Triangle ABC has the points A(1,1), B(7,2) and C(5,7). What composition transformation will result in △A"B"C"

Question

being at A" (2 ,-1), B" (-4,0) and C" (-2,5)?(1 point)
Responses

a rotation of - 270 degrees and a move right by 3 and up by 2
a rotation of - 270 degrees and a move right by 3 and up by 2

a refletion across the y-axis and a move to the left by 5 and up by 3
a refletion across the y-axis and a move to the left by 5 and up by 3

a reflection across the y-axis and a move to the right by 3 and down by 2
a reflection across the y-axis and a move to the right by 3 and down by 2

a refection across the x-axis and a move to the left by 3 and down by 2

Answer 1

To find the transformation that takes triangle ABC with vertices A(1,1), B(7,2), and C(5,7) to triangle A"(2,-1), B"(-4,0), and C"(-2,5), we need to analyze the movements of the points under various transformations.

Step 1: Determine the transformation necessary to find A" from A, B" from B, and C" from C.

1. Point A to A":

   • A(1,1) to A"(2,-1):
     • Change in x: 2 - 1 = 1 (right by 1)
     • Change in y: -1 - 1 = -2 (down by 2)

2. Point B to B":

   • B(7,2) to B"(-4,0):
     • Change in x: -4 - 7 = -11 (left by 11)
     • Change in y: 0 - 2 = -2 (down by 2)

3. Point C to C":

   • C(5,7) to C"(-2,5):
     • Change in x: -2 - 5 = -7 (left by 7)
     • Change in y: 5 - 7 = -2 (down by 2)

Step 2: Analyze the type of transformations

The results show a need for both a translation and potentially a reflection or rotation.

• Analyzing for reflection across the y-axis:

  • Reflect A(1,1) to get A'(-1,1); then translate A'(-1,1) to A"(2,-1) requires moving right by 3 and down by 2 (which is inconsistent).
  • Reflect B(7,2) to get B'(-7,2); then we would need to translate by the vector that gives B" (-4,0) after reflection. This doesn't match the shift needed.

• Analyzing for reflection across the x-axis:

  • Reflect A(1,1) to get A'(1,-1) leads to a necessary translation of right by 1 and up by 0, which doesn’t hold.

Step 3: Translation and transformation observation

The transformations analyzed do not consistently match the movements of points from A, B, and C to A", B", and C".

Conclusion

None of the options correctly describe the transformation based on calculations, as the movements are inconsistent across the points' mapping.

However, looking closely at transformation choices, the closest logical choice based on given answer choices would relate to understanding reflective properties and translations.

Given the choices, the second option (reflection across the y-axis and a move to the left by 5 and up by 3) may serve a context if we consider movements in aggregate and think of reflective properties getting closer together; it would reflect properties consistently to get closure.

However, transformations must be explored beyond provided options to provide precise directional shifts as above.

Thus, if we are compelled to select an option based on listed transformations, the second option appears most consistent using reflective attributes.

Answer 2

Triangle ABC has the points A(1,1), B(7,2) and C(5,7). What composition transformation will result in △A"B"C"
being at A" (2 ,-1), B" (-4,0) and C" (-2,5)?(1 point)
Responses

a rotation of - 270 degrees and a move right by 3 and up by 2
a rotation of - 270 degrees and a move right by 3 and up by 2

a refletion across the y-axis and a move to the left by 5 and up by 3
a refletion across the y-axis and a move to the left by 5 and up by 3

a reflection across the y-axis and a move to the right by 3 and down by 2
a reflection across the y-axis and a move to the right by 3 and down by 2

a refection across the x-axis and a move to the left by 3 and down by 2
answer with one of the responses avalable

Answer 3

To determine the transformation from triangle ABC with points A(1,1), B(7,2), and C(5,7) to triangle A"(2,-1), B"(-4,0), and C"(-2,5), we can break it down into possible transformations based on the responses given.

Step 1: Move Points A, B, C to A", B", C"

1. Point A(1, 1) to A"(2, -1):

   • Change in x: \(2 - 1 = 1\) (moves right by 1)
   • Change in y: \(-1 - 1 = -2\) (moves down by 2)

2. Point B(7, 2) to B"(-4, 0):

   • Change in x: \(-4 - 7 = -11\) (moves left by 11)
   • Change in y: \(0 - 2 = -2\) (moves down by 2)

3. Point C(5, 7) to C"(-2, 5):

   • Change in x: \(-2 - 5 = -7\) (moves left by 7)
   • Change in y: \(5 - 7 = -2\) (moves down by 2)

Step 2: Analyze the transformations offered in the responses

Let's evaluate the given options:

1. A rotation of -270 degrees and a move right by 3 and up by 2:

   • A rotation would change the positions in a complex way; this does not match the transformations noted.

2. A reflection across the y-axis and a move to the left by 5 and up by 3:

   • Reflection across the y-axis would produce (B' becomes (-7,2) and needs to be moved left and up), but still mismatched.

3. A reflection across the y-axis and a move to the right by 3 and down by 2:

   • This option would not produce the final coordinates either considering individual translations.

4. A reflection across the x-axis and a move to the left by 3 and down by 2:

   • Reflecting across the x-axis, point movements here seem to be similarly misaligned.

Conclusion

Given the choices, the option that involves a reflection across the y-axis and a move to the left by 5 and up by 3 is the option that mathematically aligns more closely with the individual translations we've noted.

So the final response is: "a reflection across the y-axis and a move to the left by 5 and up by 3."