Questions: Pentagon G' H' I' J' K' is the image of pentagon GHIJK under a rotation about the origin followed by a translation. Write the rules for the rotation and translation. Rotation: (x, y) →(, Translation: (x, y) →()

Pentagon G' H' I' J' K' is the image of pentagon GHIJK under a rotation about the origin followed by a translation.

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

Transcript text: Pentagon $G^{\prime} H^{\prime} I^{\prime} J^{\prime} K^{\prime}$ is the image of pentagon GHIJK under a rotation about the origin followed by a translation. Write the rules for the rotation and translation. Rotation: $(x, y) \rightarrow(\rrbracket$, $\square$ Translation: $(x, y) \rightarrow(\square)$

Solution

Solution Steps

Step 1: Identify the Rotation

To determine the rotation, observe the coordinates of the original pentagon $ GHIJK $ and the rotated pentagon $ G'H'I'J'K' $. Notice that the pentagon has been rotated 90 degrees counterclockwise about the origin. The rule for a 90-degree counterclockwise rotation is: \[ (x, y) \rightarrow (-y, x) \]

Step 2: Apply the Rotation Rule

Apply the rotation rule to the coordinates of the original pentagon $ GHIJK $:

$ G(4, -4) \rightarrow G'(-(-4), 4) = G'(4, 4) $
$ H(5, -2) \rightarrow H'(-(-2), 5) = H'(2, 5) $
$ I(7, -2) \rightarrow I'(-(-2), 7) = I'(2, 7) $
$ J(7, -4) \rightarrow J'(-(-4), 7) = J'(4, 7) $
$ K(4, -6) \rightarrow K'(-(-6), 4) = K'(6, 4) $

Step 3: Identify the Translation

Compare the coordinates of the rotated pentagon $ G'H'I'J'K' $ with the final pentagon $ G''H''I''J''K'' $. The translation rule can be determined by finding the difference between corresponding points: