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?

Answer 1

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.