Question

which series of transformations shows that pentagon a is congruent to pentagon b? two coordinate grids with pentagons a (green, upper left) and b (red, lower middle) are shown. the options are: a. reflect pentagon a over the y - axis and translate it 2 units to the left and 8 units up. b. rotate pentagon a 90° clockwise about the point (1,1), reflect it over the x - axis, and translate it 3 units to the left. c. reflect pentagon a over the y - axis, rotate it 180° clockwise about the origin, and translate it 3 units up. d. rotate pentagon a 90° clockwise about the origin, reflect it over the x - axis, and translate it 6 units to the left.

Explanation:

To determine which transformation series shows pentagon A is congruent to pentagon B, we analyze each option:

• Option A: Reflecting over the y - axis and translating left 2 and up 8 would not align A with B. The direction and translation amounts are incorrect.
• Option B: Rotating 90° clockwise about (1,1), reflecting over the x - axis, and translating left 3. Let's consider the orientation and position. A 90° clockwise rotation, followed by x - axis reflection (which flips vertically), and a left translation can align the shapes. The key is the rotation center and subsequent transformations matching the position and orientation change from A to B.
• Option C: Reflecting over y - axis, rotating 180° (which is a double reflection or rotation that would flip both x and y), and translating up 3 does not match the orientation and position of B relative to A.
• Option D: Rotating about the origin, reflecting over x - axis, and translating left 6 does not result in the correct position and orientation of B.

Answer:

B. Rotate pentagon A \(90^\circ\) clockwise about the point \((1, 1)\), reflect it over the \(x\) - axis, and translate it 3 units to the left.