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Congruent Figures Practice
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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?

(1 point)
translate by (
)

1 answer

To find the translation needed to determine that the rectangle in Quadrant III is congruent to the rectangle in Quadrant I after rotating it 90 degrees clockwise about the origin, we first analyze the coordinates of the rectangles.

  1. The first rectangle in Quadrant III has vertices:

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

  2. When you rotate this rectangle 90 degrees clockwise around the origin, the coordinates become:

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

  3. The second rectangle in Quadrant I has vertices:

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

To make the rotated rectangle match the second rectangle's position, you need to translate the fourth quadrant rectangle's vertices from \((-4, 5)\) and \((-6, 5)\) into the first quadrant locations of \((3, 4)\) and \((5, 4)\).

To do this, you can find the translation vector by calculating the difference between the coordinates of vertex points of the rotated rectangle and the target rectangle:

For the point \((-4, 5)\) to translate to \((3, 4)\):

  • The translation in the x-direction is \(3 - (-4) = 7\).
  • The translation in the y-direction is \(4 - 5 = -1\).

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

Thus, the next step after rotating the rectangle 90 degrees clockwise about the origin is to translate by \( (7, -1) \).