Question

Question

△ABC
is reflected over the line x=2
to create △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?(1 point)
Responses

a dilation by a certain factor
a dilation by a certain factor

a translation in the same direction as the reflections
a translation in the same direction as the reflections

a reflection over the line
a reflection over the line

a rotation about the intersection of the two perpendicular lines

Answer 1

To analyze the composition of the two reflections described (first over the line \( x = 2 \) and then over the line \( y = -2 \)), we can consider how each reflection affects a point.

1. Reflecting over \( x = 2 \): This will move a point \( A(x, y) \) to \( A'(4 - x, y) \). The \( x \)-coordinate is transformed to its reflection across the line \( x = 2 \).

2. Reflecting over \( y = -2 \): Then, reflecting \( A'(4 - x, y) \) over \( y = -2 \) involves changing the \( y \)-coordinate. The new \( y'\)-coordinate will be \( -2 - (y - (-2)) = -2 + 2 - y = -y \). Thus, the final point would be: \[ A''(4 - x, -y) \]

After performing both reflections, we can see how this composition of transformations can be analyzed:

• Reflecting over the vertical line \( x = 2 \) and then the horizontal line \( y = -2 \) results in a transformation that can be interpreted as a rotation around the point where these two lines intersect. This point of intersection is \( (2, -2) \). The transformation overall is equivalent to a rotation of 180 degrees about this intersection point.

Based on this reasoning, the correct description of the composition transformation is:

a rotation about the intersection of the two perpendicular lines.