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The triangle ๐ด๐ต๐ถ has been transformed onto triangle ๐ด prime ๐ต prime ๐ถ prime, which has then been transformed onto triangle ๐ด double prime ๐ต double prime ๐ถ double prime, then transformed onto ๐ด triple prime ๐ต triple prime ๐ถ triple prime as seen in the figure. Describe the single transformation that would map ๐ด๐ต๐ถ onto ๐ด prime ๐ต prime ๐ถ prime. Describe the single transformation that would map ๐ด prime ๐ต prime ๐ถ prime onto ๐ด double prime ๐ต double prime ๐ถ double prime. Describe the single transformation that would map ๐ด double prime ๐ต double prime ๐ถ double prime onto ๐ด triple prime ๐ต triple prime ๐ถ triple prime. Hence, are triangles ๐ด๐ต๐ถ and ๐ด triple prime ๐ต triple prime ๐ถ triple prime congruent?

Thereโs quite a lot of information going on here. So letโs begin by simply describing the transformation that maps ๐ด๐ต๐ถ onto ๐ด prime ๐ต prime ๐ถ prime. Thatโs 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 weโll consider the other three.

Notice next that the orientation of ๐ด๐ต๐ถ is different to the orientation of ๐ด prime ๐ต prime ๐ถ 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 letโs see if we can figure out a way to rotate it.

There are three possible points we could rotate about. Thereโs a point ๐ท, point ๐ธ, or point ๐น. Weโll consider point ๐ท first. Letโs begin by drawing a line directly from ๐ท to vertex ๐ด. Weโll then draw a line from the same center, this time to the image of ๐ด; thatโs ๐ด prime. We might deduce by the angle that these two lines make that the triangle has been rotated counterclockwise by 90 degrees. But letโs check by looking at vertex ๐ต. This time, weโll join ๐ท to ๐ต and then weโll join ๐ท to the image of ๐ต, ๐ต 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 ๐ท. The single transformation that maps ๐ด๐ต๐ถ onto the image of ๐ด๐ต๐ถ is a 90 degrees counterclockwise rotation about ๐ท.

Now, in fact, itโs worth noting the use of the word โsingleโ 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 โsingle.โ

Letโs now look at the second question, the single transformation that maps ๐ด prime ๐ต prime ๐ถ prime onto ๐ด double prime ๐ต double prime ๐ถ double prime. Thatโs 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 ๐ด prime ๐ต prime ๐ถ 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. Thatโs this line here. If we extend it, we see that we get the line ๐ธ๐น. The single transformation that maps ๐ด prime ๐ต prime ๐ถ prime onto ๐ด double prime ๐ต double prime ๐ถ double prime is a reflection in the line ๐ธ๐น.

Letโs now consider question three, โdescribe the single transformation that would map ๐ด double prime ๐ต double prime ๐ถ double prime onto ๐ด triple prime ๐ต triple prime ๐ถ triple prime.โ Okay, so weโre 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. Letโs pick a vertex on our original shape and its image and work out how far this shape has slid.

If we pick the point ๐ด here, we know that weโre going to its image ๐ด 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 ๐ด double prime ๐ต double prime ๐ถ 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, โHence, are triangles ๐ด๐ต๐ถ and ๐ด triple prime ๐ต triple prime ๐ถ triple prime congruent?โ 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 ๐ด๐ต๐ถ onto triangle ๐ด triple prime ๐ต triple prime ๐ถ triple prime, weโve done a rotation followed by a reflection followed by a translation. We have not performed a single enlargement. And so the triangle hasnโt changed in size. Rotating a triangle doesnโt change it in size, nor does reflecting or translating it. And so we can say yes, triangles ๐ด๐ต๐ถ and ๐ด triple prime ๐ต triple prime ๐ถ triple prime are congruent.