### Video Transcript

The triangle π΄π΅πΆ has been transformed onto triangle π΄ prime π΅ prime πΆ prime, which has then been transformed onto triangle π΄ double prime π΅ double prime πΆ double prime as seen in the figure. Describe the single transformation that would map π΄π΅πΆ onto π΄ prime π΅ prime πΆ prime. Describe the single transformation that would map π΄ prime π΅ prime πΆ prime onto π΄ double prime π΅ double prime πΆ double prime. Hence, are triangles π΄π΅πΆ and π΄ double prime π΅ double prime πΆ double prime congruent?

Letβs begin by looking at the transformation that maps π΄π΅πΆ onto π΄ prime π΅ prime πΆ 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 letβs identify the triangles weβre interested in. Triangle π΄π΅πΆ has vertices π΄π΅πΆ as shown, and triangle π΄ prime π΅ prime πΆ prime is this one.

Now we can instantly disregard two transformations. Firstly, π΄π΅πΆ has not changed in size, so we can disregard dilations or enlargements. Remember, these are when we make the shape bigger or smaller. Similarly, weβre able to disregard a translation. When we translate a shape, we slide it, and so its orientation doesnβt 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 π΄π΅πΆ appears to have turned about a specific angle onto π΄ prime π΅ prime πΆ prime.

When we turn a shape, we rotate it. Remember, reflecting involves flipping a shape in a mirror line. So we know weβve 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 π·. And if we have some tracing paper, we could verify this. If we then join the center of rotation π· to the point π΄ and its image π΄ prime, we see that we can rotate by 180 degrees to map from π΄ to π΄ prime. The same can be said for πΆ and also point π΅. The single transformation that maps π΄π΅πΆ onto π΄ prime π΅ prime πΆ prime is a rotation of 180 degrees about point π·.

Weβll now consider the second part of this question. We need to describe the single transformation that maps π΄ prime π΅ prime πΆ prime onto π΄ double prime π΅ double prime πΆ double prime. π΄ prime π΅ prime πΆ prime is this one, and π΄ double prime π΅ double prime πΆ 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 π΄ prime π΅ prime πΆ prime to π΄ double prime π΅ double prime πΆ 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.

Letβs choose the vertex π΄ prime and see how to map it onto π΄ double prime. We count one, two, three, four, five, six, seven, eight units down. And then we count one, two units right. Itβs more common to state the left and right direction before the upperβdown direction. And so the single transformation weβre 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 π΄π΅πΆ is congruent to π΄ double prime π΅ double prime πΆ double prime. And so we recall that if two shapes are congruent, they are identical; theyβre exactly the same. Their angles are exactly the same and their sides are exactly the same. And so letβs consider this in turn. Firstly, we know that to map from π΄π΅πΆ onto π΄ prime π΅ prime πΆ prime, we rotated the shape. Rotating does not change the size of the shape, just its orientation and location. And so π΄ prime π΅ prime πΆ prime and π΄ double prime π΅ double prime and πΆ double prime are congruent.

Letβs now consider the transformation that maps π΄ prime π΅ prime πΆ prime onto our third triangle. Itβs a translation and, once again, a translation doesnβt change the size of a shape. In fact, it merely changes its location, meaning that these two triangles are also congruent. And so if π΄π΅πΆ is identical to π΄ prime π΅ prime πΆ prime and π΄ prime π΅ prime πΆ prime is identical to π΄ double prime π΅ double prime πΆ double prime, then π΄π΅πΆ must itself be congruent to π΄ double prime π΅ double prime πΆ double prime.

And so the answer to this third part is yes. The triangles π΄π΅πΆ and π΄ double prime π΅ double prime πΆ double prime are congruent.