Question

answer each question below.
are figure a and figure b congruent?
yes no
which transformation will map figure a onto figure b exactly?
translate figure a to the left 8 units
translate figure a down 8 units
reflect figure a over the x - axis
reflect figure a over the y - axis
rotate figure a clockwise 180° about the origin
rotate figure a counterclockwise 90° about the origin
none of these
are figure c and figure d congruent?
yes no
which transformation will map figure c onto figure d exactly?
translate figure c to the right 6 units
translate figure c up 6 units
reflect figure c over the x - axis
reflect figure c over the y - axis
rotate figure c clockwise 90° about the origin
rotate figure c counterclockwise 180° about the origin
none of these

Explanation:

Left - hand side (Figure A and Figure B):

Congruence Check:

Congruent figures have the same shape and size. By visual inspection (or assuming standard geometric figure properties in a coordinate - plane context where no scaling is indicated), Figure A and Figure B have the same shape and size.

Transformation Check:

• Translation: Translating left or down changes the \(x\) or \(y\) coordinates without the symmetry required.
• Reflection over \(x\) - axis: If we reflect a point \((x,y)\) over the \(x\) - axis, we get \((x, - y)\). But Figure A and Figure B are not symmetric about the \(x\) - axis.
• Reflection over \(y\) - axis: If we reflect a point \((x,y)\) over the \(y\) - axis, we get \((-x,y)\). Figure A and Figure B are symmetric about the \(y\) - axis.
• Rotation \(180^{\circ}\): Rotating a point \((x,y)\) \(180^{\circ}\) about the origin gives \((-x, - y)\), which is not the case here.
• Rotation \(90^{\circ}\): Rotating a point \((x,y)\) \(90^{\circ}\) clockwise about the origin gives \((y,-x)\) (and counter - clockwise gives \((-y,x)\)), which is not applicable here.

Right - hand side (Figure C and Figure D):

Congruence Check:

Congruent figures have the same shape and size. By visual inspection (or assuming standard geometric figure properties in a coordinate - plane context where no scaling is indicated), Figure C and Figure D have the same shape and size.

Transformation Check:

• Translation: Translating right or up changes the \(x\) or \(y\) coordinates without the symmetry required.
• Reflection over \(x\) - axis: If we reflect a point \((x,y)\) over the \(x\) - axis, we get \((x, - y)\). But Figure C and Figure D are not symmetric about the \(x\) - axis.
• Reflection over \(y\) - axis: If we reflect a point \((x,y)\) over the \(y\) - axis, we get \((-x,y)\). Figure C and Figure D are symmetric about the \(y\) - axis.
• Rotation \(90^{\circ}\): Rotating a point \((x,y)\) \(90^{\circ}\) clockwise about the origin gives \((y,-x)\) (and counter - clockwise gives \((-y,x)\)), which is not applicable here.
• Rotation \(180^{\circ}\): Rotating a point \((x,y)\) \(180^{\circ}\) about the origin gives \((-x, - y)\), which is not the case here.

Answer:

Left - hand side:

• Are Figure A and Figure B congruent? Yes
• Which transformation will map Figure A onto Figure B exactly? Reflect Figure A over the \(y\) - axis

Right - hand side:

• Are Figure C and Figure D congruent? Yes
• Which transformation will map Figure C onto Figure D exactly? Reflect Figure C over the \(y\) - axis