Treasure

In searching through some old papers in the attic, you found a faded note from your great-great-grandfather saying that he'd buried his fortune right in the middle of a nearby field. The note identified the four corners of the field, and you have located them. From your Global Positioning System (GPS) you have found that the corner at the old oak tree is 638 feet due north of the corner thatŐs next to the large rock. A third corner is at a post 138 feet south and 550 feet east of the tree. The fourth corner is at the tip of a pond. The rock is 400 feet west and 100 feet north of the corner by the pond. Where should you dig for the treasure? (Adapted from Discovering Algebra)

Approaches

Choose origin and find coordinates of corners.

Why choose that point as origin? Can you plot the points on a calculator?

Make a scale drawing.

How accurate is it?

Find intersection of diagonals.

How do you find the coordinates of that point? What do you learn? Is that really "right in the middle"?

Connect middles of opposite edges.

How might you find midpoints knowing the coordinates of the endpoints? Is that point "right in the middle"? How does it compare with the points you found in other ways?

Cut out scale drawing, hang from corners, and drop vertical lines to find center of mass.

Is that point "right in the middle"? How might you find its coordinates?

Note that the shape is approximately a (right) trapezoid, so find the middle of the line halfway between the parallel edges.

How does that point compare with those you found before?

Simulation

A short demonstration simulation is available to show how students might work on this problem.