To determine the congruence transformation that resulted in triangle \( \triangle ABC \) changing to triangle \( \triangle DEF \), we first analyze the coordinates of the vertices:
- \( A(-1, 3) \)
- \( B(-2, -1) \)
- \( C(1, 0) \)
and
- \( D(3, -1) \)
- \( E(4, 3) \)
- \( F(1, 2) \)
Step 1: Translate Point A to Point D
Let's start by translating point \( A \) to point \( D \).
The translation vector can be found by:
\[ D - A = (3 - (-1), -1 - 3) = (3 + 1, -1 - 3) = (4, -4) \]
Step 2: Apply the Translation
Applying the translation \( (4, -4) \):
-
For \( A(-1, 3) \): \[ A' = (-1 + 4, 3 - 4) = (3, -1) = D \]
-
For \( B(-2, -1) \): \[ B' = (-2 + 4, -1 - 4) = (2, -5) \]
-
For \( C(1, 0) \): \[ C' = (1 + 4, 0 - 4) = (5, -4) \]
Step 3: Compare New Points to D, E, F
After translating \( \triangle ABC \) by \( (4, -4) \), we find \( A' = D \), but the points \( B' \) and \( C' \) are not matching with points \( E \) and \( F \).
Step 4: Identify the Next Transformation
Now, let’s check for further transformations such as rotation or reflection around the axes that might be needed to map from the current positions \( (2, -5) \) and \( (5, -4) \) to points \( E(4, 3) \) and \( F(1, 2) \).
Step 5: Check for Rotation
Since a translation didn't completely work, check the angles:
Compute the side lengths of the triangles and potentially identify a rotation.
- Calculate lengths of \( \triangle ABC \):
- \( AB = \sqrt{(B_x - A_x)^2 + (B_y - A_y)^2} = \sqrt{((-2) - (-1))^2 + ((-1) - 3)^2} = \sqrt{(-1)^2 + (-4)^2} = \sqrt{1 + 16} = \sqrt{17} \)
- \( AC = \sqrt{(C_x - A_x)^2 + (C_y - A_y)^2} = \sqrt{(1 - (-1))^2 + (0 - 3)^2} = \sqrt{(2)^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13} \)
- \( BC = \sqrt{(C_x - B_x)^2 + (C_y - B_y)^2} = \sqrt{(1 - (-2))^2 + (0 - (-1))^2} = \sqrt{(3)^2 + (1)^2} = \sqrt{9 + 1} = \sqrt{10} \)
Compute lengths for \( \triangle DEF \):
- \( DE = \sqrt{(E_x - D_x)^2 + (E_y - D_y)^2} = \sqrt{(4 - 3)^2 + (3 - (-1))^2} = \sqrt{(1)^2 + (4)^2} = \sqrt{1 + 16} = \sqrt{17} \)
- \( DF = \sqrt{(F_x - D_x)^2 + (F_y - D_y)^2} = \sqrt{(1 - 3)^2 + (2 - (-1))^2} = \sqrt{(-2)^2 + (3)^2} = \sqrt{4 + 9} = \sqrt{13} \)
- \( EF = \sqrt{(F_x - E_x)^2 + (F_y - E_y)^2} = \sqrt{(1 - 4)^2 + (2 - 3)^2} = \sqrt{(-3)^2 + (-1)^2} = \sqrt{9 + 1} = \sqrt{10} \)
Conclusion
Since the lengths match after a translation and verify that all configurations of all three points follow congruency properties, the transformation from triangle \( ABC \) to triangle \( DEF \) can be determined as:
Congruence Transformation: A translation by the vector \( (4, -4) \) followed by a rotation.
This reasoning strongly implies a combination of translation followed by rotation and/or reflection to finalize the vertex placements to match \( DEF \).