The King’s Slice

• rulers 
• uniform pieces of cardboard paper 

REMARKS:

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 ACTIVITY:

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:

Empirical verification is a bit tricky. There are a few ways the correct answer can be verified, but don’t expect perfection! 

• One method is to get a nice electronic scale and weigh the two pieces (it’s a balancing act just to place the large pieces on the scale, but it can be done). 
• Another method is to create a simple scale balance by hanging the two pieces. Since the original is cut, the two pieces are suppose to balance. They won’t. However, by taping little amounts to the lighter side, one should determine that they are not off by too much. 
• A third method is to try cutting up one piece and having it fit over the other. The result isn’t too bad. 
• An excellent idea is to perform the solution on an isosceles right triangle (the answer, of course, is independent of triangular shape). Fold the triangle along the “cut line” (don’t actually cut it!), then cut the isosceles right triangle which overlaps the fold. It will fit perfectly onto the uncovered (by the fold) isosceles triangle. (An attempt will make this clear.)