To determine the transformation that shows congruence between triangle ABC and triangle A'B'C', we can analyze the coordinates of the triangles.
Triangle ABC has vertices:
- A(2, 1)
- B(7, 1)
- C(4, 3)
Triangle A'B'C' has vertices:
- A'(-1, 2)
- B'(-1, 7)
- C'(-3, 4)
We want to see if there is a transformation involving a rotation that maps triangle ABC to triangle A'B'C'.
Step 1: Identify the Centroid of Both Triangles
The centroid of a triangle given by vertices (x1, y1), (x2, y2), and (x3, y3) is calculated as: \[ G = \left( \frac{x1+x2+x3}{3}, \frac{y1+y2+y3}{3} \right) \]
For triangle ABC: \[ G_{ABC} = \left( \frac{2 + 7 + 4}{3}, \frac{1 + 1 + 3}{3} \right) = \left( \frac{13}{3}, \frac{5}{3} \right) \]
For triangle A'B'C': \[ G_{A'B'C'} = \left( \frac{-1 -1 -3}{3}, \frac{2 + 7 + 4}{3} \right) = \left( \frac{-5}{3}, \frac{13}{3} \right) \]
Step 2: Find the Necessary Rotation
To see if a rotation will map the centroid of triangle ABC to that of triangle A'B'C', we need to determine how to rotate the coordinates.
By trial or examination using the rotation formula can help. For example, if we rotate triangle ABC around point (0, 0) by 90 degrees counterclockwise or any angle, we can check if the points match.
Conclusion
After examining symmetry and the relationship between the points, it's indeed a 90-degree counterclockwise rotation from triangle ABC centered around the origin that would align triangle ABC with triangle A'B'C'.
Thus, the transformation that shows congruence between triangle ABC and triangle A'B'C' is a 90-degree counterclockwise rotation.
Answer: 90-degree counterclockwise rotation.