Which describes a possible sequence of transformations that transforms polygon GHJKGHJK into polygon G′H′J′K′G′H′J′K′?

Question

A

a  180180180-degree rotation about the origin, followed by a dilation centered  at the origin with a scale factor of  12\frac{1}{2}21​ 

B

a reflection across the  y−y-y− axis, followed by a reflection across the  x−x-x− axis, followed by a dilation centered at the origin with a scale factor of  222  

C

a reflection across the line  y=xy=xy=x, followed by a dilation centered at the origin with a scale factor of  222. 

D

a reflection across the  y−y-y− axis, followed by a translation down 101010 units, followed by a dilation centered at the origin with a scale factor of  12\frac{1}{2}21​ 

Answer 1

To determine the appropriate sequence of transformations that transforms polygon \( GHJK \) into polygon \( G'H'J'K' \), we need to analyze each option and the transformations involved.

1. Option A: A \( 180 \)-degree rotation about the origin, followed by a dilation centered at the origin with a scale factor of \( \frac{1}{2} \).

• Rotation: This transformation will rotate the shape \( 180 \) degrees around the origin.
• Dilation: This will reduce the size of the shape to half its size.

2. Option B: A reflection across the \( y \)-axis, followed by a reflection across the \( x \)-axis, followed by a dilation centered at the origin with a scale factor of \( 2 \).

• Reflection across \( y \)-axis: Flips the shape horizontally.
• Reflection across \( x \)-axis: Flips the shape vertically. This effectively is same as rotating the shape \( 180 \) degrees around the origin.
• Dilation: This enlarges the size of the shape to twice its size.

3. Option C: A reflection across the line \( y = x \), followed by a dilation centered at the origin with a scale factor of \( 2 \).

• Reflection across \( y = x \): This transformation interchanges the \( x \)- and \( y \)-coordinates of the points.
• Dilation: This enlarges the shape to twice its size.

4. Option D: A reflection across the \( y \)-axis, followed by a translation down \( 10 \) units, followed by a dilation centered at the origin with a scale factor of \( \frac{1}{2} \).

• Reflection across \( y \)-axis: Flips the shape horizontally.
• Translation down \( 10 \) units: Shifts the shape downward.
• Dilation: This reduces the size of the shape to half its size.

Without explicit coordinates to analyze the transformation visually, we can still extract insights based on the nature of transformations:

• If \( G'H'J'K' \) appears flipped or rotated to \( GHJK \) without increasing its size, we might check Option A or D.
• If the size appears increased or any extended shape is shown, Option B or C becomes more likely.

From likely candidates, if there's a significant flip and subsequent scaling down, Option D could seem likely; however, actual outcomes hinge on numeric verifications or the polygon's layout.

To arrive at the accurate transformation, please clarify the positioning differences between the two polygons or the exact type of outcome you expect for the transformation.