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Triangle XYZ has vertices at X(−3, −3), Y(5, 5), Z(4, −6).

Part A: If triangle XYZ is rotated 90° clockwise about the origin, determine the coordinates of the vertices for the image, triangle X′Y′Z′. Show each step of your work. (4 points)

Part B: If triangle XYZ is rotated 270° counterclockwise about the origin, determine the coordinates of the vertices for the image, triangle X′Y′Z′. Show each step of your work. (4 points)

Part C: Describe the similarities and differences between the two rotations in parts A and B. (4 points)
(do only 3 sentences and dumb it down a lot and make it seem like an 8th grader wrote it)

1 answer

Part A: 90° Clockwise Rotation

To rotate a point (x, y) 90° clockwise about the origin, we use the rule (x, y) → (y, -x).

  1. For point X(-3, -3):

    • Apply the rule: (-3, -3) → (-3, 3)
    • New coordinates for X' are (-3, 3).

  2. For point Y(5, 5):

    • Apply the rule: (5, 5) → (5, -5)
    • New coordinates for Y' are (5, -5).

  3. For point Z(4, -6):

    • Apply the rule: (4, -6) → (-6, -4)
    • New coordinates for Z' are (-6, -4).

So, after the rotation, the coordinates of triangle X'Y'Z' are:

  • X'(-3, 3)
  • Y'(5, -5)
  • Z'(-6, -4)

Part B: 270° Counterclockwise Rotation

To rotate a point (x, y) 270° counterclockwise about the origin, we can use the same rule for a 90° clockwise rotation, but we reverse the points. The rule is: (x, y) → (-y, x).

  1. For point X(-3, -3):

    • Apply the rule: (-3, -3) → (3, -3)
    • New coordinates for X' are (3, -3).

  2. For point Y(5, 5):

    • Apply the rule: (5, 5) → (-5, 5)
    • New coordinates for Y' are (-5, 5).

  3. For point Z(4, -6):

    • Apply the rule: (4, -6) → (6, 4)
    • New coordinates for Z' are (6, 4).

So, after the rotation, the coordinates of triangle X'Y'Z' are:

  • X'(3, -3)
  • Y'(-5, 5)
  • Z'(6, 4)

Part C: Similarities and Differences

Both rotations move the points around the origin but in different directions. Part A turns the triangle 90 degrees to the right, while part B turns it 270 degrees to the left. The final positions of the points are different, showing how the same shape can look different depending on how you rotate it.