1. How to draw a right isosceles triangle

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How to draw a right isosceles triangle



OK, so now we're gonna consider how to draw an isosceles right triangle using the turtle. This triangle is isosceles because these two sides are the same length, and it's a right triangle because we have a right angle right here.

Let's put our turtle right over here, and for now I'm not gonna draw anything on the screen, I just want to see if I can get the turtle to follow the path of this edge.

So, um I happen to know that this particular triangle is 100 steps on this side, and 100 on this side. So, the start is easy, I can just say forward 100, and that takes me up to the corner.

Now I want to figure out how much of a turn I have to make here. Well, let's consider for a minute how we make a right angle isosceles triangle.

If we look at this square, we see that this triangle is made by starting with a square and drawing the diagonal. Well, this diagonal cuts this angle right in half, doesn't it?

We know that each corner of the square is 90 degrees. If we cut that angle in half, we have 45 degrees, right here.

So we know that this angle is 45 degrees. How much of a turn do we have to make? How much of a turn does the turtle have to make?

Let's look here at the turn, OK? Now in general, when we're turn a corner, you see we're still facing in this direction, this, this entire angle, if we take the blue corner of the triangle together with this little sector is 180 degrees. And we know that this little blue part here, this little angle is 45 degrees. So the total amount of turn, the angle in this whole sector here is 180 minus 45, or 135 degrees.

So let's turn right 135 degrees and see how that looks. OK, that looks about right. Now we have to figure out the length of this line right here, and that's a little bit trickier. Um, let's consider for a minute, what we're gonna have to look at is, well we're gonna start with two squares here, and think for a minute about the area of squares, and then it'll be clear in a minute why we have to worry about the areas. But, we said this is 100 steps, actually let's make up our own unit, we'll say this is one hectostep, OK, a hectometer is 100 meters, a hectostep is 100 steps. So this is one hectostep, so this square is one by one.

So if each side is 1, then the area as a whole is 1 hectostep. And these two squares make up two square hectosteps.

Now let's rearrange these pieces for a minute.

OK, we know the area of this square is two hectosteps, because the area of this is one hectostep, the area of this square is one hectostep squared, so this has to be two hectosteps squared.

So what does that mean about the length of this line? The length of this line times the length of this line is 2 hectosteps, So that means that one of these lines is the square root of two hectosteps. And since a hectostep is 100 steps, that means we want to go forward 100 times the square root of 2.

So if we watch this turtle when I hit return and we go forward, we should go all the way to the other corner. Let's see.

Yes, that worked, OK?

So now, let's see. Let's put our original triangle back in place and now we want to figure out how much of a turn we need to make on this side. Remember we turned 135 degrees here.

Let's take a look at that turn. Well, this is also 45 degrees, so the turn we need to make here is also 180 minus 45, or 135 degrees.

Let's go right 135. That looks right. And now, we knew that this was 100 steps. When we went from here to here it was 100 steps, and this is also 100 steps.

And finally, just to get ourselves back to our initial state, to our starting position, we'll go right 90. And that draws it.

So let's just double check. We'll move this guy down here, and put the pen down. Now I'll just back up and walk through all the steps we did before. We went forward 100, right 135, forward 100 times the square root of 2, right 135, forward 100, and right 90.

And that's how you make an isosceles right triangle.