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: $(x, y) \rightarrow(\square)$
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Solution

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Solution Steps

Step 1: Identify the Rotation

To determine the rotation, observe the coordinates of the original pentagon GHIJK GHIJK and the rotated pentagon GHIJK 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)(y,x) (x, y) \rightarrow (-y, x)

Step 2: Apply the Rotation Rule

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

  • G(4,4)G((4),4)=G(4,4) G(4, -4) \rightarrow G'(-(-4), 4) = G'(4, 4)
  • H(5,2)H((2),5)=H(2,5) H(5, -2) \rightarrow H'(-(-2), 5) = H'(2, 5)
  • I(7,2)I((2),7)=I(2,7) I(7, -2) \rightarrow I'(-(-2), 7) = I'(2, 7)
  • J(7,4)J((4),7)=J(4,7) J(7, -4) \rightarrow J'(-(-4), 7) = J'(4, 7)
  • K(4,6)K((6),4)=K(6,4) K(4, -6) \rightarrow K'(-(-6), 4) = K'(6, 4)
Step 3: Identify the Translation

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

  • G(4,4)G(6,8) G'(4, 4) \rightarrow G''(6, 8)
  • H(2,5)H(4,9) H'(2, 5) \rightarrow H''(4, 9)
  • I(2,7)I(4,11) I'(2, 7) \rightarrow I''(4, 11)
  • J(4,7)J(6,11) J'(4, 7) \rightarrow J''(6, 11)
  • K(6,4)K(8,8) K'(6, 4) \rightarrow K''(8, 8)

The translation rule is: (x,y)(x+2,y+4) (x, y) \rightarrow (x + 2, y + 4)

Final Answer

  • Rotation: (x,y)(y,x)(x, y) \rightarrow (-y, x)
  • Translation: (x,y)(x+2,y+4)(x, y) \rightarrow (x + 2, y + 4)
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