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.

You can see that I ended up with a tan triangle on the left and a plot relating the two remaining side lengths. The following videos step you through the process of creating the sketch and using it to investigate some very interesting questions about the boundary of the region on the right, isosceles triangles, right triangles, and maximal areas.

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

Investigating Area

## 2 comments:

I thought it would be really hard to come up with the equation of the U-shaped curve of all the (x,y) pairs creating a triangle with a given area - like 5 or 1 or 0. It is actually a pretty neat exercise.

I have created a Web Sketchpad version of this in the next post.

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