It looks like some people have memorized a ratio, some people know it’s Pythagorean, and some people think it’s isosceles, along with one or two mistakes that I can’t really figure out like that last one (though it looks close to dividing by sqrt(3) instead of multiplying?

I think I would go back to constructing equilateral triangles (with straightedge and compass, probably) and then cutting them in half and measuring the lengths and angles, and see where that got me.

I’d also give some other triangles whose lengths can be determined by drawing an altitude and using Pythag once or twice, to reinforce that part of the idea.

Not sure how I’d get people to start understanding similarity and thinking in terms of ratios, though. That’s probably the hardest part in all of this. A bunch of 3-4-5 triangles and so on would probably help with that. I also enjoy giving the legs as 1/3 and 1/4 and seeing how many people want to make the hypotenuse 1/5. When they see why it’s 5/12, that may be one step closer to understanding fractions as well as ratio and similarity.

## One reply on “30/60/90 Mistakes”

It looks like some people have memorized a ratio, some people know it’s Pythagorean, and some people think it’s isosceles, along with one or two mistakes that I can’t really figure out like that last one (though it looks close to dividing by sqrt(3) instead of multiplying?

I think I would go back to constructing equilateral triangles (with straightedge and compass, probably) and then cutting them in half and measuring the lengths and angles, and see where that got me.

I’d also give some other triangles whose lengths can be determined by drawing an altitude and using Pythag once or twice, to reinforce that part of the idea.

Not sure how I’d get people to start understanding similarity and thinking in terms of ratios, though. That’s probably the hardest part in all of this. A bunch of 3-4-5 triangles and so on would probably help with that. I also enjoy giving the legs as 1/3 and 1/4 and seeing how many people want to make the hypotenuse 1/5. When they see why it’s 5/12, that may be one step closer to understanding fractions as well as ratio and similarity.