Question

question
which transformation would take figure a to figure b?
answer
a counterclockwise rotation of 180 about the origin
a counterclockwise rotation of 90 about the origin
a reflection over the line y = x
a reflection over the line y = -x

Explanation:

Step1: Recall rotation rules

The rule for a counter - clockwise rotation of \(180^{\circ}\) about the origin is \((x,y)\to(-x, -y)\).

Step2: Analyze reflection rules

The rule for a reflection over the line \(y = x\) is \((x,y)\to(y,x)\). The rule for a reflection over the line \(y=-x\) is \((x,y)\to(-y,-x)\). The rule for a counter - clockwise rotation of \(90^{\circ}\) about the origin is \((x,y)\to(-y,x)\).

Step3: Check each transformation

If we assume a point \((x,y)\) in Figure A.

• For a \(180^{\circ}\) rotation: If Figure A has a point \((a,b)\), after \(180^{\circ}\) rotation about the origin, it becomes \((-a,-b)\).
• For a \(90^{\circ}\) rotation: If Figure A has a point \((a,b)\), after \(90^{\circ}\) counter - clockwise rotation about the origin, it becomes \((-b,a)\).
• For reflection over \(y = x\): If Figure A has a point \((a,b)\), it becomes \((b,a)\).
• For reflection over \(y=-x\): If Figure A has a point \((a,b)\), it becomes \((-b,-a)\).

By visual inspection of the coordinate positions of the vertices of Figure A and Figure B (assuming we can pick a vertex \((x,y)\) in Figure A and check its image in Figure B), a counter - clockwise rotation of \(180^{\circ}\) about the origin will map Figure A to Figure B.

Answer:

A counterclockwise rotation of \(180^{\circ}\) about the origin.