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A coordinate plane with 4 quadrants shows x and y axes ranging from negative 8 to 8 in increments of 1. Two rectangles are formed by lines joining four plotted points each. The coordinates of the plotted points for first rectangle joined by solid lines are left parenthesis negative 1 comma negative 2 right parenthesis, left parenthesis negative 3 comma negative 2 right parenthesis, left parenthesis negative 3 comma negative 7 right parenthesis, and left parenthesis negative 1 comma negative 7 right parenthesis. The coordinates for the second rectangle joined by lines made of dashes and dots are left parenthesis 2 comma 3 right parenthesis, left parenthesis 7 comma 3 right parenthesis, left parenthesis 7 comma 5 right parenthesis, and left parenthesis 2 comma 5 right parenthesis.

To show that the rectangle in Quadrant III is congruent to the rectangle in Quadrant I, Darnell rotates the first rectangle 90 degrees clockwise into Quadrant II. What is his next step?

(1 point)
translate (
)

1 answer

To rotate the rectangle from Quadrant III (which has vertices at \((-1, -2)\), \((-3, -2)\), \((-3, -7)\), and \((-1, -7)\)) 90 degrees clockwise into Quadrant II, Darnell would need to translate the rectangle to the correct position.

The rotation of 90 degrees clockwise moves points from \((x, y)\) to \((y, -x)\). Thus, after the rotation, the new coordinates of the rectangle would be:

  • \((-2, 1)\)
  • \((-2, 3)\)
  • \((-7, 3)\)
  • \((-7, 1)\)

However, this step places the rectangle in Quadrant IV. To translate the rotated rectangle into Quadrant II, Darnell needs to move it leftward and upward.

The first rectangle's dimensions must correspond to the second rectangle, which has vertices located at:

  • \((2, 3)\)
  • \((7, 3)\)
  • \((7, 5)\)
  • \((2, 5)\)

The next step of Darnell, after rotating the first rectangle, would then be to translate these coordinates appropriately to align with the configuration of the second rectangle, ensuring that they fit correctly within the coordinate plane of Quadrant II.

Thus, the translation can be described as moving the rectangle leftward by 4 units and upward by 2 units.

In notation, it would be translated as follows:

Translate \((-4, +2)\).

So the final answer for your question "translate ( )" is:

translate (-4, 2).