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?

Answer 1

The next step after rotating the first rectangle (in Quadrant III) 90 degrees clockwise about the origin is to determine the new coordinates of the rectangle after the rotation.

For a rotation of 90 degrees clockwise, you can convert the coordinates of each point \((x, y)\) to their new position \((y, -x)\).

For the first rectangle with coordinates:

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

After finding the new coordinates, compare these coordinates with the coordinates of the second rectangle in Quadrant I, which are:

1. \((3, 4)\)
2. \((5, 4)\)
3. \((5, 8)\)
4. \((3, 8)\)

This will help to confirm whether the rectangles are congruent by checking if they have the same dimensions (lengths and widths).

Lastly, check for alignment and dimensions to ensure they match, confirming their congruence after rotation.