Question

write the coordinates of the vertices after a reflection over the line $y = x$.\
$b(\square,\square)$\
$c(\square,\square)$\
$d(\square,\square)$\
$e(\square,\square)$

Explanation:

Step1: Find original coordinates

First, we determine the original coordinates of points \( B \), \( C \), \( D \), and \( E \) from the graph.

• For point \( B \): Looking at the grid, the \( x \)-coordinate is \(-5\) (since it's 5 units left of the origin on the \( x \)-axis) and the \( y \)-coordinate is \( 0 \) (on the \( x \)-axis), so \( B(-5, 0) \).
• For point \( C \): The \( x \)-coordinate is \(-5\) and the \( y \)-coordinate is \( 8 \), so \( C(-5, 8) \).
• For point \( D \): The \( x \)-coordinate is \(-4\) and the \( y \)-coordinate is \( 8 \), so \( D(-4, 8) \).
• For point \( E \): The \( x \)-coordinate is \(-4\) and the \( y \)-coordinate is \( 0 \), so \( E(-4, 0) \).

Step2: Apply reflection over \( y = x \)

The rule for reflecting a point \((x, y)\) over the line \( y = x \) is to swap the \( x \)- and \( y \)-coordinates, so the new coordinates \((x', y')\) are given by \( (y, x) \).

• For \( B(-5, 0) \): Swapping \( x \) and \( y \), we get \( B'(0, -5) \).
• For \( C(-5, 8) \): Swapping \( x \) and \( y \), we get \( C'(8, -5) \).
• For \( D(-4, 8) \): Swapping \( x \) and \( y \), we get \( D'(8, -4) \).
• For \( E(-4, 0) \): Swapping \( x \) and \( y \), we get \( E'(0, -4) \).

Answer:

\( B'(0, -5) \)
\( C'(8, -5) \)
\( D'(8, -4) \)
\( E'(0, -4) \)