Question

which sequence of rigid motions shows $\triangle abc\cong\triangle rst$? circle your answer.
first rigid motion
reflection across the $x$-axis
$90^{\circ}$ clockwise rotation around the origin
translation 2 units up
translation 10 units up and 2 units left
second rigid motion
translation 2 units up
reflection across the $x$-axis
translation 10 units up and 2 units left
$90^{\circ}$ clockwise rotation around the origin

Explanation:

Step1: Reflect across the x - axis

When we reflect a point \((x,y)\) across the \(x\) - axis, the transformation rule is \((x,y)\to(x, - y)\).
For example, if \(A=( - 4,-4)\), after reflection across the \(x\) - axis, \(A'\) has coordinates \((-4,4)\). If \(B=( - 2,-2)\), then \(B'\) has coordinates \((-2,2)\). If \(C=( - 2,-6)\), then \(C'\) has coordinates \((-2,6)\).

Step2: Translate 2 units up

The translation rule for moving a point \((x,y)\) \(2\) units up is \((x,y)\to(x,y + 2)\).
For \(A'\) with coordinates \((-4,4)\), after translation \(A''\) has coordinates \((-4,6)\). For \(B'\) with coordinates \((-2,2)\), after translation \(B''\) has coordinates \((-2,4)\). For \(C'\) with coordinates \((-2,6)\), after translation \(C''\) has coordinates \((-2,8)\).

Answer:

First Rigid Motion: Reflection across the \(x\) - axis; Second Rigid Motion: Translation \(2\) units up.