Question

which pair of rigid transformations will map △efg to △efg?
a. a reflection across the x - axis, then a reflection across the y - axis.
b. a rotation of 90° counterclockwise around the origin, then a reflection across the x - axis.
c. a reflection across line y = x, then a reflection across y = - x.
d. a rotation of 90° clockwise around the origin, then a rotation of 90° counterclockwise around the ori

Explanation:

Step1: Analyze Option A

• Reflection across \(x -\)axis: \((x,y)\to(x, - y)\).
• Then reflection across \(y -\)axis: \((x,-y)\to(-x,-y)\).
• Let's assume a point \(E(2,1)\) in \(\triangle EFG\). After reflection across \(x -\)axis: \((2, - 1)\), then after reflection across \(y -\)axis: \((-2,-1)\) which is \(E'\) in the figure.
• Let's take point \(F(3,3)\) in \(\triangle EFG\). After reflection across \(x -\)axis: \((3,-3)\), then after reflection across \(y -\)axis: \((-3,-3)\) which is \(F'\) in the figure.
• Let's take point \(G(7,2)\) in \(\triangle EFG\). After reflection across \(x -\)axis: \((7,-2)\), then after reflection across \(y -\)axis: \((-7,-2)\) which is \(G'\) in the figure.

Step2: Analyze Option B

• Rotation of \(90^{\circ}\) counter - clockwise around the origin: \((x,y)\to(-y,x)\).
• Then reflection across \(x -\)axis: \((-y,x)\to(-y,-x)\).
• For point \(E(2,1)\): After rotation \(90^{\circ}\) counter - clockwise \((-1,2)\), then reflection across \(x -\)axis \((-1,-2)

eq E'\).

Step3: Analyze Option C

• Reflection across \(y = x\): \((x,y)\to(y,x)\).
• Then reflection across \(y=-x\): \((y,x)\to(-x,-y)\).
• For point \(E(2,1)\): After reflection across \(y = x\) \((1,2)\), then reflection across \(y=-x\) \((-2,-1)\). But the order of mapping is not correct as per the figure's orientation.

Step4: Analyze Option D

• Rotation of \(90^{\circ}\) clockwise around the origin: \((x,y)\to(y,-x)\).
• Then rotation of \(90^{\circ}\) counter - clockwise around the origin: \((y,-x)\to(x,y)\). This is equivalent to no net transformation.

Answer:

A. A reflection across the \(x -\)axis, then a reflection across the \(y -\)axis.