a
a rotation of 180 degrees counterclockwise about vertex X followed by a dilation through vertex X by scale factor less than 1
b
a reflection over line n followed by a dilation through vertex X by scale factor greater than 1
c
a rotation of 180 degrees counterclockwise about vertex X followed by a dilation through vertex X by scale factor greater than 1
d
a reflection over line n followed by a dilation through vertex X by scale factor less than 1
d) a reflection over line n followed by a dilation through vertex X by scale factor less than 1
how do you know
We need to first understand the effect of each transformation on the figure:
a) A rotation of 180 degrees counterclockwise about vertex X will flip the figure upside down. Followed by a dilation through vertex X by a scale factor less than 1 will result in a smaller figure that is also upside down.
b) A reflection over line n will flip the figure horizontally. Followed by a dilation through vertex X by a scale factor greater than 1 will result in a bigger figure that is also horizontally flipped.
c) A rotation of 180 degrees counterclockwise about vertex X will flip the figure upside down. Followed by a dilation through vertex X by a scale factor greater than 1 will result in a bigger figure that is also upside down.
d) A reflection over line n will flip the figure horizontally. Followed by a dilation through vertex X by a scale factor less than 1 will result in a smaller figure that is not flipped.
So, option d) is the answer, as it accurately describes the sequence of transformation that will result in a smaller figure that is not flipped.
To determine the description that matches each given transformation sequence, let's break down each option and determine the correct choice.
a) a rotation of 180 degrees counterclockwise about vertex X followed by a dilation through vertex X by scale factor less than 1
To find the correct option, we first need to understand what each transformation means. A rotation of 180 degrees counterclockwise means that the shape is turned 180 degrees in a counterclockwise direction. This transformation does not involve any change in size. A dilation through vertex X by a scale factor less than 1 means that the shape is shrunk down, maintaining point X as the center of dilation, and the scale factor is less than 1.
b) a reflection over line n followed by a dilation through vertex X by scale factor greater than 1
In this scenario, we have to perform a reflection over line n before a dilation through vertex X. The reflection over line n means that the shape is flipped across line n, maintaining line n as the axis of reflection. This transformation does not involve any change in size. Following the reflection, a dilation through vertex X with a scale factor greater than 1 means that the shape is enlarged while maintaining point X as the center of dilation, and the scale factor is greater than 1.
c) a rotation of 180 degrees counterclockwise about vertex X followed by a dilation through vertex X by scale factor greater than 1
This option is similar to option a, except for the scale factor. The rotation of 180 degrees counterclockwise remains the same, but this time, the shape is dilated through vertex X with a scale factor greater than 1. This means that the shape is enlarged while maintaining point X as the center of dilation, and the scale factor is greater than 1.
d) a reflection over line n followed by a dilation through vertex X by scale factor less than 1
Lastly, we have a reflection over line n followed by a dilation through vertex X with a scale factor less than 1. The reflection over line n is the same as option b, meaning the shape is flipped across line n while maintaining line n as the axis of reflection. Following the reflection, a dilation through vertex X with a scale factor less than 1 means that the shape is shrunk down while maintaining point X as the center of dilation, and the scale factor is less than 1.
Therefore, based on the descriptions provided, we can conclude that:
a) matches with d
b) matches with c
c) matches with a
d) matches with b