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Question

A coordinate plane with 4 quadrants shows x and y axes ranging from negative 9 to 9 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 3 comma 6 right parenthesis, upper B is left parenthesis negative 3 comma 2 right parenthesis, and upper C is left parenthesis negative 6 comma 2 right parenthesis. The coordinates for the second triangle upper D upper E upper F joined by dotted lines are as follows: upper D at left parenthesis 3 comma 6 right parenthesis, upper E at left parenthesis 5 comma 2 right parenthesis, and upper F at left parenthesis 3 comma 2 right parenthesis. The coordinates of the plotted points for the third triangle upper H upper I upper G joined by lines made of dashes and dots are as follows: upper G at left parenthesis negative 3 comma negative 6 right parenthesis, upper H at left parenthesis negative 5 comma negative 2 right parenthesis, and upper I at left parenthesis negative 3 comma negative 2 right parenthesis.

Which two triangles are congruent, and which transformation(s) can you use to determine this?

(1 point)
Responses

Triangles DEF
and GHI
are congruent because they line up if you rotate DEF
180 degrees about the origin.
Triangles upper D upper E upper F and upper G upper H upper I are congruent because they line up if you rotate upper D upper E upper F 180 degrees about the origin.

These three triangles are not congruent because ABC
is larger and triangles DEF
and GHI
have different orientations.
These three triangles are not congruent because upper A upper B Upper C is larger and triangles upper D upper E upper F and upper G upper H upper I have different orientations.

Triangles ABC
and GHI
are congruent because they line up if you reflect ABC
over the x
-axis.
Triangles upper A upper B Upper C and upper G upper H upper I are congruent because they line up if you reflect upper A upper B Upper C over the x -axis.

All three triangles are congruent because ABC
is reflected over the y
-axis and then rotated 180 degrees about the origin.

Answer 1

To determine which triangles are congruent and the transformation(s) involved, let's analyze the coordinates of each triangle:

1. Triangle ABC:

   • A = (-3, 6)
   • B = (-3, 2)
   • C = (-6, 2)

2. Triangle DEF:

   • D = (3, 6)
   • E = (5, 2)
   • F = (3, 2)

3. Triangle GHI:

   • G = (-3, -6)
   • H = (-5, -2)
   • I = (-3, -2)

By examining the triangles, we will find that:

• Triangle ABC and Triangle GHI both have the same side lengths (AB = GH, AC = GI, BC = HI), but they are oriented in different quadrants.
• Triangle DEF has different lengths compared to ABC and GHI.

However, if we consider triangles ABC and GHI:

• Triangle GHI can be obtained from triangle ABC by reflecting triangle ABC over the x-axis.

According to the transformations:

• Triangles ABC and GHI are congruent because they line up if you reflect ABC over the x-axis.

Hence, the correct response is:

Triangles ABC and GHI are congruent because they line up if you reflect ABC over the x-axis.