Write-up

Title
ALL SWIMMED OUT

Problem Statement
Suppose Sammy the swimmer at the tip of pier H wants to swim to the tip of pier I. Pier H is 2 km long and pier I is 1 km long. Since the swim is very long from H to I, Sammy thinks he will need to stop off at the beach to take a break (at point J). Sammy can stop at any point on the beach between the two piers (drag point J around once you construct this in a geometry software or view the java applet below). If Sammy takes the break, where should he stop on the beach if he wants to swim the least distance (the blue path) for the entire trip? What is the shortest distance Sammy can swim for the entire trip?

Problem setup

This is the typical shortest distance using two triangles problem. We are trying to evaluate at which point on the common leg (beach) of the triangles which the swimmer must touch in order to swim the shortest distance.

Plans to Solve/Investigate the Problem

I hypothesized that the shortest distance for Sammy to swim would be the distance created by the base legs of two similar right triangles. In this case, the heights of the triangles have a ratio of 2 to 1. Therefore, if my hypothesis is correct, the legs would also have a ratio of 2 to one, and the angles would be congruent at the point of least distance.

Investigation/Exploration of the Problem

I constructed a diagram similar to the one given in the Investigation. Once placed the common point at the vertices of the two right angles, I began moving the point and investigating both the measurement changes and the angle variances from point to point. Although it is not exact due to the measurement limitations of GeoSketchPad, it does appear that the least distance from H to I running through J is indeed at the point that the triangles have proportional relationships.

Extensions of the Problem

To continue on with a proof, you could construct transverse triangles and, using the Pythagorean Theorem (as well as typical minimum distance theory) and prove that indeed the minimum distance is at the point of similarity.

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Author & Contact
Jim Taylor
jtaylor1@rockdale.k12.ga.us