Question

incorrect your answer is incorrect. 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 9 units, and then reflect that result over the y - axis. reflect figure a over the y - axis, and then translate that result down 9 units. reflect figure a over the y - axis, and then rotate that result clockwise 180° about the origin. rotate figure a counterclockwise 180° about the origin, and then reflect that result over the x - 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 (transformations like translation and reflection do not change shape or size), they are congruent.

Step2: Analyze Transformations

• For "Translate Figure A down 9 units, then reflect over \(y -\)axis":
• Translation (down 9 units) moves the figure vertically. Then reflection over \(y -\)axis ( \(x\to - x\)) flips it horizontally. This maps Figure A to Figure B.
• For "Reflect Figure A over \(y -\)axis, then translate down 9 units":
• Reflection over \(y -\)axis first ( \(x\to - x\)), then translation (down 9 units). Order of reflection and translation (when one is vertical and one is horizontal) can sometimes be interchanged. Here it maps Figure A to Figure B.
• For "Reflect over \(y -\)axis, then rotate \(180^{\circ}\) clockwise":
• Reflection over \(y -\)axis: \((x,y)\to(-x,y)\). Rotation \(180^{\circ}\) clockwise: \((x,y)\to(-x,-y)\). The combined transformation \((x,y)\to(x, - y)\) (since \((-(-x),-y)=(x, - y)\)) does not map Figure A to Figure B.
• For "Rotate \(180^{\circ}\) counter - clockwise, then reflect over \(x -\)axis":
• Rotation \(180^{\circ}\) counter - clockwise: \((x,y)\to(-x,-y)\). Reflection over \(x -\)axis: \((x,y)\to(x, - y)\). The combined transformation \((-x,-y)\to(-x,y)\) does not map Figure A to Figure B.

Answer:

(a) Yes
(b) Translate Figure A down 9 units, and then reflect that result over the \(y -\)axis; Reflect Figure A over the \(y -\)axis, and then translate that result down 9 units.