Question Video: Determining Whether Two Triangles Are Similar Using Transformations Mathematics • 8th Grade

Does there exist a series of similarity transformations that would map triangle ๐ด๐ต๐ถ to triangle ๐ธ๐น๐ท? If yes, explain your answer.

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Video Transcript

Does there exist a series of similarity transformations that would map triangle ๐ด๐ต๐ถ to triangle ๐ธ๐น๐ท? If yes, explain your answer.

Firstly, letโ€™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, letโ€™s ask ourselves what series of transformations would map triangle ๐ด๐ต๐ถ, thatโ€™s the smaller one, onto ๐ธ๐น๐ท.

Well, firstly, we just said that triangle ๐ด๐ต๐ถ is smaller than ๐ธ๐น๐ท. And so, the first thing that we could do is dilate or enlarge triangle ๐ด๐ต๐ถ. 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 ๐ท๐ธ on the new shape, we see that the corresponding length on the old shape is length ๐ด๐ถ. ๐ท๐ธ is three units in length, and ๐ด๐ถ 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 havenโ€™t defined a center of dilation or a center of enlargement, and it doesnโ€™t really matter. So, letโ€™s just enlarge ๐ด๐ต๐ถ onto its image ๐ด prime ๐ต prime ๐ถ prime, as shown. So, what are we going to need to do next? Well, letโ€™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, itโ€™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 ๐ด double prime ๐ต double prime ๐ถ double prime onto triangle ๐ท๐ธ๐น or ๐ธ๐น๐ท. 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 ๐ด๐ต๐ถ to ๐ธ๐น๐ท. And so, the answer is yes, triangle ๐ด๐ต๐ถ 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.

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