Pythagorean triples are a major part of every high school geometry class. In order to explore Pythagorean triples we created a spreadsheet that would generate Pythagorean triples. The spreadsheet allows us to generate and use a large number of triples for our investigation of various ratios in the triangles. We found some interesting results from the rations and then proceeded to use GSP to clarify the results.
Our first spreadsheet only locates triples. We numbered across the top row and first column and then ask the computer to compare and only list the ones that satisfied the formula for Pythagorean Triples.
This spreadsheet generated a small finite number of Pythagorean triples. To get many the spreadsheet would need to be extremely large. Using these few triples, we looked at several ratios.
Using these ratios we created a line graph to assist in interpreting the results.
The graphs illustrated several properties of these relationships.
BUT this spreadsheet contains only nineteen triples.
To assist in drawing conclusions about Pythagorean triples we needed to generate a larger number of triples. To do this we created a spreadsheet using Euclid's formula for generating Pythagorean triples.
Let x and y be the legs of the triangle and z the hypotenuse.
x = 2mn y = m2 - n2 z = m2 + n2
where m and n are positive integers, with m > n
We put m in column A and n in column B and filled down with consecutive integers. Then created the Pythagorean triples in column C, D, and E using Euclid's formula. So using the spreadsheet you can put in any number in column A row 1, and column B row 1 and create a whole new set of Pythagorean triples.
Once we generated this list of hundreds and hundreds of triples, we included the ratios from the previous spreadsheet to check for similar results.
By graphing the results we saw that
The ratio of the legs to the hypotenuse of each triangle do converge to zero and one. The graph has a point of intersection. As you input different values for m and n the graph for the ratios of the hypotenuse to the legs will change slightly ( the point of intersection will change) but eventually the two ratios converge to one and zero.
The ratios of the legs, as we vary m and n, is a constant decline approaching zero. Of course it will not reach zero, this would remove one of the sides of the triangle.
The ratio of the area to the perimeter will always be either an integer or some half. And this holds true for any integer values of m and n.
GSP helps explain these results.
We begin by constructing a right triangle on a line so we can animate a point along that line. From this we measure the legs and generate the ratios of interest.
By animating the point C along the line we see the ratios of the legs approaching zero. As segment AC gets longer the ratio of the two segments will get smaller and smaller. It will not reach zero because then the triangle would be nonexsistent.
As segment BC gets longer, segment AC also gets longer and their ratio converges to one. As the hypotenuse AC gets longer the ratio with the leg AB converges to zero since we are dividing the length of AC by an ever increasing number. This helps illustrate the findings from the spreadsheet.
By using the spreadsheet and GSP we can make a mathematical connection between the calculations and the geometry. For high school students this is a monumental event. GSP's animation dramatically impacts the results found from the use of the spreadsheet. This could be used as a demonstration for high school students or as a discovery lesson they complete themselves.