Which Transformations Were Used to Create Congruent

Job

Triangles $ABC$ and $PQR$ are shown beneath in the coordinate plane:

1. Show that $\triangle ABC$ is coinciding to $\triangle PQR$ with a reflection followed by a translation.
2. If you contrary the order of your reflection and translation in office (a) does it still map $\triangle ABC$ to $\triangle PQR$?
3. Detect a second manner, dissimilar from your work in role (a), to map $\triangle ABC$ to $\triangle PQR$ using translations, rotations, and/or reflections.

IM Commentary

This job gives students a chance to explore several issues relating to rigid motions of the aeroplane and triangle congruence. Get-go, office (a) gives students an opportunity to describe rigid motions in the coordinate plane to exhibit a congruence between two explicit triangles. In that location are two reasons to inquire students to first find a reflection followed past a translation that will show the congruence. One is that in that location are multiple combinations of reflections followed by translations that will work, giving students a hazard to see that at that place is more one fashion to practise this. The other is to prepare the conversation that can result subsequently students do part (b).

Part (b) looks more deeply into the different types of rigid motions (translations, rotations, and reflections) examining whether or not it matters in which guild these transformations are practical: this depends very much on the specific operations performed as is shown in the solution with some examples. If the students do not produce an example where the club of the reflection and translation influences the finish consequence, the teacher should be prepared to discuss an example like to the one provided in part (b) of the solution.

Finally, part (c) gives students a chance to think creatively well-nigh showing the congruence between the two triangles without having the steps prescribed. Encourage students to detect several different ways if they notice another one immediately.

This task is appropriate for instruction. It illustrates both 8.Yard.A.two because, as an instructional task, information technology can aid students build upward their understanding of the relationship between rigid motions and congruence. Information technology also addresses 8.Thou.3.A: while the students have not been prompted to explicitly draw the transformations in terms of coordinates, they will necessarily employ them at least implicitly. If it is advisable, the teacher can give actress instructions indicating that students need to depict the transformations explicitly in terms of coordinates; alternatively, the teacher can use this task could pb into a discussion about how to describe each of the transformations using coordinates. Office (b) is intended to provide extra depth of agreement to rigid motions and congruence. It provides students with an important example of a context where club of operations matters (in this instance the operations existence transformations of the plane).

Solution

1. Beneath the $y$-axis is shaded carmine and triangle $ABC$ is reflected over the $y$-axis. The image of this reflection is triangle $A^\prime number B^\prime C^\prime$. Reflecting almost the $y$-centrality leaves the $y$-coordinate of each point the same and switches the sign of the $x$-coordinate.

   

   So, for example, $A = (-5,ii)$ and so $A^\prime = (v,2)$. We tin can now come across that translating triangle $A^\prime number B^\prime C^\prime$ downwardly past $6$ units puts it on top of triangle $PQR$:

   

   To find the coordinates after applying this translation, the $x$-coordinate stays the aforementioned and we subtract 6 from the $y$-coordinate of each bespeak.

2. The respond here will depend on which reflection and translation have been called in part (a). For the reflection and translation called above, we reverse the club by outset translating $\triangle ABC$ by $6$ units downward and so reflecting over the $y$-axis. Below, the translated triangle is triangle $A^\prime B^\prime C^\prime$ and its reflection over the $y$-centrality is $\triangle PQR$:

   

   Below is a different reflection through the vertical line through vertex $A$, which can exist followed past the translation indicated by the blue arrows to prove the congruence of $\triangle ABC$ with $\triangle PQR$:

   

   Different in the previous instance, if we perform the translation first, giving the greenish triangle $A^\prime number B^\prime C^\prime$, and then the reflection, giving the imperial triangle $A^{\prime \prime} B^{\prime \prime number} C^{\prime \prime number}$, this does not produce the triangle $PQR$. And so in this case, performing the translation and reflection in a different order produces a different outcome.

   

3. Ane manner to show the triangle congruence would be to align one vertex at a time. Graphically this is shown below:

   

   First a translation is used to move $C$ to $R$ with the new triangle shown in green. If $B^\prime$ is the image of $B$ nether this translation, then a rotation, by the directed angle indicated in regal, moves $B^\prime number$ to $Q$: the triangle after this transformation is indicated in blue, sharing ane side with triangle $PQR$. If $A”$ is the image of $A$ after the translation and rotation, and then a reflection beyond $\overline{QR}$ moves $A”$ to $P$.