Open Middle®
Directions: Given triangle ABC with vertices (-8,2), (-2,2), and (-2, 8), create triangle DEF in quadrant one that uses a translation, rotation, and reflection (in any order) to take that triangle to triangle ABC and show congruence.
Hint
How can you use the image to find the pre-image?
Answer
One solution would be:
Triangle DEF with vertices (1, 1), (7, 1), and (7, 7).
1. 90° rotation clockwise
2. Reflection across the Y axis
3. A translation of (x-1, y+9)
Triangle DEF with vertices (1, 1), (7, 1), and (7, 7).
1. 90° rotation clockwise
2. Reflection across the Y axis
3. A translation of (x-1, y+9)
There are many solutions.
Source: Jon Henderson
Does Triangle ABC have to map sequentially onto DEF…ie, Does A have to map to D, B to E, and C to F? OR not necessarily…just has to correctly be mapped onto the preimage to show congruence with any order of points?
I love using open ended tasks in my classroom. I think this is a good one that I will use after students have learned about reflections, rotations and translations to assess students understanding. I will have them work in pairs and provide graph paper, rulers and patty paper. I will add an exit ticket on Canvas or paper where students will need to describe why the transformations worked and what properties exist in the triangle that allow us to move it and maintain congruence.
Also, to make sure I understand students draw DEF in quadrant one and then translate, rotate and reflect that triangle onto ABC correct? That is how I read it but it was a little confusing so just wanted to make sure.
This is an interesting question! I think that is definitely one way to do it, but I think students could also perform whatever translation, rotation, and reflection they want on ABC and then use the resulting triangle as their answer and assign D, E, and F accordingly.
Great question. This is the most super open-ended question I’ve encountered across the entire Open Middle site. Natalie’s comment does a great job explaining how a student can backwards design a sample DEF using *any* translation, rotation, and reflection in any order, but this question’s potential is far deeper than that.
Not only can any of D, E, and F be the right angle, any Quadrant I isosceles right triangle DEF with length 6 legs located anywhere in Quadrant I can solve this question. In fact, even the Quadrant I requirement is superfluous.
Pushing even farther using any isosceles right triangle DEF with length 6 legs, there is a guaranteed mapping of DEF onto ABC using fewer requirements than the problem’s initial requirements. Any one of these scenarios will also work:
1) Any reflection + either a translation or a rotation will do (both not required)
2) A translation + a rotation (reflection not required)
3) A single translation or rotation (only one required and no reflection needed)
Sample solution to variation #3: Because ABC is isosceles and assuming E is the right angle in DEF, the orientations of ABC and DEF can be considered identical. This identical orientation means only one of the translation or rotation is needed, and the reflection isn’t required at all. Here’s how: Draw segments AD, BE, and CF. If all three segments are parallel, then there is a guaranteed translation between the triangles. Otherwise, there is guaranteed to be a rotation that will map DEF onto ABC.
This single task could be a phenomenal and flexible exploration of all transformations for students of all levels.
NOTE: If anyone wants to seriously deepen their (or their students’) understanding of transformations, I strongly suggest taking a peek at the transformations chapter of a UCSMP Geometry text. Early in my teaching career, I gained a DEEP and student-accessible understanding of transformations through that source.