Question

which series of transformations demonstrates that the figures are congruent?
answer:

Explanation:

Step1: Analyze option c

• Reflection over the y - axis:
• For a point \((x,y)\) reflected over the \(y\) - axis, the transformation is \((x,y)\to(-x,y)\).
• Let's assume \(Q=(2,1)\), \(R=(2, - 1)\), \(S=(4,-1)\). After reflection over the \(y\) - axis, \(Q\to Q'=(-2,1)\), \(R\to R'=(-2,-1)\), \(S\to S'=(-4,-1)\).
• Translation of 4 units down:
• The transformation for a translation of 4 units down is \((x,y)\to(x,y - 4)\).
• For \(Q'=(-2,1)\), after translation \(Q''=(-2,1 - 4)=(-2,-3)\); for \(R'=(-2,-1)\), \(R''=(-2,-1 - 4)=(-2,-5)\); for \(S'=(-4,-1)\), \(S''=(-4,-1 - 4)=(-4,-5)\) (This is incorrect as per the graph).

Step2: Analyze option d

• Reflection over the \(x\) - axis:
• For a point \((x,y)\) reflected over the \(x\) - axis, the transformation is \((x,y)\to(x,-y)\). Let \(Q=(2,1)\), \(R=(2,-1)\), \(S=(4,-1)\). After reflection over the \(x\) - axis, \(Q\to Q'=(2,-1)\), \(R\to R'=(2,1)\), \(S\to S'=(4,1)\) (Incorrect as per the graph).

Step3: Analyze option b

• Rotation of \(90^{\circ}\) clockwise about the origin:
• The transformation for a point \((x,y)\) rotated \(90^{\circ}\) clockwise about the origin is \((x,y)\to(y,-x)\). Let \(Q=(2,1)\), \(R=(2,-1)\), \(S=(4,-1)\). After rotation, \(Q\to(1,-2)\), \(R\to(-1,-2)\), \(S\to(-1,-4)\) (Incorrect as per the graph).

Step4: Analyze option a

• Rotation of \(180^{\circ}\) about the origin:
• The transformation for a point \((x,y)\) rotated \(180^{\circ}\) about the origin is \((x,y)\to(-x,-y)\). Let \(Q=(2,1)\), \(R=(2,-1)\), \(S=(4,-1)\). After rotation, \(Q\to(-2,-1)\), \(R\to(-2,1)\), \(S\to(-4,1)\).
• Translation of 2 units up:
• The transformation for a translation of 2 units up is \((x,y)\to(x,y + 2)\).
• For \(Q=(-2,-1)\) after translation \(Q''=(-2,-1 + 2)=(-2,1)\); for \(R=(-2,1)\) after translation \(R''=(-2,1+2)=(-2,3)\) (Incorrect as per the graph).

Answer:

C. A reflection over the \(y\) - axis followed by a translation of 4 units down.