Question

identifying the rule for a composition of transformations
which rule describes the composition of transformations that maps rectangle pqrs to p\q
\s\?
$r_{0,270^{circ}}cdot t_{0,2}(x,y)$
$r_{0,180^{circ}}cdot t_{2,0}(x,y)$
$t_{0,2}cdot r_{0,270^{circ}}(x,y)$
$r_{0,2}cdot t_{0,180^{circ}}(x,y)$

Explanation:

Step1: Analyze translation

First, consider the translation part. If we assume a rotation - first approach. Let's check the order of operations. The notation \(A\cdot B\) means \(B\) is applied first and then \(A\).
For a translation \(T_{a,b}(x,y)=(x + a,y + b)\), if we first rotate and then translate.

Step2: Analyze rotation

For a rotation \(R_{0,\theta}(x,y)\):

• The rotation \(R_{0,270^{\circ}}(x,y)=(y,-x)\) (using the rotation formula: when rotating a point \((x,y)\) counter - clockwise about the origin by \(270^{\circ}\), \(x'=y\) and \(y'=-x\)).
• If we first rotate \(R_{0,270^{\circ}}(x,y)\) and then translate \(T_{0,2}(x,y)=(x,y + 2)\)

Let's take a vertex of rectangle \(PQRS\). Suppose a vertex \(P(x,y)\). After \(R_{0,270^{\circ}}\), it becomes \((y,-x)\), and after \(T_{0,2}\), it becomes \((y,-x + 2)\)

Let's check the other options:

• For \(R_{0,180^{\circ}}(x,y)=(-x,-y)\) and \(T_{2,0}(x,y)=(x + 2,y)\). The combined transformation \(R_{0,180^{\circ}}\cdot T_{2,0}(x,y)=R_{0,180^{\circ}}(x + 2,y)=(-(x + 2),-y)\) does not match the transformation from \(PQRS\) to \(P''Q''R''S''\)
• For \(T_{0,2}\cdot R_{0,270^{\circ}}(x,y)\), the order is wrong. If we first translate \(T_{0,2}(x,y)=(x,y + 2)\) and then rotate \(R_{0,270^{\circ}}(x,y + 2)=(y + 2,-x)\), which is not the correct transformation.
• \(R_{0,2}\) is not a standard rotation (rotation is about an angle, not a distance) and \(T_{0,180}\) is not a standard translation (translation is \((x + a,y + b)\) where \(a,b\in\mathbb{R}\) and \(180\) is an angle measure for rotation)

Answer:

\(R_{0,270^{\circ}}\cdot T_{0,2}(x,y)\)