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A coordinate plane with 4 quadrants shows x and y axes ranging from negative 9 to 9 in increments of 1. Three 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 5 comma negative 4 right parenthesis, left parenthesis negative 1 comma negative 4 right parenthesis, left parenthesis negative 1 comma negative 6 right parenthesis, and left parenthesis negative 5 comma negative 6 right parenthesis. The coordinates for the second rectangle joined by lines made of dashes and dots are left parenthesis 3 comma 4 right parenthesis, left parenthesis 5 comma 4 right parenthesis, left parenthesis 5 comma 8 right parenthesis, and left parenthesis 3 comma 8 right parenthesis. The coordinates of the plotted points for the third rectangle joined by dotted lines are left parenthesis negative 4 comma 1 right parenthesis, left parenthesis negative 6 comma 1 right parenthesis, left parenthesis negative 6 comma 5 right parenthesis, and left parenthesis negative 4 comma 5 right parenthesis.
The first step to determine that the rectangle in Quadrant III is congruent to the rectangle in Quadrant I was to rotate the first rectangle 90 degrees clockwise about the origin. What is the next step?

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1 answer

To determine the translation needed after rotating the first rectangle (in Quadrant III) 90 degrees clockwise about the origin, we need to identify the coordinates of the resulting points after rotation and then find the translation needed to align them with the second rectangle (in Quadrant I).

After rotating the first rectangle represented by the points:

  • (−5, −4)
  • (−1, −4)
  • (−1, −6)
  • (−5, −6)

The new coordinates after a 90-degree clockwise rotation about the origin will be:

  • (−4, 5) → (x, y) → (y, -x)
  • (−4, −1)
  • (−6, −1)
  • (−6, 5)

Now, we will check the coordinates and compare them with the rectangle in Quadrant I, represented by:

  • (3, 4)
  • (5, 4)
  • (5, 8)
  • (3, 8)

To align (−4, 5) with (3, 4), we need to determine what translation is needed.

To find this translation, calculate the change in x and the change in y:

  • Translation for x: 3 - (−4) = 3 + 4 = 7
  • Translation for y: 4 - 5 = -1

Thus, the translation vector will be (7, -1).

So, the next step is: translate by (7, -1)