An illustration of a coordinate plane with four quadrants shows the x and y axes extending from negative 10 to 10 in increments of 1. Three triangles are plotted on the graph. The triangle upper A upper B upper C has its vertices marked with closed points at upper A left parenthesis 2 comma negative 2 right parenthesis, upper B left parenthesis 2 comma negative 4 right parenthesis, and upper C left parenthesis 5 comma negative 4 right parenthesis. The second triangle, placed in the second quadrant, has its vertices marked with closed points at left parenthesis negative 3 comma 6 right parenthesis, left parenthesis negative 5 comma 6 right parenthesis, and left parenthesis negative 5 comma 3 right parenthesis. The third triangle, placed in the third quadrant, has its vertices marked with closed points at left parenthesis negative 2 comma negative 2 right parenthesis, left parenthesis negative 4 comma negative 2 right parenthesis, and left parenthesis negative 4 comma negative 5 right parenthesis. An arrow goes from the point left parenthesis negative 2 comma negative 2 right parenthesis to left parenthesis negative 3 comma 6 right parenthesis; an arrow goes from the point left parenthesis negative 4 comma negative 2 right parenthesis to left parenthesis negative 5 comma 6 right parenthesis; an arrow goes from left parenthesis negative 4 comma negative 5 right parenthesis to left parenthesis negative 5 comma 3 right parenthesis.

Question

What sequence of transformations is depicted in the figure, proving the congruence of the triangles? rotation of
degrees clockwise about the origin, then translation of (
)

Answer 1

To determine the transformations between the triangles and show their congruence, we need to analyze the movements of the vertices of each triangle in relation to one another.

1. Identify the triangles:

   • Triangle ABC with vertices A(2, -2), B(2, -4), C(5, -4).
   • Triangle DEF with vertices D(-3, 6), E(-5, 6), F(-5, 3).
   • Triangle GHI with vertices G(-2, -2), H(-4, -2), I(-4, -5).

2. Observing the arrows:

   • The arrow from G(-2, -2) to D(-3, 6) indicates a transformation.
   • The arrow from H(-4, -2) to E(-5, 6) suggests a similar transformation.
   • The arrow from I(-4, -5) to F(-5, 3) implies consistent movement.

Step 1: Determine Rotation To understand the transformation from triangle GHI to triangle DEF:

• The point G(-2, -2) moves to D(-3, 6).
• The point H(-4, -2) moves to E(-5, 6).
• The point I(-4, -5) moves to F(-5, 3).

Calculating the distance and confirming angles can help determine if there is a specific rotation. The points appear to involve a point movement fundamentally altering their placements across quadrants.

Let's explore them in terms of rotation about the origin.

• Triangle GHI is in the third quadrant, while in the second quadrant, triangle DEF appears to be rotated clockwise.
• The systematic angle can be assessed by checking the changes in coordinates.

Step 2: Calculate Translation After establishing the rotation, any movement from triangle GHI to triangle DEF can also be characterized by translation:

If we analyze each pair of vertices:

1. From G to D, we see the change involves moving left (negative transformation) by 1 unit and upward significantly (a vertical increase).
2. The shifts in x-coordinates suggest translation facts that can align transformations between triangles.

Conclusion and Sequence of Transformations:

• The depiction shows a rotation of \(90\) degrees clockwise about the origin, as the triangles switch quadrants adapting accordingly.
• Followed by a translation unit of \( (-1, 8) \).

Thus, the transformations establishing the congruence of the triangles would be:

Rotation of 90 degrees clockwise about the origin, then translation of (-1, 8).