They said they couldn't tell - that we needed to know the side lengths. I showed them the side lengths and moved the vertices around to create different types of triangles. I then asked how they could tell if they only had the coordinates of the vertices. A little brainstorming in their groups led them to say that they could find the distance between points. Bingo! There were still some confused looks so I picked some random points and they told me how to find the length of that segment (see shot of whiteboard, below).
Then I asked how we could tell it was a right triangle. Measure the angles, of course! When I asked if they could really tell the difference between an 89° angle and a 90° angle when measuring with a protractor, all but one student said no. More brainstorming followed and they came up with the strategy of finding the lengths of the three sides and seeing if the Pythagorean theorem holds true. I made my triangle in GeoGebra have a right angle and asked if that strategy worked. It looked something like this:
It was very close to working, but not exact because the side lengths had been rounded. If students had used the distance formula to find the side lengths and kept the lengths in exact form, then they could definitely show whether there was a right angle. But what if they didn't want to calculate all the side lengths? How could they show that two lines crossed at a right angle? I added lines in GeoGebra to help clarify what I was asking but that didn't seem to help. So I opened Desmos and entered two equations in standard form (which I then hid) and asked them to talk in their groups again.
As I circulated and asked what they had figured out, groups slowly began saying words like "perpendicular" and "slope" and "negative reciprocal". I showed them my equations which I rearranged into slope-intercept form and they could see that one had a slope of 2/3 while the other had a slope of -3/2. Okay - back to our triangles... we can find the slope of each line segment and compare them to see if there are any negative reciprocals to determine if it is a right triangle. This was the work on the whiteboard along the way:
We also talked about whether you could use more than one descriptor for a triangle. Can you have an equilateral right triangle? An isosceles right triangle? How many are possible? They seemed to indicate that they now understood what to do for homework set 16 so I told them that they would get a second try at it tonight (no new homework).
Our little (ha!) aside taken care of, we turned our attention back to similar triangles. I had students do a little recap of yesterday's work to help those who had missed yesterday's class. Then they worked through two questions that asked whether the triangles were similar:
And then two questions where they had to find missing information:
They had a little time to start working on this similar triangle handout, which we will continue tomorrow before jumping into trig.