Question
The coordinates of the vertices of △DEF are D(2, −1) , E(7, −1) , and F(2, −3) .
The coordinates of the vertices of △D′E′F′ are D′(0, −1) , E′(−5, −1) , and F′(0, −3) .
What is the sequence of transformations that maps △DEF to △D′E′F′ ?
Drag and drop the answers into the boxes to correctly complete the statement.
A sequence of transformations that maps △DEF to △D′E′F′ is a ______________ followed by a _____________
A. translation 2 units right
B. rotaion of 180 degrees about the origin
C. relfection across the y-axis
D. translation 2 units up
The coordinates of the vertices of △D′E′F′ are D′(0, −1) , E′(−5, −1) , and F′(0, −3) .
What is the sequence of transformations that maps △DEF to △D′E′F′ ?
Drag and drop the answers into the boxes to correctly complete the statement.
A sequence of transformations that maps △DEF to △D′E′F′ is a ______________ followed by a _____________
A. translation 2 units right
B. rotaion of 180 degrees about the origin
C. relfection across the y-axis
D. translation 2 units up
Answers
GPT-4o mini
To find the transformations that map triangle \( \triangle DEF \) to triangle \( \triangle D'E'F' \), let's analyze the coordinates of the vertices:
- Points of \( \triangle DEF \):
- \( D(2, -1) \)
- \( E(7, -1) \)
- \( F(2, -3) \)
- Points of \( \triangle D'E'F' \):
- \( D'(0, -1) \)
- \( E'(-5, -1) \)
- \( F'(0, -3) \)
### Step 1: Determine the transformations
**Translation Check:**
- Observing the x-coordinates:
- \( D(2, -1) \) to \( D'(0, -1) \): \( (2, -1) \to (0, -1) \) (2 units left)
- \( E(7, -1) \) to \( E'(-5, -1) \): \( (7, -1) \to (-5, -1) \) (12 units left)
- \( F(2, -3) \) to \( F'(0, -3) \): \( (2, -3) \to (0, -3) \) (2 units left)
The transformation certainly isn't just a simple translation since the translations vary.
**Reflection Check:**
- Reflecting across the y-axis:
- Reflection of \( D(2, -1) \) across the y-axis gives \( D'(-2, -1) \), not a match.
- Reflection of \( E(7, -1) \) gives \( E'(-7, -1) \), not a match.
- Reflection of \( F(2, -3) \) gives \( F'(-2, -3) \), not a match.
**Rotation Check:**
- A rotation of 180 degrees about the origin will change signs of both coordinates:
- \( D(2, -1) \) becomes \( D'(-2, 1) \)
- \( E(7, -1) \) becomes \( E'(-7, 1) \)
- \( F(2, -3) \) becomes \( F'(-2, 3) \)
This does not match with \( D', E', F' \). So 180 degrees rotation won't work either.
### Step 2: Combine transformations
- Let's consider a reflection across the y-axis followed by a translation:
1. **Reflect across the y-axis**:
- \( D(2, -1) \to D'(-2, -1) \)
- \( E(7, -1) \to E'(-7, -1) \)
- \( F(2, -3) \to F'(-2, -3) \)
2. **Now translate the reflection**:
- Applying a translation of 2 units to the right:
- \( D'(-2, -1) \to D'(0, -1) \)
- \( E'(-7, -1) \to E'(-5, -1) \)
- \( F'(-2, -3) \to F'(0, -3) \)
This sequence works perfectly.
### Final Sequence of Transformations
A sequence of transformations that maps \( \triangle DEF \) to \( \triangle D'E'F' \) is a **reflection across the y-axis** followed by a **translation 2 units right**.
Thus, the final answer is:
**A sequence of transformations that maps \( \triangle DEF \) to \( \triangle D'E'F' \) is a** **C. reflection across the y-axis** **followed by a** **A. translation 2 units right**.
- Points of \( \triangle DEF \):
- \( D(2, -1) \)
- \( E(7, -1) \)
- \( F(2, -3) \)
- Points of \( \triangle D'E'F' \):
- \( D'(0, -1) \)
- \( E'(-5, -1) \)
- \( F'(0, -3) \)
### Step 1: Determine the transformations
**Translation Check:**
- Observing the x-coordinates:
- \( D(2, -1) \) to \( D'(0, -1) \): \( (2, -1) \to (0, -1) \) (2 units left)
- \( E(7, -1) \) to \( E'(-5, -1) \): \( (7, -1) \to (-5, -1) \) (12 units left)
- \( F(2, -3) \) to \( F'(0, -3) \): \( (2, -3) \to (0, -3) \) (2 units left)
The transformation certainly isn't just a simple translation since the translations vary.
**Reflection Check:**
- Reflecting across the y-axis:
- Reflection of \( D(2, -1) \) across the y-axis gives \( D'(-2, -1) \), not a match.
- Reflection of \( E(7, -1) \) gives \( E'(-7, -1) \), not a match.
- Reflection of \( F(2, -3) \) gives \( F'(-2, -3) \), not a match.
**Rotation Check:**
- A rotation of 180 degrees about the origin will change signs of both coordinates:
- \( D(2, -1) \) becomes \( D'(-2, 1) \)
- \( E(7, -1) \) becomes \( E'(-7, 1) \)
- \( F(2, -3) \) becomes \( F'(-2, 3) \)
This does not match with \( D', E', F' \). So 180 degrees rotation won't work either.
### Step 2: Combine transformations
- Let's consider a reflection across the y-axis followed by a translation:
1. **Reflect across the y-axis**:
- \( D(2, -1) \to D'(-2, -1) \)
- \( E(7, -1) \to E'(-7, -1) \)
- \( F(2, -3) \to F'(-2, -3) \)
2. **Now translate the reflection**:
- Applying a translation of 2 units to the right:
- \( D'(-2, -1) \to D'(0, -1) \)
- \( E'(-7, -1) \to E'(-5, -1) \)
- \( F'(-2, -3) \to F'(0, -3) \)
This sequence works perfectly.
### Final Sequence of Transformations
A sequence of transformations that maps \( \triangle DEF \) to \( \triangle D'E'F' \) is a **reflection across the y-axis** followed by a **translation 2 units right**.
Thus, the final answer is:
**A sequence of transformations that maps \( \triangle DEF \) to \( \triangle D'E'F' \) is a** **C. reflection across the y-axis** **followed by a** **A. translation 2 units right**.