Question

Triangle ABC is reflected over the line x = 2 to create triangle A'B'C', which is then reflected over the line y = -2. Which of the following could describe this composition transformation using a single transformation?
A. a dilation by a certain factor
B. a rotation about the intersection of the two perpendicular lines

Answer 1

To analyze the composition of the transformations involving reflections, let's first break down what happens when triangle ABC is reflected over the line \( x = 2 \) and then reflected again over the line \( y = -2 \).

1. **First Reflection over the line \( x = 2 \)**:
When a point \( (x, y) \) is reflected over the line \( x = 2 \), the new coordinates \( (x', y') \) become:
\[
x' = 2 + (2 - x) = 4 - x
\]
The \( y \)-coordinate remains unchanged, so:
\[
y' = y
\]
Therefore, any point \( (x, y) \) transforms to \( (4 - x, y) \).

2. **Second Reflection over the line \( y = -2 \)**:
Next, when reflecting a point \( (x', y') \) over the line \( y = -2 \), the transformation is given by:
\[
x'' = x' \quad (\text{remains unchanged})
\]
\[
y'' = -2 + (-2 - y') = -4 - y'
\]
Since \( y' = y \) from the previous transformation, we get:
\[
y'' = -4 - y
\]
Therefore, the point \( (x', y') \) transforms to \( (x', -4 - y) \).

So now, substituting our expression for \( x' \) gives the combined transformation of the initial point \( (x, y) \):
\[
(x'', y'') = (4 - x, -4 - y)
\]

Now, let's analyze how these transformations can be described systematically.

Both reflections across vertical and horizontal lines do not create a dilation but create a transformation resulting from reflections. In general, when you reflect about two perpendicular lines, the result is a rotation. In this case, the reflections over \( x = 2 \) and then \( y = -2 \) can be viewed as a rotation about a point that is located at the intersection of these lines, which is point \( (2, -2) \).

Thus, the resulting transformation can be described as:

**B. a rotation about the intersection of the two perpendicular lines.**

So the appropriate answer is **B**.