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Question

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?

(1 point)
translate by what?

Answer 1

After rotating the rectangle in Quadrant III (the first rectangle with vertices at \((-5, -4)\), \((-1, -4)\), \((-1, -6)\), and \((-5, -6)\)) 90 degrees clockwise about the origin, the next step is to translate it to match the position of the rectangle in Quadrant I (the second rectangle with vertices at \((3, 4)\), \((5, 4)\), \((5, 8)\), and \((3, 8)\)).

To find the translation, we need to determine the vector that moves the center of the rotated rectangle to the center of the rectangle in Quadrant I.

1. Find the center of the rotated rectangle. The original rectangle has corners at \((-5, -4)\) and \((-1, -6)\); rotating these points will give new coordinates. After rotating 90 degrees clockwise, the following transformations occur:

   • \((-5, -4) \rightarrow (4, -5)\)
   • \((-1, -4) \rightarrow (4, -1)\)
   • \((-1, -6) \rightarrow (6, -1)\)
   • \((-5, -6) \rightarrow (6, -5)\)

   The center can be calculated as the average of the x-coordinates and the average of the y-coordinates:

   • Center after rotation = \(\left(\frac{(4 + 4 + 6 + 6)}{4}, \frac{(-5 - 1 - 1 - 5)}{4}\right) = \left(5, -3\right)\).

2. Find the center of the rectangle in Quadrant I:

   • The center of the rectangle \((3, 4)\) to \((5, 8)\) can be calculated:
   • Center = \(\left(\frac{(3 + 5)}{2}, \frac{(4 + 8)}{2}\right) = \left(4, 6\right)\).

3. Now, determine the translation vector that moves the rotated rectangle center \((5, -3)\) to the rectangle's center in Quadrant I \((4, 6)\):

   • Translation vector = \((4 - 5, 6 - (-3)) = (-1, 9)\).

So, you would translate the rotated rectangle by the vector \((-1, 9)\).