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Does there exist a series of similarity transformations that would map triangle 𝐴𝐵𝐶 to triangle 𝐸𝐹𝐷?

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If yes, explain your answer.

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Firstly, let’s recall what we mean by the term similarity transformation.

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A similarity transformation transforms an object in space to a similar object.

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And, in fact, really, a similarity transformation is just one of the four key transformations that we use.

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A translation, rotation, reflection, or a dilation will all map an object onto a similar or even congruent object.

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So, let’s ask ourselves what series of transformations would map triangle 𝐴𝐵𝐶, that’s the smaller one, onto 𝐸𝐹𝐷.

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Well, firstly, we just said that triangle 𝐴𝐵𝐶 is smaller than 𝐸𝐹𝐷.

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And so, the first thing that we could do is dilate or enlarge triangle 𝐴𝐵𝐶.

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To dilate or enlarge a shape, we need to identify a scale factor for enlargement.

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And to find this, we divide a length on the new shape by the corresponding length on the old shape.

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If we take side 𝐷𝐸 on the new shape, we see that the corresponding length on the old shape is length 𝐴𝐶.

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𝐷𝐸 is three units in length, and 𝐴𝐶 is one unit in length.

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And so, the scale factor here must be three divided by one, which is simply three.

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So, we could dilate the shape by a scale factor of three.

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That would certainly achieve the right size.

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But what else would we need to do?

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Now, we haven’t defined a center of dilation or a center of enlargement, and it doesn’t really matter.

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So, let’s just enlarge 𝐴𝐵𝐶 onto its image 𝐴 prime 𝐵 prime 𝐶 prime, as shown.

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So, what are we going to need to do next?

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Well, let’s consider a rotation.

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Now, if we rotate this shape, say 90 degrees, in a counterclockwise direction, our shape will still be in the wrong orientation.

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Essentially, if we perform this rotation, it’s going to be upside down.

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So, what else do we need to do?

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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.

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And another word for that, in fact, a mathematical word is to reflect the shape.

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And so, in fact, we can perform a series of similarity transformations that map 𝐴𝐵𝐶 to 𝐸𝐹𝐷.

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And so, the answer is yes, triangle 𝐴𝐵𝐶 would need to be dilated by a scale factor of three, rotated, and then reflected.

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And of course, we can do these in any order.