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 to the left 7 units, and then rotate that result counterclockwise ( 90^{circ} ) about the origin. reflect figure a over the ( y )-axis, and then rotate that result clockwise ( 180^{circ} ) about the origin. rotate figure a clockwise ( 90^{circ} ) about the origin, and then translate that result to the left 10 units. rotate figure a counterclockwise ( 180^{circ} ) 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. By visual inspection (or assuming side - length and angle measurements are the same as they are just transformed), Figure A and Figure B are congruent.

Step2: Analyze transformation 1

• Translation left 7 units: If we translate Figure A (original \(x\) - coordinates of vertices, say a vertex at \((x,y)\) becomes \((x - 7,y)\)).
• Rotation counter - clockwise \(90^{\circ}\) about the origin: The rule for a counter - clockwise \(90^{\circ}\) rotation about the origin is \((x,y)\to(-y,x)\). Let's assume a vertex of Figure A is \((3,3)\). After translation left 7 units, it is \((- 4,3)\). After rotation, it is \((-3,-4)\) which does not match the position of vertices of Figure B.

Step3: Analyze transformation 2

• Reflection over the \(y\) - axis: The rule is \((x,y)\to(-x,y)\).
• Rotation clockwise \(180^{\circ}\) about the origin: The rule for a clockwise \(180^{\circ}\) rotation about the origin is \((x,y)\to(-x,-y)\). Let a vertex of Figure A be \((3,3)\). After reflection over \(y\) - axis, it is \((-3,3)\). After rotation, it is \((3, - 3)\) which does not match the vertices of Figure B.

Step4: Analyze transformation 3

• Rotation clockwise \(90^{\circ}\) about the origin: The rule is \((x,y)\to(y,-x)\). Let a vertex of Figure A be \((3,3)\), after rotation it is \((3,-3)\).
• Translation left 10 units: It becomes \((3 - 10,-3)=(-7,-3)\) which matches the position of vertices of Figure B.

Step5: Analyze transformation 4

• Rotation counter - clockwise \(180^{\circ}\) about the origin: The rule is \((x,y)\to(-x,-y)\). Let a vertex of Figure A be \((3,3)\), after rotation it is \((-3,-3)\).
• Reflection over the \(x\) - axis: The rule is \((x,y)\to(x,-y)\). After reflection, it is \((-3,3)\) which does not match the vertices of Figure B.

Answer:

(a) Yes
(b) Rotate Figure A clockwise \(90^{\circ}\) about the origin, and then translate that result to the left 10 units.