Question

a triangle with points a(3, - 4), b(6, - 4), and c(- 4, - 3) is reflected over y - axis to form triangle abc. which of these transformaions will also carry triangle abc into abc? select all that apply.
a rotating the triangle 90 degree clockwise about the origin and then reflecting it across x - axis
b rotating the triangle 180 degree about the origin, and then reflecting it across x - axis
c rotating the triangle 90 degree clockwise about the origin and then reflecting it across x - axis
d reflection of the triangle across the x - axis and rotating 180 degree clockwise about the origin.
e rotating the triangle 180 degree about the origin and then reflecting it across the line y = x

Explanation:

Step1: Analyze option A

Rotation \(90^{\circ}\) clockwise about the origin: \((x,y)\to(y, -x)\). Then reflection across \(x -\)axis: \((y,-x)\to(y,x)\).

Step2: Analyze option B

Rotation \(180^{\circ}\) about the origin: \((x,y)\to(-x,-y)\). Then reflection across \(x -\)axis: \((-x,-y)\to(-x,y)\).

Step3: Analyze option C

Rotation \(90^{\circ}\) counter - clockwise about the origin: \((x,y)\to(-y,x)\). Then reflection across \(x -\)axis: \((-y,x)\to(-y,-x)\).

Step4: Analyze option D

Reflection across \(x -\)axis: \((x,y)\to(x,-y)\). Then rotation \(180^{\circ}\) clockwise about the origin: \((x,-y)\to(-x,y)\).

Step5: Analyze option E

Rotation \(180^{\circ}\) about the origin: \((x,y)\to(-x,-y)\). Then reflection across \(y = x\): \((-x,-y)\to(-y,-x)\).

Let \(A(3,4)\), \(B(6,4)\), \(C(4,8)\) be the original points.
For a reflection over the \(y -\)axis, the transformation is \((x,y)\to(-x,y)\). So \(A(3,4)\to A'(- 3,4)\), \(B(6,4)\to B'(-6,4)\), \(C(4,8)\to C'(-4,8)\)

For option A:
Take \(A(3,4)\):
First rotation \(90^{\circ}\) clockwise \((3,4)\to(4,-3)\), then reflection across \(x -\)axis \((4,-3)\to(4,3)
eq(-3,4)\)

For option B:
Take \(A(3,4)\):
First rotation \(180^{\circ}\) \((3,4)\to(-3,-4)\), then reflection across \(x -\)axis \((-3,-4)\to(-3,4)\)
Take \(B(6,4)\):
First rotation \(180^{\circ}\) \((6,4)\to(-6,-4)\), then reflection across \(x -\)axis \((-6,-4)\to(-6,4)\)
Take \(C(4,8)\):
First rotation \(180^{\circ}\) \((4,8)\to(-4,-8)\), then reflection across \(x -\)axis \((-4,-8)\to(-4,8)\)

For option C:
Take \(A(3,4)\):
First rotation \(90^{\circ}\) counter - clockwise \((3,4)\to(-4,3)\), then reflection across \(x -\)axis \((-4,3)\to(-4,-3)
eq(-3,4)\)

For option D:
Take \(A(3,4)\):
First reflection across \(x -\)axis \((3,4)\to(3,-4)\), then rotation \(180^{\circ}\) \((3,-4)\to(-3,4)\)
Take \(B(6,4)\):
First reflection across \(x -\)axis \((6,4)\to(6,-4)\), then rotation \(180^{\circ}\) \((6,-4)\to(-6,4)\)
Take \(C(4,8)\):
First reflection across \(x -\)axis \((4,8)\to(4,-8)\), then rotation \(180^{\circ}\) \((4,-8)\to(-4,8)\)

For option E:
Take \(A(3,4)\):
First rotation \(180^{\circ}\) \((3,4)\to(-3,-4)\), then reflection across \(y = x\) \((-3,-4)\to(-4,-3)
eq(-3,4)\)

Answer:

B. Rotating the triangle \(180\) degrees about the origin and then reflecting it across \(x -\)axis, D. Reflection of the triangle across the \(x -\)axis and rotating \(180\) degrees clockwise about the origin.