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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?

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To determine if the rectangle in Quadrant III is congruent to the rectangle in Quadrant I after rotating the first rectangle 90 degrees clockwise about the origin, the next step would be to find the new coordinates of the vertices of the first rectangle after the rotation.

The coordinates of the first rectangle are:

  1. \((-5, -4)\)
  2. \((-1, -4)\)
  3. \((-1, -6)\)
  4. \((-5, -6)\)

When a point \((x, y)\) is rotated 90 degrees clockwise about the origin, the new coordinates become \((y, -x)\).

Applying this transformation to each vertex results in:

  1. \((-4, 5)\) from \((-5, -4)\)
  2. \((-4, 1)\) from \((-1, -4)\)
  3. \((-6, 1)\) from \((-1, -6)\)
  4. \((-6, 5)\) from \((-5, -6)\)

The next step would be to verify whether these new coordinates form a rectangle that has the same dimensions (length and width) as the rectangle defined by the coordinates in Quadrant I, which are given as:

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

You would check:

  1. The lengths of the sides of the rectangles.
  2. The respective angles between sides (which should be right angles).

If both conditions are satisfied (same side lengths and right angles), then the rectangles are congruent.