Question

△abc has vertices at a(1,2), b(4,6), and c(4,2) in the coordinate plane. the triangle will be reflected over the x - axis and then rotated 180° about the origin to form △abc. what are the vertices of △abc?
a a(1, - 2), b(4, - 6), c(4, - 2)
b a(-1, - 2), b(-4, - 6), c(-4, - 2)
c a(-1,2), b(-4,6), c(-4,2)
d a(1,2), b(4,6), c(4,2)

Explanation:

Step1: Reflect over the \(x -\)axis

When reflecting a point \((x,y)\) over the \(x -\)axis, the rule is \((x,y)\to(x, - y)\).
For \(A(1,2)\), after reflection over the \(x -\)axis: \(A'(1,-2)\)
For \(B(4,6)\), after reflection over the \(x -\)axis: \(B'(4,-6)\)
For \(C(4,2)\), after reflection over the \(x -\)axis: \(C'(4,-2)\)

Step2: Rotate \(180^{\circ}\) about the origin

When rotating a point \((x,y)\) \(180^{\circ}\) about the origin, the rule is \((x,y)\to(-x,-y)\)
For \(A'(1,-2)\), after rotation: \(A''(-1,2)\)
For \(B'(4,-6)\), after rotation: \(B''(-4,6)\)
For \(C'(4,-2)\), after rotation: \(C''(-4,2)\)

Answer:

C. \(A'(-1,2)\), \(B'(-4,6)\), \(C'(-4,2)\)