Question

which rule describes the composition of transformations that maps △abc to △abc?
○ r₀, 90° ∘ rₓ - axis(x, y)
○ rₓ - axis ∘ r₀, 90°(x, y)
○ r₀, 180° ∘ rₓ - axis(x, y)
○ rₓ - axis ∘ r₀, 180°(x, y)

Explanation:

Step1: Analyze rotation and reflection

First, observe the orientation - change of the triangle. A rotation of 180° about the origin \(R_{O,180^{\circ}}(x,y)=(-x,-y)\) and a reflection over the x - axis \(r_{x - axis}(x,y)=(x, - y)\).
If we first rotate a point \((x,y)\) 180° about the origin, we get \((-x,-y)\). Then reflecting \((-x,-y)\) over the x - axis gives \((-x,y)\).
If we first reflect \((x,y)\) over the x - axis to get \((x, - y)\) and then rotate \((x, - y)\) 180° about the origin, we get \((-x,y)\).
By observing the transformation of \(\triangle ABC\) to \(\triangle A'B'C'\), we can see that the correct order is first rotation about the origin by 180° and then reflection over the x - axis. The composition of transformations is written as \(r_{x - axis}\circ R_{O,180^{\circ}}(x,y)\).

Answer:

\(r_{x - axis}\circ R_{O,180^{\circ}}(x,y)\)