Pythagorean triples are a major part of every high school geometry class. To explore Pythagorean triples we started by creating a spreadsheet that would show us Pythagorean triples. This enables us to use a large number of triples to explore different ratios in the ttriangles. From the ratios we saw interesting results and then looked at GSP to try and make sense of these things.
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 does show us Pythagorean triples but to get many of them, the spreadsheet would need to be very large. From the triples we had generated we looked at several ratios. The ratio of area to the perimeter of the triangles, the ratio of the legs and the ratios of the legs to the hypotenuse of each triangle. Using the ratios we created a line graph to observe the results.
From this it appears that there are some very interesting things happening in these ratios. It seems that the ratio of the legs and the ratio of the legs to the hypotenuse is staying between zero and one. From looking at the spreadsheet we see that the ratio of the legs keeps returning to 0.75. The ratio of the area to the perimeter is constantly rising and it is always an integer or some half. BUT this spreadsheet only has nineteen triples in it.
To draw conclusions about the triples we need to generate a very large number of triples. We did this by creating a spreadsheet using Euclid's formula for creating 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 have a way of creating hundreds and hundreds of triples we added the same ratios from the previous spreadsheet to see if the same results occured.
By graphing the results we see that the ratio of the legs to the hypotenuse of each triangle converge to zero and one. On the graph there is 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 a little ( the point of intersections will change) but eventually the two ratios will converge to one and zero.
By graphing the results of the ratios of the legs as we vary m and n we see that the ratio is a constant decline that approaches zero. Of course it will not reach zero because then one leg of your triangle will not exist.
From looking at the ratio of the area to the perimeter on the first spreadsheet will thought that it would always be an integer or some half. We see that still holds true no matter what numbers we put in for m and n.
By looking at GSP we can make some sense of these things that appear to be happening.
We start by constructing a right triangle on a line so we can animate one pointalong the line. From this we can measure the legs and our ratios to see theresults.
As we animate the point C along the line we see the ratios of the legs does approach zero. As segmet AC gets longer the ratio of the two segments will get smaller and smaller. It will not reach zero because then the triangle would not exist.
When looking at the ratios of the legs to the hypotenuse our spreadsheet findings still hold. As segment BC gets longer segment AC also gets longer and their ratio goes toward one. As the hypotenuse AC gets longer the ratio with the leg AB goes to zero because you are dividing the length of AC by a larger and larger number. This give explaination to the findings from the spreadsheet.
By using the spreadsheet and GSP we can make a mathematical connections between the calculations and the geometry. For high school students this is a monumental event. By using the GSP and animation we can see meaning for the results on the spreadsheet. This could be used as a demonstration for high school students or as a discovery lesson they could complete themselves.