This is a nice experiment intended, originally, to provide a real world experiment which involves the square root of 2. The problem posed can be solved mathematically in many different ways.
The instructor should cut variously shaped, large triangles from the cardboard pieces. Each team of students is given one of these triangles. The following story is now told:
Once upon a time there was a certain kingdom which was in the shape of the triangle you have before you. One day, it was the King’s birthday. As part of the celebration, the king’s chef ordered that a slice of cake be brought to the ruler in the shape of his kingdom, and this was the size of the piece before you.
However, the Queen-who happened to be a skilled mathematician-was very interested in keeping down the king’s weight. By coincidence, she came upon the bearers of the King’s slice of cake and demanded of them, “No! This is too great a piece. The king shall have only half this amount.”
But the bearers exclaimed, “Your highness, with all due respect, the chef has molded this slice in the proportions of his lands and we fear for what might be done to us should we alter this sacred gift.”
Alas the Queen would hear none of this and told them, “Nonsense! With a single straight cut you can both preserve the design of the kingdom and honor your queen’s desires. Now get to it, ... or you shall pay a price greater than any mere chef can impose!”
The question, of course, is to determine the precise distance from one vertex (along a perpendicular to the opposite side) at which one should draw and cut a line, parallel to the opposite side, so that the smaller triangle is similar to the original and precisely half the area.
Empirical verification is a bit tricky. There are a few ways the correct answer can be verified, but don’t expect perfection!
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