Question

quadrilateral abcd is transformed according to the rule (x, y) → (y, -x). which is another way to state the transformation?
○ ( r_{0, 90^circ} )
○ ( r_{0, 180^circ} )
○ ( r_{0, 270^circ} )
○ ( r_{0, 360^circ} )

Explanation:

Step1: Recall rotation rules

The rotation rules about the origin \( O \) are:

• \( 90^\circ \) rotation: \( (x,y)\to(-y,x) \) (Wait, no, correction: \( 90^\circ \) counterclockwise: \( (x,y)\to(-y,x) \), \( 90^\circ \) clockwise: \( (x,y)\to(y, -x) \))
• \( 180^\circ \) rotation: \( (x,y)\to(-x,-y) \)
• \( 270^\circ \) rotation: \( (x,y)\to(x, -y) \) (counterclockwise) or \( (x,y)\to(-y,x) \) (clockwise? Wait, let's re - establish:

The standard rotation rules (counterclockwise about the origin):

• \( R_{O,90^\circ} \): \( (x,y)\to(-y,x) \)
• \( R_{O,180^\circ} \): \( (x,y)\to(-x,-y) \)
• \( R_{O,270^\circ} \): \( (x,y)\to(y,-x) \)

Wait, actually, the rule \( (x,y)\to(y, - x) \) corresponds to a \( 270^\circ \) counterclockwise rotation about the origin or a \( 90^\circ \) clockwise rotation. But among the given options, \( R_{O,270^\circ} \) has the transformation rule \( (x,y)\to(y, - x) \) (when rotating counterclockwise by \( 270^\circ \), which is equivalent to rotating clockwise by \( 90^\circ \)). Let's verify with a point. Take \( (1,0) \). Applying \( (x,y)\to(y, - x) \), we get \( (0, - 1) \). Rotating \( (1,0) \) \( 270^\circ \) counterclockwise about the origin: The formula for counterclockwise rotation by \( \theta \) is \( x'=x\cos\theta - y\sin\theta \), \( y'=x\sin\theta + y\cos\theta \). For \( \theta = 270^\circ \), \( \cos270^\circ=0 \), \( \sin270^\circ=- 1 \). So \( x'=x\times0 - y\times(-1)=y \), \( y'=x\times(-1)+y\times0=-x \). So \( (x,y)\to(y, - x) \) which matches the given transformation. The \( 90^\circ \) counterclockwise rotation gives \( (x,y)\to(-y,x) \), \( 180^\circ \) gives \( (-x,-y) \), \( 360^\circ \) gives \( (x,y) \) (identity transformation).

Step2: Match with the given options

The transformation rule \( (x,y)\to(y, - x) \) is the same as the rule for \( R_{O,270^\circ} \) (rotation about the origin by \( 270^\circ \) counterclockwise).

Answer:

\( R_{O,270^\circ} \) (the option corresponding to \( R_{0,270^\circ} \) in the given choices)