Answer: The correct set of transformations are:
(b) a reflection of shape I across the x-axis followed by a 90° counterclockwise rotation about the origin.
(d) a reflection of shape I across the y-axis followed by a 90° clockwise rotation about the origin.
Step-by-step explanation: We are given to select the correct set of transformations that confirm the congruence of shape II and shape I in the figure.
The co-ordinates of the vertices of shape I are (0, -10), (2, -8), (12,-4) and (10, -8).
And the co-ordinates of the vertices of shape II are (-10, 0), (-8, 2), (-4, 12) and (-8, 10).
There can be two possible sets of transformations:
<u>Case I :</u> First reflection across X-axis and then rotation of 90° in counterclockwise direction about the origin.
After reflection across X-axis, the the y co-ordinate of each vertex will change its sign (y = -y). So, the vertices becomes
(0, 10), (2, 8), (12,4) and (10, 8).
Now, after rotation of 90° counterclockwise about the origin, the rule which is to be applied to the vertices is
(x, y) ⇒ (-y, x).
So, the final vertices are
(0, 10) ⇒ (-10, 0),
(2, 8) ⇒ (-8, 2),
(12, 4) ⇒ (-4, 12),
(10, 8) ⇒ (-8, 10).
These are the vertices of shape II. So, option (b) is correct.
<u>Case II :</u> First reflection across Y-axis and then rotation of 90° in clockwise direction about the origin.
After reflection across Y-axis, the the x co-ordinate of each vertex will change its sign (x = -x). So, the vertices becomes
(0, -10), (-2, -8), (-12, -4) and (-10, -8).
Now, after rotation of 90° clockwise about the origin, the rule which is to be applied to the vertices is
(x, y) ⇒ (y, -x).
So, the final vertices are
(0, -10) ⇒ (-10, 0),
(-2, -8) ⇒ (-8, 2),
(-12, -4) ⇒ (-4, 12),
(-10, -8) ⇒ (-8, 10).
These are the vertices of shape II. So, option (d) is correct.
Shape I lie in Quadrant IV and shape II lie in Quadrant II.
According to option (a), if we reflect shape I across the x-axis followed by a 90° clockwise rotation about the origin, then the final shape lie in Quadrant IV, does not lie in Quadrant II, where shape II is lying.
So, option (a) is incorrect.
According to option (c), if we reflect shape I across the y-axis followed by a 90° counterclockwise rotation about the origin, then the final shape will again lie in Quadrant IV.
So, option (c) is incorrect.
According to option (e), if we reflect shape I across the x-axis followed by a 180° rotation about the origin, then the final shape lie in Quadrant III, does not lie in Quadrant II.
So, option (e) is incorrect.
Thus, (b) and (d) are correct options.
