FINAL

BY

DEBORAH ECKSTEIN

BOUNCING BARNEY

Barney is in the triangular room shown here. He walks from a point on BC parallel to AC. When he reaches AB, he turns and walks parallel to BC. When he reaches AC, he turns and walks parallel to AB. Prove that Barney will eventually return to his starting point. How many times will Barney reach a wall before returning to his staring point? Explore and discuss for various starting points on line BC, including points exterior to segment BC. Discuss and prove any mathematical conjectures you find in the situation.

I. Let's see if we can get Barney back to his original starting place.Lets let Barney start at point D. Then travel parallel to AC landing at point F and then traveling parallel to BC landing at point G and then traveling parallel to AB landing at point E.

From his walking Barney created two parallelograms, BFGE and DFGE. So now we can say,

Now Barney will continue on his journey and he will walk from point E parallel to AC and landing at point H, then turing and walking parallel to BC landing at point I.

Looking at his walking picture we can see that he has formed two new parallelograms, AGEH and CEHI. So we can conclude,

YAY BARNEY !! Since AIFD is a parallelogram we know that Barney ended in the same place that he started, PointD.

II. Now let's start barney out at a different place! What if Barney started on the vertex of our triangle?

Notice that Barney will end up walking around the triangle, maybe he wanted to calculate our perimeter for us. I wonder if he ended up walking more by starting at the vertex than he did when he started at our point D.

WOW the distance comes out to be the same! That is pretty neat. The only difference would be that in starting at our point D Barney hits our wall 6 times and with the vertex he stays on the wall.

III. Let's see what happens when Barney starts out at the midpoint of BC. How many times do you think he will hit the wall? Where will he end up hitting the wall at on each side of the triangle?

He hits the wall three times at each midpoint and returns to his starting point! Lets see why this is. We know that if we have two parallel lines and midpoints joining each other then the segments are congruent so we can say

And we can Note the Barney will travel half of the distance as he has before!

IV. Now let's see what happens to Barney when he starts somewhere outside of our triangle.

Once again Barney returns to his starting point but why? Let's investigate...

Since Barney's path is IHGFED we can see that he will return to his starting point!

There are many more locations that you could choose to look at. Use your imagination and see what you can come up with.

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BUTTEFLY THEOREM

Through the midpoint M of any chord PQ of a circle, any chords AB and CD are drawn; lines AD and BC meet PQ at points X and Y. You may want to construct a line PQ containing the chord PQ. Prove that M is the midpoint of XY.

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