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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.

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?

2 answers

To rotate the first rectangle (in Quadrant III) 90 degrees clockwise into Quadrant II, Darnell needs to perform a transformation on the coordinates of the rectangle.

The coordinates of the rectangle in Quadrant III are:

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

When rotating a point \( (x, y) \) 90 degrees clockwise, the transformation can be described by the formula: \[ (x, y) \rightarrow (y, -x) \]

Applying this transformation to each coordinate of the rectangle:

  1. For \( (-1, -2) \): \[ (-1, -2) \rightarrow (-2, 1) \]
  2. For \( (-3, -2) \): \[ (-3, -2) \rightarrow (-2, 3) \]
  3. For \( (-3, -7) \): \[ (-3, -7) \rightarrow (-7, 3) \]
  4. For \( (-1, -7) \): \[ (-1, -7) \rightarrow (-7, 1) \]

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

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

The next step after the rotation would be to check if this new rectangle (in Quadrant II) matches the dimensions of the original rectangle in Quadrant III and confirm if the two rectangles are indeed congruent by measuring the lengths of their sides. If the widths and heights are equal, it demonstrates that the rectangles are congruent, which confirms the task.

Thus, Darnell's next step is to verify the congruence of the rectangles by possibly comparing the lengths of their corresponding sides or calculating their dimensions.

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