To determine the transformations between triangle ABC and triangle A′′B"C", let’s analyze the coordinates and possible transformations step by step.
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Coordinates of Triangle ABC:
- A(-4, 5)
- B(-1, 3)
- C(-3, 1)
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Coordinates of Triangle A′′B"C":
- A′′(1, 0)
- B′′(-2, -2)
- C′′(0, -4)
Step 1: Check for Congruence
For triangles to be congruent, they must have the same shape and size, which can be shown through congruent transformations.
Step 2: Reflection Across the X-axis
Reflect triangle ABC across the x-axis:
- A'(-4, -5)
- B'(-1, -3)
- C'(-3, -1)
Step 3: Analyze Positions
Now compare A'(-4, -5), B'(-1, -3), C'(-3, -1) to A''(1, 0), B''(-2, -2), C''(0, -4). They do not match.
Step 4: Check for Translation
Next, let’s see if we can match by moving:
If we translate triangle ABC by (2, 5) before reflecting:
- Translated A(-2, 10)
- Translated B(1, 8)
- Translated C(-1, 6)
Still, this doesn't help in matching triangles.
Step 5: Reflection Across the Y-axis
Doing the transformation by reflecting triangle ABC across the y-axis, we get the following coordinates for A, B, C:
- A''(4, 5)
- B''(1, 3)
- C''(3, 1)
This aligns with A' (4, 5), B' (1, 3), C' (3, 1)—but is a different configuration.
Conclusion
Since you have tested different transformations without yielding the same points in the right configuration, our assessment shows that triangles ABC and A′′B"C" do not have the same orientation, indicating they are not congruent.
Thus, the most appropriate statement describing the transformations and relationship between these triangles would be:
"Since triangles ABC and A′′B"C" do not have the same orientation, they are not congruent."