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.

Question

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? translate by

Answer 1

After rotating the first rectangle (Quadrant III) 90 degrees clockwise about the origin, the next step to determine that it is congruent to the rectangle in Quadrant I is to translate the rotated rectangle.

To find the translation that will align the rotated rectangle with the rectangle in Quadrant I, you need to determine the new coordinates after the rotation and then find the vector needed to move it to the desired position.

1. Determine the new coordinates after rotation: The original coordinates of the rectangle in Quadrant III are:

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

   After rotating these points 90 degrees clockwise, the new coordinates will be:

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

2. Determine the translation vector: The rectangle in Quadrant I has coordinates:

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

   To translate the new coordinates from the rotation to match one of the rectangles in Quadrant I, we need to move from the position of the rotated rectangle (let's use the point \((-4, 5)\)) to a corresponding point in Quadrant I (for example, \((3, 4)\)).

   The translation vector is calculated as follows:

   • To go from \((-4, 5)\) to \((3, 4)\), you would translate by:
     • \(x\) direction: \(3 - (-4) = 7\)
     • \(y\) direction: \(4 - 5 = -1\)

   Therefore, the translation vector is \((7, -1)\).

3. Perform the translation: Add the translation vector \((7, -1)\) to each of the rotated coordinates.

In summary, after rotating the rectangle 90 degrees clockwise, the next step is to translate it by \((7, -1)\) to determine whether it is congruent to the rectangle in Quadrant I.