Video Transcript
If there exists a combination of rotations, reflections, and translations that would map one shape to another, would the two shapes be congruent?
Let’s think about what we’re being asked here. We’re being asked if any shape was rotated, reflected, or translated with the object and its image be congruent. We can remember that congruent means the same shape and size. More mathematically speaking, we define congruent as having all corresponding pairs of angles equal and all corresponding pairs of sides equal. So let’s take any shape. Let’s say we have this triangle and then we rotate it.
We could, for example, rotate it 90 degrees clockwise about this center of rotation to create the image of this original shape. In this case, we’re asking, would the object and its image be congruent? We would still have corresponding pairs of angles equal and all the corresponding sides are equal. So, yes, after rotation, the object and its image are congruent.
Let’s say we then take the image and reflect it. We could choose any line of reflection and create a second image. We would still have corresponding angles equal and corresponding sides equal. So our reflection will produce a congruent object and its image. Let’s see what happens with a translation which is when we simply move an object left or right and up or down. We could translate this image two to the image three by moving five to the right and three down. Once again, all corresponding pairs of angles are equal and all corresponding pairs of sides are equal, meaning that translation creates a congruent object and its image.
If we wanted, we could go ahead and rotate, reflect, or translate this third image as many times as we wanted. What we’re really being asked here is, if we take any original shape, would it be congruent with the final image after a combination of transformations, as each of these individual transformations produce a congruent object and image that even with a series of these the original shape would still be congruent with the final image?
And so our answer is yes. You might notice that there’s one transformation that isn’t included in this question. It’s dilation. A dilation is when a shape will get larger or smaller, depending on the scale factor. This means that the object and its image would be a different size and, therefore, dilation would not produce a congruent shape.
