Question

part 1 of 2
(a) are figure a and figure b congruent?
yes no
correct answer:
yes
progress: 1/2
part 2 of 2
(b) which sequence(s) of transformations will map figure a onto figure b exactly? choose all that apply.
translate figure a down 6 units, and then rotate that result clockwise ( 180^{circ} ) about the origin.
translate figure a down 9 units, and then reflect that result over the y - axis.
reflect figure a over the y - axis, and then translate that result down 9 units.
rotate figure a counterclockwise ( 90^{circ} ) about the origin, and then translate that result down 7 units.
none of these

Explanation:

Step1: Analyze the first option

Let's assume a point \((x,y)\) on Figure A. Translating down 6 units gives \((x,y - 6)\). Rotating \((x,y-6)\) clockwise \(180^{\circ}\) about the origin: the rule for a \(180^{\circ}\) clockwise rotation about the origin is \((a,b)\to(-a,-b)\). So \((x,y - 6)\to(-x,-(y - 6))=(-x,-y + 6)\). This does not map Figure A onto Figure B.

Step2: Analyze the second option

Take a point \((x,y)\) on Figure A. Translating down 9 units gives \((x,y-9)\). Reflecting \((x,y - 9)\) over the \(y\) - axis: the rule for a reflection over the \(y\) - axis is \((a,b)\to(-a,b)\). So \((x,y-9)\to(-x,y - 9)\).

Step3: Analyze the third option

Take a point \((x,y)\) on Figure A. Reflecting over the \(y\) - axis gives \((-x,y)\). Translating \((-x,y)\) down 9 units: using the rule \((a,b)\to(a,b - k)\) (where \(k = 9\)), we get \((-x,y-9)\).

Step4: Analyze the fourth option

Take a point \((x,y)\) on Figure A. Rotating counter - clockwise \(90^{\circ}\) about the origin: the rule for a \(90^{\circ}\) counter - clockwise rotation about the origin is \((a,b)\to(-b,a)\). So \((x,y)\to(-y,x)\). Translating down 7 units: \((-y,x)\to(-y,x - 7)\). This does not map Figure A onto Figure B.

Answer:

The second (Translate Figure A down 9 units, and then reflect that result over the y - axis) and the third (Reflect Figure A over the y - axis, and then translate that result down 9 units) options apply.