Question
B is 4,4 C is 4,1 A is 2,1 B" is -1,2 C" is -1,-1 A" is -1,1 (3 points)Two transformations were performed on Triangle ABC. The first transformation was a reflection over the . The second transformation was a . The second transformation was and . Question 2(1 point)Are the shapes congruent? How do you know?
Answers
GPT-4o mini
To analyze the transformations performed on Triangle ABC and the subsequent vertices given, let's clarify the information provided:
1. **Vertices of the original triangle ABC:**
- \( A(2, 1) \)
- \( B(4, 4) \)
- \( C(4, 1) \)
2. **Transformed vertices:**
- \( A'(-1, 1) \)
- \( B'(-1, 2) \)
- \( C'(-1, -1) \)
### First Transformation: Reflection
To determine the line of reflection, let’s look at the original and transformed coordinates:
- The x-coordinates of points A and B change from positive to negative, indicating a reflection across the y-axis or another vertical line.
### Second Transformation
Since the question leaves the second transformation undefined, we need to observe how the remaining coordinates have changed after both transformations.
- After reflection (if we assume reflection across the y-axis), the x-coordinates of A and B would change (2 to -2 and 4 to -4), but instead, we observe that they are being translated to a different position. If it’s a reflection over the y-axis, points would transform to:
- \( A(2, 1) \) reflects to \( A'(-2, 1) \)
- \( B(4, 4) \) reflects to \( B'(-4, 4) \)
- \( C(4, 1) \) reflects to \( C'(-4, 1) \)
After the reflection, we can see that further transformations must lead to the coordinates given (noting a translation).
### Checking Congruency
To check if the shapes are congruent after the transformations, the following points should hold:
1. **Same Distances:**
- Calculate the lengths of sides in Triangle ABC:
- \( AB = \sqrt{(4-2)^2 + (4-1)^2} = \sqrt{4 + 9} = \sqrt{13} \)
- \( AC = \sqrt{(4-2)^2 + (1-1)^2} = \sqrt{4} = 2 \)
- \( BC = \sqrt{(4-4)^2 + (4-1)^2} = \sqrt{9} = 3 \)
- Check reflected figures:
- \( A'(-1, 1) \), \( B'(-1, 2) \), \( C'(-1, -1) \)
- Similarly, distances can be computed here.
2. **Equal angles:**
- Congruent triangles not only have equal lengths but also have equal angles.
3. **Final determination:**
If the original triangle and the transformed triangle share the same lengths, angles, and properties, they are congruent.
### Conclusion
Thus, if the second transformation between the triangle and its transformed version maintains the point distances and inclinations of the triangle, **yes, the shapes are congruent** because congruency preserves length and angles regardless of the transformations applied (reflection, translation, rotation).
If you know the transformations explicitly, please provide them for a clearer answer.
1. **Vertices of the original triangle ABC:**
- \( A(2, 1) \)
- \( B(4, 4) \)
- \( C(4, 1) \)
2. **Transformed vertices:**
- \( A'(-1, 1) \)
- \( B'(-1, 2) \)
- \( C'(-1, -1) \)
### First Transformation: Reflection
To determine the line of reflection, let’s look at the original and transformed coordinates:
- The x-coordinates of points A and B change from positive to negative, indicating a reflection across the y-axis or another vertical line.
### Second Transformation
Since the question leaves the second transformation undefined, we need to observe how the remaining coordinates have changed after both transformations.
- After reflection (if we assume reflection across the y-axis), the x-coordinates of A and B would change (2 to -2 and 4 to -4), but instead, we observe that they are being translated to a different position. If it’s a reflection over the y-axis, points would transform to:
- \( A(2, 1) \) reflects to \( A'(-2, 1) \)
- \( B(4, 4) \) reflects to \( B'(-4, 4) \)
- \( C(4, 1) \) reflects to \( C'(-4, 1) \)
After the reflection, we can see that further transformations must lead to the coordinates given (noting a translation).
### Checking Congruency
To check if the shapes are congruent after the transformations, the following points should hold:
1. **Same Distances:**
- Calculate the lengths of sides in Triangle ABC:
- \( AB = \sqrt{(4-2)^2 + (4-1)^2} = \sqrt{4 + 9} = \sqrt{13} \)
- \( AC = \sqrt{(4-2)^2 + (1-1)^2} = \sqrt{4} = 2 \)
- \( BC = \sqrt{(4-4)^2 + (4-1)^2} = \sqrt{9} = 3 \)
- Check reflected figures:
- \( A'(-1, 1) \), \( B'(-1, 2) \), \( C'(-1, -1) \)
- Similarly, distances can be computed here.
2. **Equal angles:**
- Congruent triangles not only have equal lengths but also have equal angles.
3. **Final determination:**
If the original triangle and the transformed triangle share the same lengths, angles, and properties, they are congruent.
### Conclusion
Thus, if the second transformation between the triangle and its transformed version maintains the point distances and inclinations of the triangle, **yes, the shapes are congruent** because congruency preserves length and angles regardless of the transformations applied (reflection, translation, rotation).
If you know the transformations explicitly, please provide them for a clearer answer.