The triangle 𝐴𝐵𝐶 has been transformed onto triangle 𝐴 prime 𝐵 prime 𝐶 prime, which has then been transformed onto triangle 𝐴 double prime 𝐵 double prime 𝐶 double prime as seen in the figure. Describe the single transformation that would map 𝐴𝐵𝐶 onto 𝐴 prime 𝐵 prime 𝐶 prime. Describe the single transformation that would map 𝐴 prime 𝐵 prime 𝐶 prime onto 𝐴 double prime 𝐵 double prime 𝐶 double prime. Hence, are triangles 𝐴𝐵𝐶 and 𝐴 double prime 𝐵 double prime 𝐶 double prime congruent?
Let’s begin by looking at the transformation that maps 𝐴𝐵𝐶 onto 𝐴 prime 𝐵 prime 𝐶 prime. We do need to be a little bit careful. We want to describe this single transformation. A common mistake here is to describe two transformations such as a rotation followed by a translation. In order to correctly answer a question of this type, we must include one transformation only. So let’s identify the triangles we’re interested in. Triangle 𝐴𝐵𝐶 has vertices 𝐴𝐵𝐶 as shown, and triangle 𝐴 prime 𝐵 prime 𝐶 prime is this one.
Now we can instantly disregard two transformations. Firstly, 𝐴𝐵𝐶 has not changed in size, so we can disregard dilations or enlargements. Remember, these are when we make the shape bigger or smaller. Similarly, we’re able to disregard a translation. When we translate a shape, we slide it, and so its orientation doesn’t change. We can see our shapes are in a different orientation. They appear to be a different way round. One is a rotation and the other is a reflection. In fact, we can see that 𝐴𝐵𝐶 appears to have turned about a specific angle onto 𝐴 prime 𝐵 prime 𝐶 prime.
When we turn a shape, we rotate it. Remember, reflecting involves flipping a shape in a mirror line. So we know we’ve rotated the shape. There are two more things we need to decide. Firstly, we decide the center of rotation and then the angle about which it is turned. Well, if we look carefully, we see it absolutely must have rotated about point 𝐷. And if we have some tracing paper, we could verify this. If we then join the center of rotation 𝐷 to the point 𝐴 and its image 𝐴 prime, we see that we can rotate by 180 degrees to map from 𝐴 to 𝐴 prime. The same can be said for 𝐶 and also point 𝐵. The single transformation that maps 𝐴𝐵𝐶 onto 𝐴 prime 𝐵 prime 𝐶 prime is a rotation of 180 degrees about point 𝐷.
We’ll now consider the second part of this question. We need to describe the single transformation that maps 𝐴 prime 𝐵 prime 𝐶 prime onto 𝐴 double prime 𝐵 double prime 𝐶 double prime. 𝐴 prime 𝐵 prime 𝐶 prime is this one, and 𝐴 double prime 𝐵 double prime 𝐶 double prime is this one. Now, if we look carefully at these two shapes, we see that they are in the same orientation. They seem to be the same way round. And so we can assume that to get from 𝐴 prime 𝐵 prime 𝐶 prime to 𝐴 double prime 𝐵 double prime 𝐶 double prime, we need to slide the shape. When we slide a shape, formally we say we translate it. And to describe the translation, we need to decide by how many units and in which direction the shape has slid.
Let’s choose the vertex 𝐴 prime and see how to map it onto 𝐴 double prime. We count one, two, three, four, five, six, seven, eight units down. And then we count one, two units right. It’s more common to state the left and right direction before the upper–down direction. And so the single transformation we’re interested in is a translation by two units right and eight units down. Note also that this is regularly written in column vector form as the vector two, negative eight.
The third part of this question asked us to identify whether triangle 𝐴𝐵𝐶 is congruent to 𝐴 double prime 𝐵 double prime 𝐶 double prime. And so we recall that if two shapes are congruent, they are identical; they’re exactly the same. Their angles are exactly the same and their sides are exactly the same. And so let’s consider this in turn. Firstly, we know that to map from 𝐴𝐵𝐶 onto 𝐴 prime 𝐵 prime 𝐶 prime, we rotated the shape. Rotating does not change the size of the shape, just its orientation and location. And so 𝐴 prime 𝐵 prime 𝐶 prime and 𝐴 double prime 𝐵 double prime and 𝐶 double prime are congruent.
Let’s now consider the transformation that maps 𝐴 prime 𝐵 prime 𝐶 prime onto our third triangle. It’s a translation and, once again, a translation doesn’t change the size of a shape. In fact, it merely changes its location, meaning that these two triangles are also congruent. And so if 𝐴𝐵𝐶 is identical to 𝐴 prime 𝐵 prime 𝐶 prime and 𝐴 prime 𝐵 prime 𝐶 prime is identical to 𝐴 double prime 𝐵 double prime 𝐶 double prime, then 𝐴𝐵𝐶 must itself be congruent to 𝐴 double prime 𝐵 double prime 𝐶 double prime.
And so the answer to this third part is yes. The triangles 𝐴𝐵𝐶 and 𝐴 double prime 𝐵 double prime 𝐶 double prime are congruent.