“Bouncing Barney”

by

Christopher R. Whitworth

Barney is in the triangular room shown here.  He walks from a point on AC parallel to BC.  When he reaches AB, he turns and walks parallel to AC.  When he reaches BC, he turns and walks parallel to AB.  How many times will Barney reach a wall before returning to his starting point?

To solve this problem, one could use Geometers Sketchpad 4.0.  After opening the program, one could simply place three points anywhere on the blank document and link them with line segments.  This will create a triangle.

Place a point anywhere on one of the lines and create lines that will run parallel to one of the two line segments that the point does not lie on.  It does not matter which of the two line segments you choose.

Once the first line has been created, use the ‘arrow’ tool on Geometers Sketchpad 4.0 to place a point where the line and line segment intersects.  Then, continue to create lines that will make two small congruent triangles in the corner of each triangle.

Continue the process until you reach Barney’s starting point.

Barney will reach a wall six times before returning to his starting point.

What happens if Barney's first direction is NOT parallel to any of the sides?

Note that Barney never returned to his starting point.  However, when Barney traveled in directions that would create perpendicular lines to the line segments opposite to the points, then parallel lines are created within the triangle.  But, since Barney did NOT travel in parallel lines to the sides, he did not return to his starting point.

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