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The rule T1, -4 CompositionRO, 180°(x, y) is applied to rectangle KLMN.

On a coordinate plane, 5 rectangles are shown. Rectangle K L M N has points (3, negative 4), (3, negative 1), (5, negative 1), (5, negative 4). Rectangle 1 has points (negative 4, 6), (negative 4, 4), (negative 1, 4), (negative 1, 6). Rectangle 2 has points (0, 0), (negative 2, 0), (negative 2, 3), (0, 3). Rectangle 3 has points (negative 2, 0), (negative 4, 0), (negative 2, negative 3), (negative (4, negative 3). Rectangle 4 has points (1, 0), (4, 0), (1, 2), (4, 2).
Which rectangle shows the final image?

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1 answer

To apply the transformation specified by the rule T1, -4 CompositionRO, 180°(x, y) to rectangle KLMN, we'll first break it down into its components.

  1. 180° Rotation: This transformation rotates the rectangle KLMN around the origin (0, 0) by 180 degrees. To rotate a point (x, y) by 180 degrees, the new coordinates will be (-x, -y).

    The points of rectangle KLMN are:

    • K(3, -4)
    • L(3, -1)
    • M(5, -1)
    • N(5, -4)

    Applying 180° rotation:

    • K'(3, -4) → K''(-3, 4)
    • L'(3, -1) → L''(-3, 1)
    • M'(5, -1) → M''(-5, 1)
    • N'(5, -4) → N''(-5, 4)

    So the points after rotation are:

    • K''(-3, 4)
    • L''(-3, 1)
    • M''(-5, 1)
    • N''(-5, 4)

  2. Translation: Now we apply the translation T1, -4. This means we will be translating the shape by 1 unit in the x-direction (no change since a positive x translation shifts right) and -4 units in the y-direction (down).

    We will translate the rotated points:

    • K''(-3, 4) → K''(-3 + 1, 4 - 4) → K'''(-2, 0)
    • L''(-3, 1) → L''(-3 + 1, 1 - 4) → L'''(-2, -3)
    • M''(-5, 1) → M''(-5 + 1, 1 - 4) → M'''(-4, -3)
    • N''(-5, 4) → N''(-5 + 1, 4 - 4) → N'''(-4, 0)

    The final coordinates after the rotation and translation are:

    • K'''(-2, 0)
    • L'''(-2, -3)
    • M'''(-4, -3)
    • N'''(-4, 0)

Now, we can compare these coordinates with the given rectangles:

Rectangle 3 has points:

  • (-2, 0)
  • (-4, 0)
  • (-2, -3)
  • (-4, -3)

Seeing that the final coordinates of rectangle KLMN after applying the transformations match the points of Rectangle 3, the final image is Rectangle 3.