A double reflection of a pre-image across perpendicular lines produce the same result, as
1:scaled dilation
180 rotation
90-degree rotation
270-degree rotation
This statement is incorrect. A double reflection of a pre-image across perpendicular lines does not always produce the same result as the other transformation options mentioned (scaled dilation and rotations).
A double reflection across perpendicular lines results in a 180-degree rotation about the point of intersection of the two lines. This is different from a scaled dilation, 90-degree rotation, or 270-degree rotation, which would produce different images.
A double reflection of a pre-image across perpendicular lines can be thought of as a combination of two reflections. Let's break it down step-by-step:
Step 1: Reflection across the first line
The first reflection flips the pre-image across the first perpendicular line.
Step 2: Reflection across the second line
The second reflection flips the image obtained from step 1 across the second perpendicular line.
Overall, this double reflection produces the same result as a 180-degree rotation.
To summarize:
Double reflection across perpendicular lines = 180-degree rotation
To understand why a double reflection of a pre-image across perpendicular lines produces the same result as a scaled dilation, 180-degree rotation, 90-degree rotation, or 270-degree rotation, we need to delve into the properties of these transformations.
1. Scaled Dilation:
A scaled dilation is a transformation that enlarges or shrinks an object while maintaining the proportions between its points. It is defined by a scale factor that determines the degree of enlargement or shrinkage. However, when a pre-image is reflected twice across perpendicular lines, it does not change its size, only its orientation.
2. 180-Degree Rotation:
A 180-degree rotation is a transformation that rotates the pre-image half a revolution, or 180 degrees, in a counterclockwise direction around a fixed center point. When a pre-image is reflected twice across perpendicular lines, it essentially rotates by 180 degrees, resulting in the same orientation as a single 180-degree rotation.
3. 90-Degree Rotation:
Similar to a 180-degree rotation, a 90-degree rotation involves rotating the pre-image by a quarter of a revolution, or 90 degrees, counterclockwise around a fixed center point. When a pre-image is reflected twice across perpendicular lines, it essentially rotates by 180 degrees, which is the same as performing two consecutive 90-degree rotations. As a result, the final image has the same orientation as a double reflection.
4. 270-Degree Rotation:
A 270-degree rotation involves rotating the pre-image by three-quarters of a revolution or 270 degrees counterclockwise around a fixed center point. Similarly, when a pre-image is reflected twice across perpendicular lines, it rotates by 180 degrees, equivalent to two 270-degree rotations. Therefore, the final image has the same orientation as a double reflection.
In summary, a double reflection of a pre-image across perpendicular lines produces the same result as a scaled dilation, 180-degree rotation, 90-degree rotation, or 270-degree rotation because in all cases, the final orientation of the image remains the same.