Question

find the coordinates of the vertices after the reflection over the line.
\\(g = (\quad, \quad)\\)
\\(h = (\quad, \quad)\\)
\\(i = (\quad, \quad)\\)
\\(f = (\quad, \quad)\\)

Explanation:

⚡ Using what you learned: reflections · transformations in the coordinate plane

Step 1: Identify original coordinates

Identify the coordinates of the vertices of the rectangle from the graph:

• \( E = (-2, 3) \)
• \( F = (5, 2) \)
• \( G = (5, -3) \)
• \( H = (-2, -3) \)

Step 2: Identify the line of reflection

The dashed line \( k \) passes through points such as \( (5, 0) \), \( (0, -5) \), and \( (-5, -10) \).
The equation of this line is:

$$ y = x - 5 $$

Step 3: Apply reflection rule

To reflect a point \( (x, y) \) over a line of the form \( y = x + c \):

• The new coordinates \( (x', y') \) are given by:

$$ x' = y - c $$

$$ y' = x + c $$

For the line \( y = x - 5 \), we have \( c = -5 \):

$$ x' = y + 5 $$

$$ y' = x - 5 $$

Apply this rule to each vertex:

• For \( G(5, -3) \):

$$ G' = (-3 + 5, 5 - 5) = (2, 0) $$

• For \( H(-2, -3) \):

$$ H' = (-3 + 5, -2 - 5) = (2, -7) $$

• For \( E(-2, 3) \):

$$ E' = (3 + 5, -2 - 5) = (8, -7) $$

• For \( F(5, 2) \):

$$ F' = (2 + 5, 5 - 5) = (7, 0) $$

Answer:

• \( G' = (2, 0) \)
• \( H' = (2, -7) \)
• \( E' = (8, -7) \)
• \( F' = (7, 0) \)