Video Transcript
Does there exist a series of similarity transformations that would map triangle ๐ด๐ต๐ถ to triangle ๐ธ๐น๐ท? If yes, explain your answer.
Firstly, letโs recall what we mean by the term similarity transformation. A similarity transformation transforms an object in space to a similar object. And, in fact, really, a similarity transformation is just one of the four key transformations that we use. A translation, rotation, reflection, or a dilation will all map an object onto a similar or even congruent object. So, letโs ask ourselves what series of transformations would map triangle ๐ด๐ต๐ถ, thatโs the smaller one, onto ๐ธ๐น๐ท.
Well, firstly, we just said that triangle ๐ด๐ต๐ถ is smaller than ๐ธ๐น๐ท. And so, the first thing that we could do is dilate or enlarge triangle ๐ด๐ต๐ถ. To dilate or enlarge a shape, we need to identify a scale factor for enlargement. And to find this, we divide a length on the new shape by the corresponding length on the old shape. If we take side ๐ท๐ธ on the new shape, we see that the corresponding length on the old shape is length ๐ด๐ถ. ๐ท๐ธ is three units in length, and ๐ด๐ถ is one unit in length. And so, the scale factor here must be three divided by one, which is simply three. So, we could dilate the shape by a scale factor of three. That would certainly achieve the right size. But what else would we need to do?
Now, we havenโt defined a center of dilation or a center of enlargement, and it doesnโt really matter. So, letโs just enlarge ๐ด๐ต๐ถ onto its image ๐ด prime ๐ต prime ๐ถ prime, as shown. So, what are we going to need to do next? Well, letโs consider a rotation. Now, if we rotate this shape, say 90 degrees, in a counterclockwise direction, our shape will still be in the wrong orientation. Essentially, if we perform this rotation, itโs going to be upside down. So, what else do we need to do? Well, we need to essentially flip the shape to get from ๐ด double prime ๐ต double prime ๐ถ double prime onto triangle ๐ท๐ธ๐น or ๐ธ๐น๐ท. And another word for that, in fact, a mathematical word is to reflect the shape.
And so, in fact, we can perform a series of similarity transformations that map ๐ด๐ต๐ถ to ๐ธ๐น๐ท. And so, the answer is yes, triangle ๐ด๐ต๐ถ would need to be dilated by a scale factor of three, rotated, and then reflected. And of course, we can do these in any order.
