Question

which composition of two rigid motions maps \\( \triangle a b c \\) to \\( \triangle a ^ { prime } b ^ { prime } c ^ { prime } \\)?

Explanation:

Step1: Analyze reflection over \(x -\)axis

When a point \((x,y)\) is reflected over the \(x -\)axis, the transformation rule is \((x,y)\to(x, - y)\).

Step2: Analyze reflection over \(y = 0\) (same as \(x -\)axis reflection in terms of rule) and then translation

After reflection over \(x -\)axis, then we need to check the translation. But let's first check the composition of two reflections.
The composition of a reflection over \(x = 0\) (where the rule for reflection over \(x = 0\) is \((x,y)\to(-x,y)\)) and then reflection over \(y = 0\) (rule \((x,y)\to(x, - y)\)):
Let's take a point \(A(3,5)\) (from the graph - assume coordinates \(A(3,5)\), \(B(1,2)\), \(C(5,1)\)).
First, reflect over \(x = 0\): \(A(3,5)\to A'(- 3,5)\), then reflect over \(y = 0\): \(A'(-3,5)\to A''(-3,-5)\)
Another way: composition of \(R_{y = 0}\circ R_{x = 0}\) (first reflect over \(x = 0\), then over \(y = 0\))
Take \(A(3,5)\):
First \(R_{x = 0}\): \((3,5)\to(-3,5)\)
Then \(R_{y = 0}\): \((-3,5)\to(-3,-5)\)
If we first reflect over \(y = 0\) (rule \((x,y)\to(x, - y)\)): \(A(3,5)\to(3,-5)\), then reflect over \(x = 0\) (rule \((x,y)\to(-x,y)\)): \((3,-5)\to(-3,-5)\)

Answer:

B. \(R_{y = 0}\circ R_{x = 0}\)