A coordinate plane with 4 quadrants shows x and y axes ranging from negative 6 to 6 in increments of 1. Three triangles are formed by joining three plotted points each. The coordinates of the plotted points for the first triangle upper A upper B upper C joined by solid lines are upper A is left parenthesis negative 4 comma 5 right parenthesis, upper B is left parenthesis negative 1 comma 3 right parenthesis, and upper C is left parenthesis negative 3 comma 1 right parenthesis. The coordinates for the second triangle upper A prime upper B prime upper C prime joined by dotted lines are as follows: upper A prime at left parenthesis 4 comma 5 right parenthesis, upper B prime at left parenthesis 1 comma 3 right parenthesis, and upper C prime at left parenthesis 3 comma 1 right parenthesis. The coordinates of the plotted points for the third triangle upper A double prime upper B double prime upper C double prime joined by lines made of dashes and dots are as follows: upper A double prime at left parenthesis 1 comma 0 right parenthesis, upper B double prime at left parenthesis negative 2 comma negative 2 right parenthesis, and upper C double prime at left parenthesis 0 comma negative 4 right parenthesis.

Question

how would you describe this series of transformations?

Rotation of 90 degrees clockwise and then reflection across the x-axis shows that triangle ABC is congruent to triangle A′′B"C"

Reflection across the y-axis and then a translation of (−3,−5) shows that triangle ABC is congruent to triangle A′′B"C"

Translation of (2,0) and then reflection across the x-axis shows that triangle ABC is congruent to triangle A′′B"C"

Since triangles ABC and A′′B"C" do not have the same orientation, they are not congruent.

Answer 1

To determine the transformations between triangle ABC and triangle A′′B"C", let’s analyze the coordinates and possible transformations step by step.

1. Coordinates of Triangle ABC:

   • A(-4, 5)
   • B(-1, 3)
   • C(-3, 1)

2. 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."