Triangle HGI, H = (4, 0), G = (3, -3), I = (0, -4) is reflected over the line x−axis to produce G'H'I'. then G'H'I' is rotated 90° to produce G''H''I''. Determine the coordinates of G'H'I' and G''H''I''. Support how you found those new coordinates with mathematical calculations and/or explanations.

Answer 1

To find the coordinates of the reflected triangle \( G'H'I' \) after reflecting triangle \( HGI \) over the x-axis, we will apply the reflection formula. When reflecting a point \( (x, y) \) over the x-axis, the new coordinates become \( (x, -y) \).

Step 1: Reflection over the x-axis

• Point H = (4, 0):

  • After reflection: \( H' = (4, -0) = (4, 0) \) (remains the same since it lies on the x-axis).

• Point G = (3, -3):

  • After reflection: \( G' = (3, 3) \).

• Point I = (0, -4):

  • After reflection: \( I' = (0, 4) \).

Thus, after reflection over the x-axis, the coordinates are:

• \( G' = (3, 3) \)
• \( H' = (4, 0) \)
• \( I' = (0, 4) \)

Coordinates of G'H'I':

\[ G'H'I' = (3, 3), (4, 0), (0, 4) \]

Step 2: Rotation by 90°

To rotate a point \( (x, y) \) by 90 degrees counterclockwise about the origin, the new coordinates are given by \( (-y, x) \).

• Point G' = (3, 3):

  • After 90° rotation: \[ G'' = (-3, 3) \]

• Point H' = (4, 0):

  • After 90° rotation: \[ H'' = (0, 4) \]

• Point I' = (0, 4):

  • After 90° rotation: \[ I'' = (-4, 0) \]

Thus, after the 90° rotation, the coordinates are:

• \( G'' = (-3, 3) \)
• \( H'' = (0, 4) \)
• \( I'' = (-4, 0) \)

Coordinates of G''H''I'' :

\[ G''H''I'' = (-3, 3), (0, 4), (-4, 0) \]

Summary of Results

After completing the reflection and rotation operations, the final coordinates of triangles are:

• \( G'H'I' = (3, 3), (4, 0), (0, 4) \)
• \( G''H''I'' = (-3, 3), (0, 4), (-4, 0) \)