Triangle inequality workshop Name: ________________________

1. Draw a convex quadrilateral ABCD. In this context, convex means that the diagonals of the quadrilateral intersect at a point, M, inside the figure.

Determine for which point in ABCD the sum of the distances from the point to the vertices is as small as possible.

Outline:

· Choose an arbitrary point, P, inside ABCD. (Sketch)

· Use the triangle inequality to compare the sum of the distances from P to the vertices and the sum of the distances from M to the vertices.

· Conclusion …

· Can you think of an application for this?

2. In the super fun game, Math-Tag, you start at point A, run to the line, touch the line at a single point P, and run to point B. If point A is (11, 5), point B is

(1, 15), and the line is the x-axis, where will you touch the x-axis to make your total trip as short as possible?

Outline:

• Sketch the setup.
• What point is just as far from the x-axis as point A? Alternately, from what other point, A’, will the trip to B be just as far? Think symmetrically.
• Once you have the answer, consider how the triangle inequality figures into this.
• Can you think of an application for this?