suspended with the triangular surface parallel to the ground. From what point should it hang? Sketch (not on the computer) and explain your solution.

c) You are an architect. You are designing a swimming pool and surrounding lanai. The client wants the lanai to be circular. The property on which it will be built is near a street, and has an existing house and a gazebo. (The property line is shown as a dotted line) Where would you place the center of the lanai, to make the lanai as large as possible? Sketch (not on the computer) and explain your solution below:

What is the center of a triangle? Let's explore this further.

3) Open the Geometer's SketchPad to a New Sketch. Construct a triangle ABC. Make the segments thick. Construct the midpoint of side AB and then a line through the midpoint perpendicular to the side AB. Make the line red and thin. Construct a point P anywhere on the perpendicular bisector. What is true about this point? ________________________ Measure the distance from this point to the two endpoints of the line (A and B). What is true about these distances, PA and PB? ____ Drag point P. Is this still true? ____ Delete point P.

Construct the perpendicular bisectors of the remaining sides. Make these lines red and thin. Do all three perpendicular bisectors meet at a point? ______ What is true about this point? ______________________________ Is this the center of a triangle? Why or why not?

Drag any vertex of the triangle and make some conjectures about where this point falls in different triangles. Sketch an example of each of the following types of triangles: scalene, isosceles, equilateral, acute, right, and obtuse triangles. Sketch the perpendicular bisectors for each triangle, and the point where they meet. Does this point seem like it could be called the center for any of these triangles? Discuss this, briefly.

4) Undo all the way back to your triangle, or Save this one and open a New Sketch. (To Undo, use either Command Z back to where you want to go, or Option Z back to the very start, then Command R to redo the triangle construction.) Now construct an angle bisector. Construct a point P on the angle bisector. What do you think might be true about this point? Is it equidistant from the vertices of the triangle as the perpendicular bisector was? Is this point equally distant from anything?

When you have answered this question, delete point P and go on. Construct the remaining two angle bisectors. Do they all meet at a point? _____ What is true about this point? ___________________________________________________________

Could this point be considered the center of a triangle? Why or why not? Drag any vertex of the triangle and make some conjectures about where this point falls in different triangles. Does it matter whether the triangle is scalene, isosceles, equilateral, acute, right, or obtuse? Include at least one sketch in your discussion:

5) Undo the angle bisector construction, or Save it and open a New Sketch. Construct the medians of your triangle. Construct the point of intersection. Six triangles are formed. Construct the polygon interiors for each of these triangles and make the interiors different shades and/or different colors. Explore the properties of these triangles. What can you discover about these 6 triangles? Are they congruent? Similar? Anything? Drag any vertex of the triangle, and measure anything you want. (You can measure perimeter or area by constructing the Polygon Interior, then use the Measure menu.

Could this point be considered the center of a triangle? Why or why not? Drag any vertex of the triangle and make some conjectures about where this point falls in different triangles. Be sure to consider scalene, isosceles, equilateral, acute, right, and obtuse triangles. Is this the center for any of these triangles? Include sketches in your discussion:

6) This time, undo until you have a blank screen (Option Z) or Save and open a New Sketch. Construct three non-collinear points. Select the points 2 at a time and use the Construct menu to construct lines until you have a triangle. Construct the triangle's interior and make it a light shade of any color. Construct the 3 altitudes. Do they meet at a point? _____ could this point be considered the center of a triangle? Why or why not? Drag any vertex of the triangle and make some conjectures about where this point falls in different triangles. Sketch an example of each of the following types of triangles: scalene, isosceles, equilateral, acute, right, and obtuse triangles. Sketch the altitudes for each triangle, and the point where they meet. Does this point seem like it could be called the center for any of these triangles? Discuss this, briefly.

7) Take another look at the questions and your answers for #2 a, b and c. Discuss what you have discovered, in relation to the three questions asked in #2. Look for connections between the geometric properties you discovered in #3-6 and the "real-life" applications in #2. Explain, for each of the situations in #2, how the city planner, sculptor, or architect would find the center needed.