△ABC

Question

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 rotation about the intersection of the two perpendicular lines
a rotation about the intersection of the two perpendicular lines

a reflection over the line
a reflection over the line

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

a translation in the same direction as the reflections

Answer 1

To determine the result of the composition of the two reflections, we can analyze the effects of reflecting a triangle over two different lines.

1. Reflection over the line \(x=2\): This transformation reflects points horizontally across the line \(x=2\).

2. Reflection over the line \(y=-2\): This reflection reflects points vertically across the line \(y=-2\).

When you reflect across two perpendicular lines (the vertical line \(x=2\) and the horizontal line \(y=-2\)), this can be shown to be equivalent to a rotation. Specifically, reflecting across two perpendicular lines results in a rotation about the point of intersection of the two lines.

In this case:

• The intersection point of the lines \(x=2\) and \(y=-2\) is the point \((2, -2)\).
• The composition of the two reflections will effectively rotate the triangle \(\triangle ABC\) around the point \((2, -2)\) by 180 degrees.

Given this analysis, the correct answer in the options provided would be:

a rotation about the intersection of the two perpendicular lines.