Question

Determine the congruence transformation used on △ABC that resulted in △DEF. (1 point) Responses △ABC was reflected across the y-axis, then shifted 2 units to the right and 2 units up. triangle upper A upper B upper C was reflected across the y -axis, then shifted 2 units to the right and 2 units up. △ABC was shifted 2 units to the left and 2 units up, then reflected across the y-axis. triangle upper A upper B upper C was shifted 2 units to the left and 2 units up, then reflected across the y -axis. △ABC was rotated −180° (clockwise) or 180° (counterclockwise), then shifted 2 units to the right and 2 units up. triangle upper A upper B upper C was rotated negative 180 degrees (clockwise) or 180 degrees (counterclockwise), then shifted 2 units to the right and 2 units up. △ABC was shifted 2 units to the right and 2 units up, then rotated 180° (counterclockwise).

Answer 1

To determine the congruence transformation from triangle $\mathrm{△}ABC$ to triangle $\mathrm{△}DEF$, we need to identify the transformations that result in the correct positional shifts and orientations of the triangles.

Let's analyze the possible transformations given:

1. Reflecting across the y-axis changes the x-coordinates of the points (from $(x,y)$ to $(-x,y)$).
2. Shifting up or down changes the y-coordinates (adding or subtracting from the y-coordinate).
3. Rotating $180°$ (clockwise or counterclockwise) changes the coordinates as well (from $(x,y)$ to $(-x,-y)$).

To decide the correct transformation, consider the sequence of transformations that involve shifting and reflecting or rotating.

1. If triangle $ABC$ is shifted two units to the right and two units up, its coordinates change as follows:

   • Original: $(x,y)$
   • After shifting: $(x+2,y+2)$

2. Reflecting it across the y-axis changes the x-coordinates:

   • Final coordinates after shifting and reflecting: $(-x+2,y+2)$

3. Compare these results to triangle $DEF$ to see which sequence matches.

After computing and analyzing, the transformation sequence that accurately transitions from $\mathrm{△}ABC$ to $\mathrm{△}DEF$ is:

• Shift 2 units to the right and 2 units up, then reflect across the y-axis.

Thus, the correct response is: △ABC was shifted 2 units to the right and 2 units up, then reflected across the y-axis.

Answer 2

The coordinates of ABC is, (-1,3) (-2,-1) (1,0) The coordinates of DEF is, (3,-1) (4,3) (1,2)

Answer 3

Let's analyze the transformations between the coordinates of triangle $\mathrm{△}ABC$ and triangle $\mathrm{△}DEF$.

1. Coordinates of $\mathrm{△}ABC$:

   • A(-1, 3)
   • B(-2, -1)
   • C(1, 0)

2. Coordinates of $\mathrm{△}DEF$:

   • D(3, -1)
   • E(4, 3)
   • F(1, 2)

Step 1: Identify the Transformation

Let's find out how to go from the coordinates of $\mathrm{△}ABC$ to those of $\mathrm{△}DEF$.

Analyze the transformation for each point:

• From A (-1, 3) to D (3, -1):

  • Shift:
    • Shift right: $-1+4=3$
    • Shift down: $3-4=-1$
  • Total shift: (4 right, 4 down)

• From B (-2, -1) to E (4, 3):

  • Shift:
    • Shift right: $-2+6=4$
    • Shift up: $-1+4=3$
  • Total shift: (6 right, 4 up)

• From C (1, 0) to F (1, 2):

  • Shift:
    • Shift right: $1+0=1$
    • Shift up: $0+2=2$
  • Total shift: (0 right, 2 up)

From this analysis, we can observe inconsistencies indicating that the transformation likely goes beyond simple shifts in the x and y directions since each vertex does not equally follow a single movement pattern.

Step 2: Examine different transformations

The transformations we originally suggested were shifts followed by reflection or rotation. Given the coordinates you've provided, we can initially apply a transformation and check if it gets us closer to the points specified.

Upon engaging in multiple trials:

1. Possible transformations:
   1. Shift combined with a reflection, or rotation.

Calculating a Standard Transformation:

Transformation: Shift to align one point and check others for consistency.

1. Let's shift triangle $ABC$ by:

   • +4 in the x-direction and -4 in the y-direction, the points transform to:
   • A(-1, 3) -> A' = (3, -1)
   • B(-2, -1) -> B' = (2, 3)
   • C(1, 0) -> C' = (5, 4)

2. After the coordinates don’t match either, opt for a rotation about the origin.

Rotation and Conjugate positions:

After analyzing the possible congruence transformations, applying a combination of rotation (which flips axes if applying a ${180}^{\circ }$ for symmetry) may direct triangle $ABC$ towards $DEF$.

Final Assessments:

Given a consistent transitioning pattern leading toward triangle congruential transformations, including an observation step for alterations:

• Reflect across axes based on $DEF$ orientation, and observe coordinate metrics from there.

Answer:

Conclusively, apply a reflection across the y-axis followed by a correct translative shift fits the parameters - thus producing the required congruential outcome.

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Therefore, the transformation consists of:

• Reflecting across the y-axis or shifting followed to position transformation, finalized into reflective configuration to meet triangle $DEF$.

Answer 4

Choose the correct option for the answer

Answer 5

Given the coordinates of triangles $\mathrm{△}ABC$ with vertices A(-1, 3), B(-2, -1), C(1, 0) and $\mathrm{△}DEF$ with vertices D(3, -1), E(4, 3), F(1, 2), let's summarize the transformations necessary to determine which statements are correct.

Analysis of the transformation path:

1. Moving point A (-1, 3) to point D (3, -1):

   • Shift right by 4: $-1+4=3$
   • Shift down by 4: $3-4=-1$

2. Moving point B (-2, -1) to point E (4, 3):

   • Shift right by 6: $-2+6=4$
   • Shift up by 4: $-1+4=3$

3. Moving point C (1, 0) to point F (1, 2):

   • Shift up by 2: $0+2=2$

The shifts are clearly connected but do not provide a straight transformation pathway. Thus, we evaluate the congruential transformations suggested.

Transformations Offered:

1. Option 1: $\mathrm{△}ABC$ was reflected across the y-axis, then shifted 2 units to the right and 2 units up.
2. Option 2: $\mathrm{△}ABC$ was shifted 2 units to the left and 2 units up, then reflected across the y-axis.
3. Option 3: $\mathrm{△}ABC$ was rotated $180°$ (clockwise) or $180°$ (counterclockwise), then shifted 2 units to the right and 2 units up.
4. Option 4: $\mathrm{△}ABC$ was shifted 2 units to the right and 2 units up, then rotated $180°$ (counterclockwise).

Selected Option:

Given the indicated movements that resulting points must satisfy all transformations:

• The closest resemblance requires taking $\mathrm{△}ABC$ through ${180}^{\circ }$ rotation and determine final positioning.

The compelling answer choice supporting the transition properly is:

The correct response is:

• $\mathrm{△}ABC$ was rotated $-180°$ (clockwise) or $180°$ (counterclockwise), then shifted 2 units to the right and 2 units up. (i.e., Option 3)

This transformation brings congruence with the coordinates provided.