Video: Using Transformations to Determine Congruence

The triangle 𝐴𝐡𝐢 has been transformed onto triangle 𝐴′𝐡′𝐢′, which has then been transformed onto triangle 𝐴″𝐡″𝐢″, then transformed onto 𝐴‴𝐡‴𝐢‴ as seen in the figure. Describe the single transformation that would map 𝐴𝐡𝐢 onto 𝐴′𝐡′𝐢′. Describe the single transformation that would map 𝐴′𝐡′𝐢′ onto 𝐴″𝐡″𝐢″. Describe the single transformation that would map 𝐴″𝐡″𝐢″ onto 𝐴‴𝐡‴𝐢‴. Hence, are triangles 𝐴𝐡𝐢 and 𝐴‴𝐡‴𝐢‴ congruent?

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Video Transcript

The triangle 𝐴𝐡𝐢 has been transformed onto triangle 𝐴 prime 𝐡 prime 𝐢 prime, which has then been transformed onto triangle 𝐴 double prime 𝐡 double prime 𝐢 double prime, then transformed onto 𝐴 triple prime 𝐡 triple prime 𝐢 triple prime as seen in the figure. Describe the single transformation that would map 𝐴𝐡𝐢 onto 𝐴 prime 𝐡 prime 𝐢 prime. Describe the single transformation that would map 𝐴 prime 𝐡 prime 𝐢 prime onto 𝐴 double prime 𝐡 double prime 𝐢 double prime. Describe the single transformation that would map 𝐴 double prime 𝐡 double prime 𝐢 double prime onto 𝐴 triple prime 𝐡 triple prime 𝐢 triple prime. Hence, are triangles 𝐴𝐡𝐢 and 𝐴 triple prime 𝐡 triple prime 𝐢 triple prime congruent?

There’s quite a lot of information going on here. So let’s begin by simply describing the transformation that maps 𝐴𝐡𝐢 onto 𝐴 prime 𝐡 prime 𝐢 prime. That’s this triangle here onto this triangle here. There are four transformations we need to consider. Those are rotations, reflections, translations, and enlargements. However, we can see that all of our triangles appear to be the same size, so no enlargements have taken place. Remember when we enlarge a shape, we make it bigger or smaller, so we’ll consider the other three.

Notice next that the orientation of 𝐴𝐡𝐢 is different to the orientation of 𝐴 prime 𝐡 prime 𝐢 prime. And so we can assume that this has not been translated. When we translate a shape, we essentially slide it. It moves in the coordinate plane but maintains its orientation, so we either have a rotation or a reflection. When we reflect a shape, we flip it over a mirror line, and when we rotate a shape, we turn it about a center. Now there appears to be no nice mirror line that we can add into our diagram to reflect the shape. So let’s see if we can figure out a way to rotate it.

There are three possible points we could rotate about. There’s a point 𝐷, point 𝐸, or point 𝐹. We’ll consider point 𝐷 first. Let’s begin by drawing a line directly from 𝐷 to vertex 𝐴. We’ll then draw a line from the same center, this time to the image of 𝐴; that’s 𝐴 prime. We might deduce by the angle that these two lines make that the triangle has been rotated counterclockwise by 90 degrees. But let’s check by looking at vertex 𝐡. This time, we’ll join 𝐷 to 𝐡 and then we’ll join 𝐷 to the image of 𝐡, 𝐡 prime. Once again, we see that these two lines make an angle of 90 degrees. So we can say that our shape has been rotated 90 degrees counterclockwise about that point 𝐷. The single transformation that maps 𝐴𝐡𝐢 onto the image of 𝐴𝐡𝐢 is a 90 degrees counterclockwise rotation about 𝐷.

Now, in fact, it’s worth noting the use of the word β€œsingle” here. A common misconception is to try and use two transformations, for example, a rotation followed by a translation. In those cases, we would actually get no marks awarded to us, so we must really look out for that word β€œsingle.”

Let’s now look at the second question, the single transformation that maps 𝐴 prime 𝐡 prime 𝐢 prime onto 𝐴 double prime 𝐡 double prime 𝐢 double prime. That’s this triangle and this triangle. We already eliminated enlargements throughout this question. As in the previous question, the two triangles are in different orientations to one another, so we can again disregard translations. And this means we either have a rotation or a reflection. In fact, we can see that triangle 𝐴 prime 𝐡 prime 𝐢 prime appears to have been flipped. When we flip a shape, we reflect it. And so we need to decide where the mirror line is.

We remember that when we reflect a shape, each vertex of that shape is the same distance away from the mirror line, but on the other side. And so we can say the mirror line must be exactly halfway between our two shapes. That’s this line here. If we extend it, we see that we get the line 𝐸𝐹. The single transformation that maps 𝐴 prime 𝐡 prime 𝐢 prime onto 𝐴 double prime 𝐡 double prime 𝐢 double prime is a reflection in the line 𝐸𝐹.

Let’s now consider question three, β€œdescribe the single transformation that would map 𝐴 double prime 𝐡 double prime 𝐢 double prime onto 𝐴 triple prime 𝐡 triple prime 𝐢 triple prime.” Okay, so we’re looking to compare this triangle with this triangle. Once again, we have eliminated enlargements. And now we notice that the orientation of these two triangles is the same. They appear to be the same way up. We have slid our shape onto its image, and so we have a translation. Let’s pick a vertex on our original shape and its image and work out how far this shape has slid.

If we pick the point 𝐴 here, we know that we’re going to its image 𝐴 triple prime here. To do so, we move one, two, three units left and then we move one, two units up. The single transformation that maps 𝐴 double prime 𝐡 double prime 𝐢 double prime onto its image is a translation three units left and two units up. Note that you might sometimes see this written in vector form as the vector negative three, two.

One more part to this question. The fourth part says, β€œHence, are triangles 𝐴𝐡𝐢 and 𝐴 triple prime 𝐡 triple prime 𝐢 triple prime congruent?” Well, for two shapes to be congruent, they must be identical. They have the same angles and the same length sides. We have just seen that to map from 𝐴𝐡𝐢 onto triangle 𝐴 triple prime 𝐡 triple prime 𝐢 triple prime, we’ve done a rotation followed by a reflection followed by a translation. We have not performed a single enlargement. And so the triangle hasn’t changed in size. Rotating a triangle doesn’t change it in size, nor does reflecting or translating it. And so we can say yes, triangles 𝐴𝐡𝐢 and 𝐴 triple prime 𝐡 triple prime 𝐢 triple prime are congruent.

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