I appreciate all the responses and I feel like I have failed to communicate exactly what my issue is.
Somewhere along this math journey I have absorbed that there is a difference between enlarging an object centered at 0,0 like the unit circle for example and a circle sitting somewhere in a quadrant somewhere. In terms of the effect this has on the enlarging, Im not 100% clear on.
Anyway, I got the idea that for me to successfully complete this question I had to move the triangle first to the center somehow (as in its sitting exactly in the middle of the grid.) and then enlarge it. I have zero idea how you would do that.
So I feel at this point my assumption that I had to do this is completely wrong.
I emailed my teacher about this question, saying what I have said here and he basically said I have been taught how to transform the triangle. ie move it. and that's all he said which basically clarified nothing because essentially I wanted to know if that's what I was expected to do and how would I go about it.
Anyway I have reached a point that Im just going to enlarge it and rotate it where it is and that will be my answer.
I appreciate all the help and I feel its my teacher that is not helping clarify what I want to know which I would feel is essential to the learning process.
You're right that you made a wrong assumption. As I've said previously,
the only "center" that matters in the assignment is the "center of dilation". The enlargement you have to do (which the matrix automatically does for you) multiplies every distance from
that center by a fixed amount. Where the "center" of the object is doesn't matter.
What probably lies behind your thinking is the fact that if the object being moved has a "center" (not all objects have a single well-defined center; the "center" of a triangle can be defined in many different ways), that
center will be moved in the same was as anything else. Most people tend to naturally think of enlargement of an object as being
relative to its own center -- like a ball being inflated or a child growing.
The key idea you need to pick up, perhaps, is that the enlargement being taught is not really enlargement
of a particular object; it is a
transformation of the entire plane! The object is just caught up in this overall growth, like a boat in a current; so the dilation will not only make it bigger, it will also move it away from the center. Once you start seeing it this way, everything you have been taught should make sense. It's possible that your teacher failed to explain this; it's also very possible that he did, but it didn't make an impression on you because you can't help thinking of it in the object-centered way. I would expect your textbook to have some pictures and explanations that might say all this. Perhaps a site like
this can help; the first picture shows a dilation of a triangle that is away from the origin, and is moved further away. A later picture shows a triangle that surrounds the origin, and doesn't so clearly show that it is also moved.
Now, if you quoted the problem accurately as "
Enlarge triangle ABC by a factor of 2", then that is an unfortunate wording, because it does focus your attention on the triangle, as if you were to only enlarge that, and not do anything else. Hopefully, the context was written to make it clear that what they want you to do is to apply a matrix transformation
of the plane that will enlarge the triangle (and also move every point except the origin!).
The important thing is that you let go of your misunderstanding of the goal, and see what is actually being taught. That may take some time yet, as you get used to a new way of seeing things.
