Question

1. the figures are congruent. select all the sequences of transformations that would take figure 1 to figure 2

a translate by directed line segment ad.
b. rotate 180° around point e.
c. translate by directed line segment ae, and reflect across segment ac.
d. translate by directed line segment ce,
and rotate 90° counterclockwise around point e.
e. rotate 180° around point c, translate by directed line segment ce, then reflect across segment ef.
f reflect across segment ab, rotate clockwise by angle bfe using center f, then reflect across segment ef.

Explanation:

Step1: Analyze Option A

Translating Figure 1 by directed line segment \(AD\) will move each point of Figure 1 in the same direction and distance as from \(A\) to \(D\). Since the figures are congruent, this translation will map Figure 1 to Figure 2.

Step2: Analyze Option B

Rotating Figure 1 \(180^{\circ}\) around point \(E\). A \(180^{\circ}\) rotation about a point is a transformation that maps a point \((x,y)\) to \((-x + 2x_E,-y+ 2y_E)\) (if \(E=(x_E,y_E)\)). For congruent figures, this rotation can map Figure 1 to Figure 2.

Step3: Analyze Option C

Translating by \(AE\) changes the position, but reflecting across \(AC\) will not map Figure 1 to Figure 2 as the orientation and position after these two - step operations (translation and reflection) do not match.

Step4: Analyze Option D

Translating by \(CE\) and then rotating \(90^{\circ}\) counter - clockwise around \(E\). The combination of translation and \(90^{\circ}\) rotation (which changes the orientation) will not map Figure 1 to Figure 2.

Step5: Analyze Option E

Rotating \(180^{\circ}\) around \(C\) and then translating by \(CE\) and reflecting across \(EF\). The multiple - step operations (rotation, translation, reflection) do not result in mapping Figure 1 to Figure 2.

Step6: Analyze Option F

Reflecting across \(AB\) changes the position with respect to \(AB\), rotating clockwise by \(\angle BFE\) around \(F\) and then reflecting across \(EF\). The combination of these three non - congruent - mapping operations (in terms of mapping Figure 1 to Figure 2) will not work.

Answer:

A. Translate by directed line segment \(AD\), B. Rotate \(180^{\circ}\) around point \(E\)