Question
Determine the congruence transformation used on △ABC that resulted in △DEF. (Coordinates; A(-1, 3), B(-2, -1), C(1, 0), D(3, -1), E(4, 3), F(1, 2))
Answers
Answer
is that your definite answer?
Answered by
GPT-4o mini
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.
1. 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 \( DEF \).
- \( A(-1, 3) \)
- \( B(-2, -1) \)
- \( C(1, 0) \)
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.
1. 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 \( DEF \).
Answered by
GPT-4o mini
Based on the analysis, the congruence transformation from triangle \( \triangle ABC \) to triangle \( \triangle DEF \) involves two primary transformations: **a translation followed by a rotation.**
Here’s the summary of the transformations:
1. **Translation:** Moving triangle \( ABC \) using the translation vector \( (4, -4) \) to get point \( D \) from point \( A \).
2. **Rotation:** After translation, orientation changes are necessary to align the other points (which may involve rotating about the point \( D \) or another point).
### Final Answer:
The definite conclusion for the congruence transformation from triangle \( \triangle ABC \) to triangle \( \triangle DEF \) is a **translation followed by a rotation.**
If a specific rotation angle or center is required, further calculations based on angles formed by the points would be necessary. However, the general description of transformations in terms of congruency is accurate.
Here’s the summary of the transformations:
1. **Translation:** Moving triangle \( ABC \) using the translation vector \( (4, -4) \) to get point \( D \) from point \( A \).
2. **Rotation:** After translation, orientation changes are necessary to align the other points (which may involve rotating about the point \( D \) or another point).
### Final Answer:
The definite conclusion for the congruence transformation from triangle \( \triangle ABC \) to triangle \( \triangle DEF \) is a **translation followed by a rotation.**
If a specific rotation angle or center is required, further calculations based on angles formed by the points would be necessary. However, the general description of transformations in terms of congruency is accurate.