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"
.
Rotation of 90 degrees clockwise and then reflection across the x -axis shows that triangle upper A upper B upper C is congruent to triangle upper A double prime upper B double prime upper C double prime .

Translation of (2,0)
and then reflection across the x
-axis shows that triangle ABC
is congruent to triangle A′′B"C"
.
Translation of left parenthesis 2 comma 0 right parenthesis and then reflection across the x -axis shows that triangle upper A upper B upper C is congruent to triangle upper A double prime upper B double prime upper C double prime .

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

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

Answer 1

To analyze the transformations that relate the triangles described, we will first clarify their coordinates and then describe the transformations between the triangles.

Coordinates:

Triangle ABC:

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

Triangle A'B'C':

• A' = (4, 5)
• B' = (1, 3)
• C' = (3, 1)

Triangle A''B''C'':

• A'' = (1, 0)
• B'' = (-2, -2)
• C'' = (0, -4)

Analysis of Transformations:

1. Triangle ABC and A'B'C':

   • The points of triangle A'B'C' can be derived from triangle ABC through a reflection across the y-axis followed by a translation.
   • A is at (-4, 5) and A' is at (4, 5). This is consistent with a reflection across the y-axis.
   • B is at (-1, 3) and B' is at (1, 3), also consistent with the y-axis reflection.
   • C is at (-3, 1) and C' is at (3, 1), again consistent with the y-axis reflection.
   • Since the distances between corresponding points are equal and the angles are preserved, these two triangles are congruent.

2. Triangle ABC and A''B''C'':

   • By analyzing the coordinates, we can check if there are any transformations that can yield triangle A''B''C'' from triangle ABC.
   • The points of triangle A''B''C'' do not show a simple reflection from triangle ABC:
   • For instance, moving point A from (-4, 5) to A'' (1, 0) would require both a change in orientation and scaling.
   • It appears to require a combination of translations and reflections, probably involving moving towards the origin in some way. The points are not congruent.

Conclusion:

• Triangle ABC is congruent to triangle A'B'C' through reflection across the y-axis (and no additional translation).
• Triangle ABC is NOT congruent to triangle A''B''C'', as they do not maintain the same orientation or size: they are effectively transformed in a less straightforward manner that suggests a different relationship or transformation.

Based on these observations, we can conclude that the statements regarding the transformations leading to congruence or lack thereof between the triangles are valid. Specifically, triangles ABC and A'B'C' are congruent due to reflection, while triangles ABC and A''B''C'' are not congruent since they do not have the same orientation or relationship.