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?
(2 points)
rotation of (
) degrees clockwise about the origin, then translation of (
,
​​​​​​​)​​​​​​​

Answer 1

To determine the sequence of transformations that shows the congruence of the triangles, we will analyze the positions of the triangles.

1. **Triangle A (Triangle ABC)** has vertices at \(A(2, -2)\), \(B(2, -4)\), and \(C(5, -4)\).

2. **Triangle B** is in the second quadrant with vertices at \((-3, 6)\), \((-5, 6)\), and \((-5, 3)\).

3. **Triangle C** is in the third quadrant with vertices at \((-2, -2)\), \((-4, -2)\), and \((-4, -5)\).

### Step 1: Identify Translations

By observing the arrows in the description, it appears that there are transformations between triangle ABC (the first triangle) and the triangles in the second (Triangle B) and third quadrants (Triangle C).

- The arrow from \((-2, -2)\) to \((-3, 6)\) indicates that this is a translation. The change in coordinates is:
\[
(-2, -2) \to (-3, 6) \quad \Rightarrow \quad \Delta x = -3 + 2 = -1, \quad \Delta y = 6 + 2 = 8
\]
- The transformation from \((-4, -2)\) to \((-5, 6)\):
\[
(-4, -2) \to (-5, 6) \quad \Rightarrow \quad \Delta x = -5 + 4 = -1, \quad \Delta y = 6 + 2 = 8
\]
- The transformation from \((-4, -5)\) to \((-5, 3)\):
\[
(-4, -5) \to (-5, 3) \quad \Rightarrow \quad \Delta x = -5 + 4 = -1, \quad \Delta y = 3 + 5 = 8
\]

In all cases, the transformation is a translation of \((-1, 8)\).

### Step 2: Identify Rotation

Next, we notice that the arrows suggest that there may be a rotation around the origin. The triangles seem to retain their shape, indicating that rotation is likely involved.

To find the angle of rotation, we can look at a pair of corresponding segments. Observing Triangle A and Triangle B:
- The segment \(AB\) is vertical (along \(x = 2\)) and goes from \(A(2,-2)\) to \(B(2,-4)\).
- The orientation of Triangle B suggests that its vertical segment runs along \(x = -5\) in a similar orientation.

Since a rotation of 180 degrees would convert a positive segment in the x-direction to a negative segment in the third quadrant while maintaining the shape, we suspect a rotation of 180 degrees about the origin is being utilized.

### Conclusion

Putting these transformations together, we find:
1. A rotation of **180 degrees clockwise** around the origin.
2. Followed by a translation of **(-1, 8)**.

Thus, the complete answer to the transformations leading to congruence is:
- **Rotation of 180 degrees clockwise about the origin, then translation of (-1, 8).**