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Part A

Which describes a possible sequence of transformations that transforms polygon
into polygon



?
A
a reflection across the line y=xy=xy=x, followed by a dilation centered at the origin with a scale factor of 222.
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 180180180-degree rotation about the origin, followed by a dilation centered at the origin with a scale factor of 12\frac{1}{2}
2
1
​

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}
2
1
​

(b)

​
Part B
Which best describes the relationship between polygon
and polygon



?
​
A
They are similar because the area of polygon GHJKGHJKGHJK is the same as area of polygon G′H′J′K′G'H'J'K'G
′
H
′
J
′
K
′
.
B
They are similar because polygon G′H′J′K′G'H'J'K'G
′
H
′
J
′
K
′
can be obtained from polygon GHJKGHJKGHJK by a sequence of transformations.
C
They are NOT similar because polygon G′H′J′K′G'H'J'K'G
′
H
′
J
′
K
′
cannot be obtained from polygon GHJKGHJKGHJK in a single transformation.
D
They are NOT similar because the orientation of polygon GHJKGHJKGHJK is not the same as the orientation of polygon G′H′J′K′G'H'J'K'G
′
H
′
J
′
K
′
.

​

1 answer

Part A

To determine the most suitable sequence of transformations that transforms polygon \( GHJK \) into polygon \( G'H'J'K' \), we can analyze the options:

  • Option A: A reflection across the line \( y=x \), followed by a dilation centered at the origin with a scale factor of \( 2 \).
  • 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 \( 2 \).
  • Option C: A \( 180 \)-degree rotation about the origin, followed by a dilation centered at the origin with a scale factor of \( \frac{1}{2} \).
  • 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 \( \frac{1}{2} \).

Without specific coordinates or visual representation of the polygons, we can infer based on the given options.

  • Reflection and dilation are common transformations that can create similar shapes.
  • A \( 180 \)-degree rotation followed by dilation may change the size and shape, depending on the original dimensions.

If the polygons are similar and one can be obtained from the other via transformations, I would lean towards Option B, as it includes a series of logical reflections leading up to a dilation, which is a common method for transformations preserving similarity.

Part B

When considering the relationships between polygons based on similarity:

  • Option A: They are similar because the area of polygon \( GHJK \) is the same as the area of polygon \( G'H'J'K' \).
  • Option B: They are similar because polygon \( G'H'J'K' \) can be obtained from polygon \( GHJK \) by a sequence of transformations.
  • Option C: They are NOT similar because polygon \( G'H'J'K' \) cannot be obtained from polygon \( GHJK \) in a single transformation.
  • Option D: They are NOT similar because the orientation of polygon \( GHJK \) is not the same as the orientation of polygon \( G'H'J'K' \).

In this case, the best option would be B. The similarity of polygons rests primarily on whether one can be obtained from another through a series of transformations, which includes scaling, rotation, and reflection. Hence, if the transformations lead to a change in size or orientation while still being geometric transformations, they can be classified as similar. Thus, option B correctly describes the relationship.

In summary:

  • Part A: B
  • Part B: B