Congruent figures practice - transformations, and congruence
Sure! Here's a practice problem involving congruent figures and transformations:
Consider the following triangle ABC:
A(2, 4), B(4, 6), C(6, 2)
Perform the following transformations and determine if the resulting figure is congruent to triangle ABC:
1. Translation by vector (-1, 1)
2. Reflection across the x-axis
3. Rotation by 90 degrees counterclockwise about the origin
Solution:
1. Translation by vector (-1, 1):
The new coordinates of triangle ABC after translation will be:
A'(1, 5), B'(3, 7), C'(5, 3)
To determine if triangle ABC is congruent to A'B'C', we need to compare their corresponding sides and angles. In this case, all corresponding sides have the same length, and the corresponding angles have the same measure. Therefore, triangle ABC is congruent to A'B'C' after translation.
2. Reflection across the x-axis:
The new coordinates of triangle ABC after reflection will be:
A(2, -4), B(4, -6), C(6, -2)
Again, we compare the corresponding sides and angles of triangle ABC and A'B'C'. In this case, all corresponding sides have the same length and the corresponding angles have the same measure. Therefore, triangle ABC is congruent to A'B'C' after reflection.
3. Rotation by 90 degrees counterclockwise about the origin:
The new coordinates of triangle ABC after rotation will be:
A'(-4, 2), B'(-6, 4), C'(-2, 6)
Once again, we compare the corresponding sides and angles of triangle ABC and A'B'C'. In this case, all corresponding sides have the same length and the corresponding angles have the same measure. Therefore, triangle ABC is congruent to A'B'C' after rotation.
In summary, all of the transformations performed on triangle ABC resulted in congruent figures.