Does there exist a series of similarity transformations that would map triangle 𝐴𝐵𝐶 to triangle 𝐸𝐹𝐷? If yes, explain your answer.
Firstly, let’s 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, let’s ask ourselves what series of transformations would map triangle 𝐴𝐵𝐶, that’s the smaller one, onto 𝐸𝐹𝐷.
Well, firstly, we just said that triangle 𝐴𝐵𝐶 is smaller than 𝐸𝐹𝐷. And so, the first thing that we could do is dilate or enlarge triangle 𝐴𝐵𝐶. 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 𝐷𝐸 on the new shape, we see that the corresponding length on the old shape is length 𝐴𝐶. 𝐷𝐸 is three units in length, and 𝐴𝐶 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 haven’t defined a center of dilation or a center of enlargement, and it doesn’t really matter. So, let’s just enlarge 𝐴𝐵𝐶 onto its image 𝐴 prime 𝐵 prime 𝐶 prime, as shown. So, what are we going to need to do next? Well, let’s 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, it’s going to be upside down. So, what else do we need to do? Well, we need to essentially flip the shape to get from 𝐴 double prime 𝐵 double prime 𝐶 double prime onto triangle 𝐷𝐸𝐹 or 𝐸𝐹𝐷. 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 𝐴𝐵𝐶 to 𝐸𝐹𝐷. And so, the answer is yes, triangle 𝐴𝐵𝐶 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.