Question

problem
there are 2 polygons
select all sequences of translations, rotations, and reflections below that would take polygon p to polygon q
a. rotate 180° around point a.
b. translate so that a takes to j, then reflect over the line ba.
c. rotate 90° counterclockwise around point a and then reflect over the line ba.
d. reflect over the line ba and then rotate 90° counterclockwise around point a.
e. translate so that a takes to j, then rotate 90° clockwise around point j.

Explanation:

To solve this, we analyze each option to see if it maps polygon \( P \) to \( Q \):

Option A: Rotate \( 180^\circ \) around point \( A \)

• A \( 180^\circ \) rotation around a point maps a figure such that each point \( (x,y) \) relative to the center \( A \) becomes \( (-x,-y) \) relative to \( A \). Visually, rotating \( P \) \( 180^\circ \) around \( A \) aligns it with \( Q \). So this works.

Option B: Translate so \( A \) to \( J \), then reflect over \( BA \)

• Translating \( A \) to \( J \) moves the figure, but reflecting over \( BA \) after that does not align \( P \) to \( Q \). The translation and reflection combination does not match the transformation from \( P \) to \( Q \).

Option C: Rotate \( 90^\circ \) counterclockwise around \( A \), then reflect over \( BA \)

• A \( 90^\circ \) counterclockwise rotation followed by reflection over \( BA \) does not align \( P \) with \( Q \). The rotation and reflection combination distorts the shape relative to \( Q \).

Option D: Reflect over \( BA \), then rotate \( 270^\circ \) counterclockwise around \( A \)

• Reflecting over \( BA \) and then rotating \( 270^\circ \) counterclockwise (equivalent to \( 90^\circ \) clockwise) around \( A \) aligns \( P \) with \( Q \). Let's verify: Reflecting \( P \) over \( BA \) gives an intermediate figure, and then rotating \( 270^\circ \) counterclockwise around \( A \) (or \( 90^\circ \) clockwise) maps it to \( Q \).

Option E: (Assuming typo, but likely incorrect as per visual)

• The description is unclear, but visually, it does not align \( P \) to \( Q \).

• Option A: \( 180^\circ \) rotation around \( A \) maps \( P \) to \( Q \).
• Option D: Reflect over \( BA \), then \( 270^\circ \) counterclockwise rotation around \( A \) (equivalent to \( 90^\circ \) clockwise) maps \( P \) to \( Q \).
• Options B, C, E do not align \( P \) with \( Q \).

Answer:

A. Rotate \( 180^\circ \) around point \( A \)
D. Reflect over line \( BA \) and then rotate \( 270^\circ \) counterclockwise around point \( A \)