Which series of transformations shows that pentagon a is congruent to pentagon b? a. rotate pentagon a 90* clockwise about the point (3, 1), reflect it over the x-axis, and translate it 3 units to the left. b. rotate pentagon a 90* counterclockwise about the origin, reflect it over the x-axis, and translate it 8 units to the right and 1 unit up. c. reflect pentagon a over the y-axis, rotate it 180* clockwise about the origin, and translate it 3 units up d. reflect pentagon a over the y-axis and translate it 2 units to the left and 8 units up. ps don't answer this question if you don't know the answer for sure.

The correct answer is:

B) A 90° counterclockwise rotation about the origin, followed by a reflection across the x-axis, followed by a translation 8 units right and 1 unit up.

Explanation:

The coordinates of the points of the pre-image are:
(3, 1)
(3, 4)
(5, 7)
(6, 5)
(6, 2)

The coordinates of the points of the image are:
(7,-2)
(4,-2)
(1,-4)
(3,-5)
(6,-5)

A 90° counterclockwise rotation about the origin negates the y-coordinate and switches it and the x-coordinate.  Algebraically,
(x,y)→(-y,x).

When this is applied to our points, we get:
(3, 1)→(-1, 3)
(3, 4)→(-4, 3)
(5, 7)→(-7, 5)
(6, 5)→(-5, 6)
(6, 2)→(-2, 6)

A reflection across the x-axis negates the y-coordinate.  Algebraically,
(x, y)→(x, -y).

Applying this to our new points, we have:
(-1, 3)→(-1, -3)
(-4, 3)→(-4, -3)
(-7, 5)→(-7, -5)
(-5, 6)→(-5, -6)
(-2, 6)→(-2, -6)

A translation 8 units right and 1 unit up adds 8 to the x-coordinate and 1 to the y-coordinate.  Algebraically,
(x, y)→(x+8, y+1).

Applying this to our new points, we have:
(-1, -3)→(-1+8,-3+1) = (7, -2)
(-4, -3)→(-4+8,-3+1) = (4, -2)
(-7, -5)→(-7+8,-5+1) = (1, -4)
(-5, -6)→(-5+8,-6+1) = (3, -5)
(-2, -6)→(-2+8,-6+1) = (6, -5)

These match the coordinates of the image, so this is the correct series of transformations.