This GeoGebra model can be used to find right triangles where the length of the legs and the hypotenuse all have integer lengths. A 3-4-5 right triangle is the most used such right triangle, but there are 9 where the length of one leg is less than or equal to 15 and the length of the other leg is less than or equal to 26 having an integer hypotenuse.
The two sliders can be used to change the lengths of the two legs of the right triangle. See if you can devise a method to easily find all 9 of these special right triangles?
Some of these triangles are similar, i.e., their angles are the same. See if you can organize these 9 triangles into groups of similar triangles. How are the sides of these triangles related?
See the YouTube video http://youtu.be/jQtat4ucRG4 for more information.

Using the two sliders, find the 9 right triangles that can be built whose sides and hypotenuse have integer lengths.
Organize these 9 into groups of similar triangles (same angles) and explore the relationships between their side lengths.