For some reason, I woke up this morning thinking about triangles. Particularly triangles with longest side 10 units. I thought that Sketchpad might be an interesting way to construct said triangles and investigate relationships between the lengths of the other two sides.
In Ontario, students in the Grade 9 Applied Level are expected to do investigations like this, although they start with rectangles - which seems more complicated. They are expected to investigate figures with maximal area as well as 3D shapes.
You can download the sketch, but it is more fun to create it yourself. I have captured my investigation in case it helps in the 12 videos below. If you find that you have trouble motivating yourself to watch 12 fascinating videos, you could just watch the last one to get a sense of where the investigation ends up.
(You can click on the title to get the video in a new tab)
Constructing the Triangle
Constructing the Point representing Side Lengths
Constructing the x and y segments
Tracing the Side Lengths
Investigating the Region of Possible Side Lengths
Investigating the Boundaries of the Region
Reasoning about the Equations of the Boundaries
Investigating Isosceles Triangles
Investigating More Isosceles Triangles
Investigating Right Triangles
Triangle in a Circle