Question

are figure a and figure b congruent?
yes no
which transformation will map figure a onto
figure b exactly?
translate figure a to the right 6 units
translate figure a up 6 units
reflect figure a over the x - axis
reflect figure a over the y - axis
rotate figure a clockwise 90° about the
origin
rotate figure a counterclockwise 180°
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 left 4 units
translate figure c down 4 units
reflect figure c over the x - axis
reflect figure c over the y - axis
rotate figure c clockwise 180° about the
origin
rotate figure c counterclockwise 90° about
the origin
none of these

Explanation:

Step1: Check Congruence

Congruent figures have the same shape and size. By visual inspection, Figure A and Figure B are congruent (same shape and size), while Figure C and Figure D are also congruent (same shape and size).

Step2: Analyze Transformations for Figure A and B

• Translation: Moving a figure without rotation or reflection. Translating Figure A up 6 units won't map it to Figure B (incorrect direction). Translating right 6 units also won't (wrong orientation).
• Reflection: Reflecting over x - axis or y - axis changes orientation. Figure A and B don't have a mirror - like relationship over x or y - axis.
• Rotation: Rotating Figure A counter - clockwise 180° about the origin changes its position. Let’s assume a point \((x,y)\) in Figure A. After a 180° counter - clockwise rotation about the origin, the point becomes \((-x,-y)\). By checking corresponding vertices (if we assume coordinate - based analysis from the graph), 180° rotation about the origin maps Figure A to Figure B.

Step3: Analyze Transformations for Figure C and D

• Translation: Translating left 4 units (if we consider horizontal movement between the two figures based on their positions on the coordinate - like plane in the graph) can map Figure C to Figure D. Let’s assume a point \((x,y)\) in Figure C. Translating it to the left 4 units gives \((x - 4,y)\). By checking corresponding vertices (from visual inspection of their positions), translating Figure C to the left 4 units maps it to Figure D.

Answer:

For the first set (Figure A and B):

• Congruence: Yes
• Transformation: Rotate Figure A counterclockwise \(180^{\circ}\) about the origin

For the second set (Figure C and D):

• Congruence: Yes
• Transformation: Translate Figure C to the left 4 units