Question

identifying the rule for a composition of transformations
which rule describes the composition of
transformations that maps \\( \triangle d e f \\) to \\( \triangle d^{\prime \prime} e^{\prime \prime} f^{\prime \prime} \\)?
\\( r_{0.90^{circ}} \cdot t_{5,0}(x, y) \\)
\\( t_{-5,0} \cdot r_{0.90^{circ}}(x, y) \\)
\\( t_{5,0} \cdot r_{0.90^{circ}}(x, y) \\)
\\( r_{0.90^{circ}}(x, y) \cdot t_{-5,0} \\)

Explanation:

Step1: Analyze the rotation

First, consider the rotation \(R_{0,90^{\circ}}\). The rule for a \(90^{\circ}\) counter - clockwise rotation about the origin \((x,y)\to(-y,x)\).

Step2: Analyze the translation

Then, consider the translation. A translation \(T_{- 5,0}(x,y)=(x - 5,y)\). If we first rotate \(\triangle DEF\) by \(90^{\circ}\) counter - clockwise about the origin (\(R_{0,90^{\circ}}\)) and then translate it \(5\) units to the left (\(T_{-5,0}\)), we get the mapping from \(\triangle DEF\) to \(\triangle D''E''F''\).

Answer:

\(T_{-5,0}\circ R_{0,90^{\circ}}(x,y)\)