Determine by applying transformations whether the two triangle seen in the given figure are congruent.
Congruence transformations are those in which the position of the image and preimage may differ, but the two shapes are identical in their shape and size. The three different types of congruence transformation are translations, rotations, and reflections. Dilations are not a congruence transformation as they affect the size of the shape.
Let’s consider first whether the triangle 𝐴𝐵𝐶 could be mapped to the triangle 𝐴 prime 𝐵 prime 𝐶 prime using a translation. Remember a translation is just a shift of a shape and it preserves its orientation. We can see that therefore a translation has not been applied in this case as the two triangles are in different orientation.
Next, let’s consider a rotation. Now, this does look possible as the orientation of the two triangles is different. From a look at the diagram, it appears that the center of rotation may be somewhere around this point here — the point with coordinates negative one, zero.
If I connect this point to the point 𝐴 on the original triangle, we have to go one unit to the left and one unit down. To connect this point to 𝐴 prime on the image, we have to go one unit to the right and one unit up. So those are the complete opposite of each other.
The same is true for the points 𝐵 and 𝐵 prime. To connect the point negative one, zero to 𝐵, we need to go four units to the right and three units down. To connect instead to 𝐵 prime, we need to go four units left and three units up. Again, this is the complete opposite.
For the point 𝐶, we need to go one unit to the right and one unit down. And for 𝐶 prime, it’s one unit left and one unit up. This means that the triangle has indeed been rotated by 180 degrees about this point with coordinates negative one, zero.
Therefore, as triangle 𝐴𝐵𝐶 can be mapped onto triangle 𝐴 prime 𝐵 prime 𝐶 prime by a congruence transformation, a rotation, the two triangles are congruent.