Question

which rule describes the composition of transformations that maps $\triangle abcd$ to $\triangle abcd$?
$t_{5, - 6}circ r_{y=-x}(x,y)$
$r_{y = - x}circ t_{5, - 6}(x,y)$
$t_{6,-5}circ r_{y - axis}(x,y)$
$r_{y - axis}circ t_{6,-5}(x,y)$

Explanation:

1. First, analyze the transformation steps:

• Translation:
• Observe the movement of the vertices of \(\triangle ABCD\) to an intermediate - figure. A translation is a transformation that moves every point of a figure the same distance in the same direction. If we consider the movement of the figure before reflection, we can see that the figure is first translated. For a translation \(T_{a,b}(x,y)=(x + a,y + b)\).
• By looking at the grid, we can see that the figure is translated 5 units to the right and 6 units down. So the translation rule is \(T_{5,-6}(x,y)=(x + 5,y-6)\).
• Reflection:
• After the translation, the figure is reflected over the line \(y=-x\). The rule for reflecting a point \((x,y)\) over the line \(y =-x\) is \(r_{y=-x}(x,y)=(-y,-x)\).
• In a composition of transformations \(T\circ r\) (where \(T\) is a translation and \(r\) is a reflection), the reflection is applied first to the original points and then the translation is applied to the reflected points. But the notation \(T\circ r(x,y)=T(r(x,y))\) means we first perform \(r\) and then \(T\). Here, we first translate and then reflect. So the composition of transformations is \(r_{y =-x}\circ T_{5,-6}(x,y)\). This means we first apply the translation \(T_{5,-6}(x,y)=(x + 5,y - 6)\) to the point \((x,y)\) and then apply the reflection \(r_{y=-x}\) to the resulting point.

1. Analyze the options:

• Option 1: \(T_{5,-6}\circ r_{y=-x}(x,y)\) means we first reflect over \(y=-x\) and then translate, which is incorrect.
• Option 2: \(r_{y=-x}\circ T_{5,-6}(x,y)\) is the correct order of transformations as we first translate the figure 5 units to the right and 6 units down and then reflect it over the line \(y=-x\).
• Option 3: \(T_{6,-5}\circ r_{y - axis}(x,y)\): The translation values are wrong (\(T_{6,-5}\) instead of \(T_{5,-6}\)) and the reflection is over the \(y\) - axis (\(r_{y - axis}(x,y)=(-x,y)\)) instead of \(y=-x\), so this is incorrect.
• Option 4: \(r_{y - axis}\circ T_{6,-5}(x,y)\): The translation values are wrong and the reflection is over the \(y\) - axis, so this is incorrect.

Answer:

B. \(r_{y=-x}\circ T_{5,-6}(x,y)\)