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.

Question

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? translate ( )

Answer 1

To place the rotated rectangle into Quadrant II after Darnell rotates the rectangle in Quadrant III (first rectangle) 90 degrees clockwise, he needs to perform a translation to move the rectangle into Quadrant II.

The original rectangle in Quadrant III has corners at:

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

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

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

To translate the rotated rectangle into Quadrant II, Darnell should translate it right (positive x direction) and up (positive y direction).

To determine the translation, we can look at the position of the rectangle after rotation:

• The x-coordinates need to be increased enough to move to Quadrant II, where x-coordinates are positive.
• The y-coordinates can stay the same but need to be above the x-axis.

Assuming he wants to just move it into a visible position in Quadrant II, he could translate it by moving the figure a set distance right (for example, 3 units) and up (for example, 4 units).

So, the next step could be: translate (3, 4).

This will ensure the rectangle moves into Quadrant II with all its points having positive x-coordinates and appropriate y-coordinates.