Question

answer the questions about figure a and figure b below.
(a) are figure a and figure b congruent?
yes no
(b) which sequence(s) of transformations will map figure a onto figure b exactly?
choose all that apply.
translate figure a down 6 units, and then reflect that result over the y - axis.
translate figure a to the right 6 units, and then rotate that result counterclockwise ( 180^{circ} ) about the origin.
rotate figure a counterclockwise ( 90^{circ} ) about the origin, and then translate that result to the right 9 units.
rotate figure a clockwise ( 180^{circ} ) about the origin, and then reflect that result over the y - axis.
none of these

Explanation:

Step1: Check Congruence

Congruent figures have the same shape and size. Since Figure A and Figure B have the same shape and size (they are just transformed versions of each other), they are congruent. So for part (a), the answer is Yes.

Step2: Analyze Transformations

• Option 1:

Translate Figure A down 6 units. Then reflect over the \(y -\)axis.
Let's assume a point \((x,y)\) in Figure A. After translation down 6 units, it becomes \((x,y - 6)\). After reflection over the \(y -\)axis (\(x\to - x\)), the point is \((-x,y - 6)\).

• Option 2:

Translate Figure A to the right 6 units (\((x,y)\to(x + 6,y)\)), then rotate counter - clockwise \(180^{\circ}\) about the origin (\((x,y)\to(-x - 6,-y)\)).

• Option 3:

Rotate Figure A counter - clockwise \(90^{\circ}\) about the origin (\((x,y)\to(-y,x)\)), then translate to the right 9 units (\((x,y)\to(-y+9,x)\)).

• Option 4:

Rotate Figure A clockwise \(180^{\circ}\) about the origin (\((x,y)\to(-x,-y)\)), then reflect over the \(y -\)axis (\((x,y)\to(x,-y)\)).

By visual inspection (or assuming coordinates of vertices of the trapezoid - like figures):
Let's assume a vertex of Figure A is \((- 4,3)\)

• For Option 1: Translate down 6 units \((-4,3-6)=(-4,-3)\), reflect over \(y -\)axis \((4,-3)\)
• For Option 3: Rotate counter - clockwise \(90^{\circ}\) \((-3,-4)\), translate right 9 units \((-3 + 9,-4)=(6,-4)\) (incorrect)
• For Option 2: Translate right 6 units \((-4 + 6,3)=(2,3)\), rotate \(180^{\circ}\) \((-2,-3)\) (incorrect)
• For Option 4: Rotate \(180^{\circ}\) \((4,-3)\), reflect over \(y -\)axis \((4,-3)\) (incorrect)

Answer:

(a) Yes
(b) Translate Figure A down 6 units, and then reflect that result over the \(y -\)axis.