Question

complete the mapping of the vertices of δdef.

• d(2, −4) → d dropdown
• e(1, −1) → e dropdown
• f(5, 1) → f dropdown

what is the rule that describes a reflection across the line y = x?
( r_{x=y}(x, y) ) → dropdown

Explanation:

Step1: Reflect \( D(2, -4) \) over \( y = x \)

To reflect a point \((x, y)\) over the line \( y = x \), we swap the \( x \)- and \( y \)-coordinates. So for \( D(2, -4) \), the new coordinates \( D' \) will be \((-4, 2)\).

Step2: Reflect \( E(1, -1) \) over \( y = x \)

Using the same rule (swap \( x \) and \( y \)), for \( E(1, -1) \), the new coordinates \( E' \) will be \((-1, 1)\).

Step3: Reflect \( F(5, 1) \) over \( y = x \)

Swapping the coordinates of \( F(5, 1) \), we get \( F' = (1, 5) \).

Step4: Determine the reflection rule over \( y = x \)

The general rule for reflecting a point \((x, y)\) across the line \( y = x \) is to interchange the \( x \)- and \( y \)-coordinates, so the rule is \((x, y) \to (y, x)\).

Answer:

• \( D(2, -4) \to D'(-4, 2) \)
• \( E(1, -1) \to E'(-1, 1) \)
• \( F(5, 1) \to F'(1, 5) \)
• The rule \( r_{x = y}(x, y) \to (y, x) \)