Question

what is the rule for the reflection?
( \bigcirc r_{y=x}(x,y) \to (-y,-x) )
( \bigcirc r_{y=-x}(x,y) \to (-y,-x) )
( \bigcirc r_{y=x}(x,y) \to (y,x) )
( \bigcirc r_{y=-x}(x,y) \to (y,x) )

Explanation:

Step1: Identify coordinates of a point

Take point \( A(-2, 1) \) and its image \( A'(1, -2) \).

Step2: Analyze reflection rules

• For \( r_{y = x} \), the rule is \( (x, y) \to (y, x) \). Let's check: For \( A(-2, 1) \), applying \( r_{y = x} \) gives \( (1, -2) \), which matches \( A' \).
• For \( r_{y=-x} \), the rule is \( (x, y) \to (-y, -x) \). For \( A(-2, 1) \), applying this gives \( (-1, 2) \), which does not match \( A' \).
• The first option's rule for \( r_{y = x} \) is incorrect. The fourth option's rule for \( r_{y=-x} \) is incorrect.

Answer:

\( r_{y=x}(x, y) \to (y, x) \) (the third option: \( \boldsymbol{r_{y=x}(x, y) \to (y, x)} \))