Congruent Figures Quick Check

Question

2 of 52 of 5 Items

Question
Use the image to answer the question.
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.

How would you describe this series of transformations?

(1 point)
Responses

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"
.
Reflection across the y -axis and then a translation of left parenthesis negative 3 comma negative 5 right parenthesis 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 .

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 .
Skip to navigation

Answer 1

To determine the series of transformations that show the congruence between the triangles as described, we need to analyze the given coordinates and the transformations that can map one triangle onto another.

1. Triangles:

   • Triangle ABC: A(-4, 5), B(-1, 3), C(-3, 1)
   • Triangle A''B''C'': A''(1, 0), B''(-2, -2), C''(0, -4)

2. Transformations:

   • Translation: Changes the position of the triangles without altering their shape.
   • Reflection: Flips the triangle across a specified axis.
   • Rotation: Rotates the triangle around a point (commonly the origin).

3. Congruence Consideration:

   • To check for congruence, we look at whether one triangle can be transformed into another using the transformations mentioned.
   • The statement about the orientations of the triangles matters. Two triangles are congruent if you can perform transformations that make them overlap completely.

From the provided options:

• The first two options discuss translations and reflections in connection to the triangle ABC and A''B''C''.
• The third and fourth options also mention congruence but bring up orientation.
• The last two options introduce rotations, which were not mentioned in the transformations before, as they are less likely based on simple translations/reflections seen in the problem.

Given the details about the triangles, we look for a transformation that can show one triangle is simply a translated/reflected version of the other without losing congruence.

If we consider the transformations, the first option seems fitting:

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

This suggests that triangle ABC can be moved and flipped, preserving its size and shape, allowing it to match triangle A''B''C''. Hence, the correct response to describe the transformations is:

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