Answer:
(d) translation right 5, then rotation 180°
Step-by-step explanation:
The necessary transformations can be discerned by comparing the image figure to the original. The image figure has the same clockwise orientation, but each of the line segments is 180° from its original direction.
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<h3>rotation</h3>
The 180° change in direction from AB (east) to A'B' (west) means the figure was rotated 180°. For rotation 180°, the result is the same for clockwise and counterclockwise.
Rotation preserves the distance of each point from the center of rotation, so that center is the midpoint of AA' or BB' or CC'. The midpoint of AA' is ...
((-4, -1) +(-1, 1))/2 = (-5, 0)/2 = (-2.5, 0)
Rotation 180° about center (h, k) can be described by the transformation ...
(x, y) ⇒ (2h -x, 2k -y)
For the center (h, k) = (-2.5, 0), the transformation is ...
(x, y) ⇒ (2(-2.5) -x, 2(0) -y)
(x, y) ⇒ (-5 -x, -y)
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<h3>translation</h3>
Written as ...
(x, y) ⇒ (-5 -x, -y)
the transformation is equivalent to the sequence of transformations ...
- (x, y) ⇒ (-x, -y) . . . . . rotation 180°
- (x, y) ⇒ (x -5, y) . . . . translation left 5 units . . . (<em>not an answer choice</em>)
If we factor out the minus sign, it is also equivalent to a different sequence of transformations:
- (x, y) ⇒ (x +5, y) . . . . . translation right 5 units
- (x, y) ⇒ (-(x +5), -y) . . . rotation 180°
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The set of transformations performed on ∆ABC is ...
- translation right 5 units, followed by
- rotation 180°.
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The attachment shows the possible transformation sequences. The red figure is obtained by rotating ∆ABC 180° (first). It would need to be translated left 5 units to make ∆A'B'C'. The blue figure is obtained by shifting ∆ABC right 5 units. Rotating it 180° will give ∆A'B'C'.
