Question

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.
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).**