Question

determining if figures are congruent and related by a transformation
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 5 units
translate figure a down 5 units
reflect figure a over the x - axis
reflect figure a over the y - axis
rotate figure a clockwise 180° about the origin
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 7 units
translate figure c up 7 units
reflect figure c over the x - axis
reflect figure c over the y - axis
rotate figure c clockwise 90° about the origin

Explanation:

Step1: Check congruence for Figure A and B

Congruent figures have same shape and size. Figure A and B do not have same shape (orientation is different in a non - rotational way). So answer for first congruence is No.

Step2: Check transformation for Figure A and B

For Figure A to B: Reflecting over the \(x -\)axis changes the \(y -\)coordinate sign. If we consider a point \((x,y)\) on Figure A, after reflection over \(x -\)axis it becomes \((x, - y)\). This maps Figure A to Figure B.

Step3: Check congruence for Figure C and D

Congruent figures have same shape and size. Figure C and D have same shape and size. So answer for second congruence is Yes.

Step4: Check transformation for Figure C and D

For Figure C to D: Rotating Figure C clockwise \(90^{\circ}\) about the origin. Using rotation formula \((x,y)\to(y, - x)\) (for \(90^{\circ}\) clockwise rotation) maps Figure C to Figure D.

Answer:

For Figure A and B: No, Reflect Figure A over the \(x -\)axis.
For Figure C and D: Yes, Rotate Figure C clockwise \(90^{\circ}\) about the origin.