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Note that △ XYZ is upside down compared to △ JKL.
Does the composition of transformations change the shape or size of △ JKL?
Composition of transformations:
Reflection: in the x-axis.
Translation: (a,b) → (a+4,b)
Yes, it is a congruence transformation.
| Corresponding sides | Corresponding angles |
|---|---|
| XZ ≅ JL | ∠ X ≅ ∠ J |
| XY ≅ JK | ∠ Y ≅ ∠ K |
| YZ ≅ KL | ∠ Z ≅ ∠ L |
Examining the diagram, we notice that the orientation of △ XYZ is upside down when compared to △ JKL. Therefore, to give the triangles the same orientation, we should reflect △ JKL in the x-axis which turns it upside down. When the triangles have the same orientation, we can perform a translation to map the triangles to each other.
If (a,b) is reflected in the x-axis, then its image is the point (a,- b).
preimage (a,b) → image (a,- b)
Using this rule, we can determine the coordinates of our image.
| Point | (a,b) | (a,- b) |
|---|---|---|
| J | (- 3,2) | (- 3,- 2) |
| K | (- 2,4) | (- 2,- 4) |
| L | (0,2) | (0,- 2) |
Now we can graph the image of △ JKL after reflecting it in the x-axis.
The coordinates of △ J'K'L' has the same vertical position as △ XYZ. Therefore, we only have to translate △ J'K'L' horizontally by 4 units to the right: preimage (a,b) → image (a+4,b) By performing this translation, we can map the triangles to each other.
The composition of transformations that maps △ JKL to △ XYZ is: Reflection:& in the $x-$axis. Translation:& (a,b) → (a+4,b)
Another name for a rigid motion or a combination of rigid motions is a congruence transformation. Since both reflections and translations preserves angle and side measures, these transformations are in fact rigid motions. Therefore, we know that the composition is a congruence transformation. Let's identify the corresponding parts