Questions: Triangle G'H'I' is the image of triangle GHI under a translation followed by a rotation 180° about the origin. Write the rules for the translation and rotation. Translation: (x, y) -> Rotation: (x, y) -> ( ,

Triangle G'H'I' is the image of triangle GHI under a translation followed by a rotation 180° about the origin.

Write the rules for the translation and rotation.
Translation: (x, y) ->
Rotation: (x, y) -> ( ,

Transcript text: Triangle $G^{\prime} H^{\prime} I^{\prime}$ is the image of triangle $G H I$ under a translation followed by a rotation $180^{\circ}$ about the origin. Write the rules for the translation and rotation. Translation: $(x, y) \mapsto$ $\square$ Rotation: $(x, y) \mapsto($ $\square$ , $\square$

Solution

Solution Steps

Step 1: Identify the Coordinates of the Original and Translated Points

Original points: $ G(-6, 8) $, $ H(-4, 8) $, $ I(-5, 3) $
Translated points: $ G'(5, -4) $, $ H'(7, -4) $, $ I'(6, -9) $

Step 2: Determine the Translation Vector

Calculate the difference between the original and translated points:
- $ G \rightarrow G' $: $ (5 - (-6), -4 - 8) = (11, -12) $
- $ H \rightarrow H' $: $ (7 - (-4), -4 - 8) = (11, -12) $
- $ I \rightarrow I' $: $ (6 - (-5), -9 - 3) = (11, -12) $
Translation vector: $ (x, y) \rightarrow (x + 11, y - 12) $

Step 3: Verify the Rotation

After translation, the points should be:
- $ G \rightarrow (5, -4) $
- $ H \rightarrow (7, -4) $
- $ I \rightarrow (6, -9) $
Check if these points match the rotated points:
- Rotation 180° about the origin: $ (x, y) \rightarrow (-x, -y) $
- $ G' \rightarrow (-5, 4) $
- $ H' \rightarrow (-7, 4) $
- $ I' \rightarrow (-6, 9) $
These points match the final positions of $ G', H', I' $.