Question Video: Identifying Types of Similarity Transformation | Nagwa Question Video: Identifying Types of Similarity Transformation | Nagwa

Question Video: Identifying Types of Similarity Transformation Mathematics

A triangle has vertices at the points 𝐴(1, 1), 𝐵(3, 1), and 𝐶(1, 3). The image of this triangle under a transformation has vertices at the points 𝐴′(−1, −1), 𝐵′(−1, −3), and 𝐶′(−3, −1). Which transformation has taken place?

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Video Transcript

A triangle has vertices at the points 𝐴 one, one; 𝐵 three, one; and 𝐶 one, three. The image of this triangle under a transformation has vertices at the points 𝐴 prime which is negative one, negative one, 𝐵 prime which is negative one, negative three, and 𝐶 prime which is negative three, negative one. Which transformation has taken place?

So there are a couple of ways that we can answer this. One way is to sketch a diagram, so let’s do that. We need a simple set of coordinate axes. And we’re going to plot each point in turn. Point 𝐴 has coordinates one, one. So that’s here. Point 𝐵 lies at three, one which is here. And point 𝐶 is at one, three which is here. And so in fact, triangle 𝐴𝐵𝐶 is a right isosceles triangle. Let’s now plot the image of the triangle. We’re given the points 𝐴 prime which is negative one, negative one; that’s here. And so point 𝐴 is mapped onto 𝐴 prime. The vertex 𝐵 prime is negative one, negative three, which is here. So 𝐵 maps onto 𝐵 prime. And finally, we have 𝐶 prime, which is over here.

And so we have another right isosceles triangle in this quadrant here. Our job is to identify the transformation that’s taken place. So let’s list the transformations we know. The four similarity transformations that we know are reflections. The fl reminds us that we flip a shape in a mirror line. We have rotations where we turn the shape about a center. We then have translations, and the sl reminds us that this means we slide the shape, left or right and up or down. And finally, we have dilations or enlargements. The l reminds us that the dilation makes a shape bigger or smaller.

And so we can instantly disregard one of our transformations. We can see that triangle 𝐴𝐵𝐶 and its image are the exact same size. They’re just in different orientations. And so we don’t have a dilation. Could we have performed translation? Could we have slid the original shape? Well, no. When we slide the shape, we don’t actually change its orientation, just its position on the coordinate grid. And so we have two options. We have a reflection or a rotation. And at first glance, it might look like it could be either one of these. And so the position of each vertex is really important.

When we rotate the shape, we turn it. And so we would assume that the shape might have been rotated 180 degrees. But this can’t be the case. If that was to happen, the shape would end up essentially being upside down. And so vertex 𝐶 prime would actually be vertex 𝐵 prime. And so we disregard rotations. And we must be reflecting a shape. Let’s double check by identifying the location of the mirror line. It looks like simply by observation that the mirror line might be the line 𝑦 equals negative 𝑥. But let’s check.

We’re going to measure the perpendicular distance of each vertex on 𝐴𝐵𝐶 from the mirror line. The perpendicular distance from point 𝐴 to the mirror line is one diagonal of a square. The image of 𝐴 should be the exact same distance on the other side of the line. Well, it is; it’s one diagonal. Similarly, if we consider point 𝐵, that’s two diagonals away from the line. And the perpendicular distance of the image of 𝐵, 𝐵 prime, is the same but on the other side. We can repeat this process with point 𝐶. And we see that, in fact, 𝐴𝐵𝐶 has been reflected, and it’s been reflected in the line 𝑦 equals negative 𝑥. And so the answer here is that a reflection has taken place.

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