Question Video: Using Transformations to Determine Similarity Mathematics • 8th Grade

The triangle 𝐴𝐡𝐢 has been transformed onto triangle 𝐴′𝐡′𝐢′ which has been transformed onto triangle 𝐴′′𝐡′′𝐢′′. 1) Describe the single transformation that maps 𝐴𝐡𝐢 to 𝐴′𝐡′𝐢′. 2) Describe the single transformation that maps 𝐴′𝐡′𝐢′ to 𝐴′′𝐡′′𝐢′′. 3) Hence, are triangles 𝐴𝐡𝐢 and 𝐴′′𝐡′′𝐢′′ similar?

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Video Transcript

The triangle 𝐴𝐡𝐢 has been transformed onto triangle 𝐴 prime 𝐡 prime 𝐢 prime which has been transformed onto triangle 𝐴 double prime 𝐡 double prime 𝐢 double prime. Describe the single transformation that maps 𝐴𝐡𝐢 to 𝐴 prime 𝐡 prime 𝐢 prime. Describe the single transformation that maps 𝐴 prime 𝐡 prime 𝐢 prime to 𝐴 double prime 𝐡 double prime 𝐢 double prime. Hence, are triangles 𝐴𝐡𝐢 and 𝐴 double prime 𝐡 double prime 𝐢 double prime similar?

In this question, we have a series or combinations of transformations which begin with triangle 𝐴𝐡𝐢. The first transformation takes us to the second smaller triangle. And the second transformation takes us to the larger triangle of 𝐴 double prime 𝐡 double prime 𝐢 double prime. We can begin by finding the first transformation between the two smaller triangles.

We can recall that the four types of transformation are translation, reflection, rotation, and dilation. If we look at the two triangles 𝐴𝐡𝐢 and 𝐴 prime 𝐡 prime 𝐢 prime, we can see that they’re the same size. Therefore, this is unlikely to be a dilation, as this usually changes the size. We can see that our two triangles are at different orientation. So, we can rule out translation as this moves the shape but keeps it the same way up.

The two triangles are not a mirror image of each other. So, we can rule out reflection. Let’s see if we could describe this transformation as a rotation. Starting with triangle 𝐴𝐡𝐢, if we rotated this in a clockwise direction, we could then work out the angles for which this must be rotated. Between 𝐴 and 𝐴 prime, there’s a right angle of 90 degrees. Between 𝐢 and 𝐢 prime, we can also see a 90-degree angle. This will confirm that we have a rotation of 90 degrees.

Notice that we’ve found this rotation by moving our vertices through the same point or coordinate. This will be the center of rotation. So, in order to fully describe this transformation, we need to put together the facts that we’ve discovered β€” the center of rotation, the angle, and the direction. We can then give our answer to the first part as a 90-degrees clockwise rotation about the origin. We could, of course, also have described this as a 270-degree counterclockwise rotation about the origin. Giving the coordinate zero, zero instead of the origin would also have been valid.

Let’s look at the second question. Describe the single transformation that maps 𝐴 prime 𝐡 prime 𝐢 prime to 𝐴 double prime 𝐡 double prime 𝐢 double prime. We need to be careful that we’re using the second smaller triangle and asking how we go from this to the larger triangle. If we look at our list of possible transformations, the first three β€” translation, reflection, and rotation β€” keep the object and its image the same size. As the triangles here are different sizes, that means there’s just one possible transformation, dilation.

To describe a dilation, we need to find the center of dilation and the scale factor. We can find the scale factor relatively easily by looking at how the length on the image have increased from the length on the original shape. We can compare the lengths of 𝐴 double prime 𝐡 double prime and 𝐴 prime 𝐡 prime. We can see that on 𝐴 prime 𝐡 prime, the length is two units, and on the top length of 𝐴 double prime 𝐡 double prime, this is four units long. So, it looks like we’ll have a scale factor of two. But it’s always worth checking some of the lengths on the other sides, just to be sure. The length 𝐡 prime 𝐢 prime is three units long and the length 𝐡 double prime 𝐢 double prime is six units long. And as that’s twice as large, then we’ve confirmed that the scale factor is two.

There’s a nice, easy way to find the center of dilation. To do this, we create a ray between each vertex and its image. Here, we have a ray between 𝐡 prime and 𝐡 double prime and 𝐴 prime and 𝐴 double prime. We can do the same between 𝐢 prime and 𝐢 double prime. And the place where the rays converge will be the center of dilation, which once again will be the origin or the coordinate zero, zero. We put our answer into this statement form that this will be a dilation from the origin by a scale factor of two.

Our final question asks if our triangles 𝐴𝐡𝐢 and 𝐴 double prime 𝐡 double prime 𝐢 double prime are similar. We can recall that similar means the same shape, but a different size. The angles remain the same, but the sides will be in proportion. So, in our first transformation from 𝐴𝐡𝐢 to 𝐴 prime 𝐡 prime 𝐢 prime, we didn’t change the size of these triangles, which means that they are congruent. And when we transformed 𝐴 prime 𝐡 prime 𝐢 prime to 𝐴 double prime 𝐡 double prime 𝐢 double prime, the image here did get larger. So, these two triangles would not be congruent.

Each of the lengths in the image of 𝐴 double prime 𝐡 double prime 𝐢 double prime was in proportion to those in the triangles of 𝐴 prime 𝐡 prime 𝐢 prime. All corresponding pairs of angles are congruent. So, our triangles 𝐴𝐡𝐢 and 𝐴 double prime 𝐡 double prime 𝐢 double prime are similar. So, our answer for the final part of this question is: yes, these triangles are similar.

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