The triangle 𝐴𝐵𝐶 has been transformed onto triangle 𝐴 prime 𝐵 prime 𝐶 prime, which has been transformed onto triangle 𝐴 double prime 𝐵 double prime 𝐶 double prime. Question 1) Describe the single transformation that maps 𝐴𝐵𝐶 onto 𝐴 prime 𝐵 prime 𝐶 prime. Question 2) Describe the single transformation that maps 𝐴 prime 𝐵 prime 𝐶 prime onto 𝐴 double prime 𝐵 double prime 𝐶 double prime. Question 3) Hence, are triangles 𝐴𝐵𝐶 and 𝐴 double prime 𝐵 double prime 𝐶 double prime similar?
Let’s look at question one first. Let’s start by addressing the notation 𝐴𝐵𝐶 goes to 𝐴 prime 𝐵 prime 𝐶 prime or 𝐴 dash 𝐵 dash 𝐶 dash. In this case, the 𝐴 prime 𝐵 prime 𝐶 prime refers to the new points of 𝐴, 𝐵, and 𝐶. In the same way, 𝐴 double prime, 𝐵 double prime, and 𝐶 double prime are the new points of 𝐴 prime 𝐵 prime 𝐶 prime or the second new points of 𝐴, 𝐵, and 𝐶. In this question, we’re talking about transformations. And there are four main transformations.
The first type of transformation is a translation. A translation move the shape left or right and up and down. In a translation, the shape will stay in the same orientation. So it won’t flip or it won’t spin. The second type of transformation is a rotation. In a rotation, the shape will turn or spin about a point. To describe a rotation, we need to specify the angle, the direction, for example, clockwise or anticlockwise, and the point of rotation. That is the point about which the shape will rotate. The third type of transformation is a reflection. In a reflection, the reflected shape will look like a mirror image of the original shape. To describe a reflection, we need to also note the line of reflection. The line of reflection would be like a line of symmetry between the old shape and the new reflected shape.
The final transformation is a dilation, which will be an enlargement or reduction of the shape. To describe a dilation, we need to know the scale factor and the point of dilation. And the scale factor of the dilation if, for example, we have a scale factor of three, that would mean that the shape was three times bigger. If we had a scale factor of a half, the new shape would be half as big, meaning it was smaller and would be the same as reducing it. So let’s now go ahead and see if we can investigate the transformation from 𝐴𝐵𝐶 onto 𝐴 prime 𝐵 prime 𝐶 prime.
If we look at our two triangles, we can see that they are in a different orientation. So that means it’s not a translation. We can also see that 𝐴 prime 𝐵 prime 𝐶 prime is not a rotation of 𝐴𝐵𝐶. And that’s because if we rotated 𝐴𝐵𝐶, then on our new shape, 𝐴 prime would be on the right-hand side of 𝐵 prime. In this case, it does look like 𝐴𝐵𝐶 and 𝐴 prime 𝐵 prime 𝐶 prime are a reflection. Since it looks like if we held up a mirror between them, we would see the image on the other side. Let’s see if we can establish where the line of reflection would be.
Let’s start by drawing a line between 𝐴 and 𝐴 prime. We can see that there are eight squares between 𝐴 and 𝐴 prime. So halfway between them would be four squares along. Next, we can draw a line between 𝐶 and 𝐶 prime and see that there are two squares between those. So halfway between them would be one square away from each. Between 𝐵 and 𝐵 prime, there are six squares. So halfway would be the same point three squares away from the line. To find the line of reflection then, we draw a line that will go through the points that we have just drawn. We can see that this line also fits with line segment 𝐷𝐸 that’s already labeled.
So we can describe our transformation by saying that 𝐴𝐵𝐶 maps to 𝐴 prime 𝐵 prime 𝐶 prime by reflection in line segment 𝐷𝐸. So the answer is a reflection in line segment 𝐷𝐸.
Let’s now look at question two.
Describe the single transformation that maps 𝐴 prime 𝐵 prime 𝐶 prime onto 𝐴 double prime 𝐵 double prime 𝐶 double prime.
So here, we’re looking at how the second triangle maps onto the third triangle. Since the triangle has got larger, we can eliminate the first three transformations. Since in translation, rotation, and reflection, the shape always stays the same size. Let’s see if we can establish the scale factor that is how many times bigger is 𝐴 double prime 𝐵 double prime 𝐶 double prime than 𝐴 prime 𝐵 prime 𝐶 prime. Let’s start by taking a look at one of the lengths.
In the length 𝐵 prime to 𝐶 prime, we can see that it was two squares up. The corresponding length on our new triangle would be between 𝐵 double prime and 𝐶 double prime. And here, that’s four squares high. So it looks like it’s two times larger. It’s always worth having a look at another set of corresponding lengths just to be sure. We can see that between 𝐴 prime and 𝐵 prime, it goes one square across and one square up. Between the corresponding points 𝐴 double prime and 𝐵 double prime, it’s two squares along and two squares up, which confirms it will be two times bigger, which means that our scale factor is two.
To find the point of dilation, we can start by joining the vertex on the old triangle with its new corresponding vertex on the new triangle. So, for example, here we have drawn a line between 𝐶 prime and 𝐶 double prime. Next, we can draw a line between 𝐵 prime and 𝐵 double prime. And then, we can draw a line between 𝐴 prime and 𝐴 double prime. We noticed that there’s a point where these three lines converge. At this point is the point of dilation. Here, it’s labeled 𝐸. So now, we know that our dilation has a scale factor under point of dilation. And we just need to put that into a single statement. We can say that 𝐴 prime 𝐵 prime 𝐶 prime goes to 𝐴 double prime 𝐵 double prime 𝐶 double prime by a dilation from the point 𝐸 by a scale factor of two.
Let’s look at the final question.
Hence, are triangles 𝐴𝐵𝐶 and 𝐴 double prime 𝐵 double prime 𝐶 double prime similar?
So let’s have a look at the transformation journey between these triangles. We know that our first triangle 𝐴𝐵𝐶 was transformed to 𝐴 prime 𝐵 prime 𝐶 prime by a reflection. This means that triangle 𝐴 prime 𝐵 prime 𝐶 prime is the same shape and the same size as 𝐴𝐵𝐶. The mathematical word for this would be “congruent.” So we know that 𝐴𝐵𝐶 and 𝐴 prime 𝐵 prime 𝐶 prime are congruent. The transformation which takes 𝐴 prime 𝐵 prime 𝐶 prime to 𝐴 double prime 𝐵 double prime 𝐶 double prime was a dilation of scale factor two, meaning that this triangle is two times larger than triangle 𝐴𝐵𝐶.
However, this triangle is still the same shape. That is, all the angles are the same. The mathematical word we use for shapes that are the same shape but different sizes is “similar.” So triangle 𝐴 double prime 𝐵 double prime 𝐶 double prime is similar to 𝐴 prime 𝐵 prime 𝐶 prime. And therefore, it must also be similar to triangle 𝐴𝐵𝐶 since those two triangles are congruent. So our final answer is yes, since triangles 𝐴𝐵𝐶 and 𝐴 double prime 𝐵 double prime 𝐶 double prime are similar.