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?
rectangles vertex's
(-1,-7) (-3,-7) (-1,-2) (-3,-2)
To complete the process of showing that the rectangle in Quadrant III is congruent to the rectangle in Quadrant I, Darnell's next step would be to translate the rotated rectangle to align it with the original rectangle in Quadrant I.
In this case, the rotated rectangle's vertices after a 90 degrees clockwise rotation would be:
(-7, -3) (-7, -1) (-2, -3) (-2, -1)
Darnell would then need to translate the rotated rectangle by moving it 6 units to the right and 4 units down to align it with the original rectangle in Quadrant I. Therefore, the next step would be to add 6 to the x-coordinate and subtract 4 from the y-coordinate of each vertex:
(-1, -7) (-1, -3) (-7, -7) (-7, -3)
By performing this translation, the rotated rectangle would now be congruent and aligned with the original rectangle in Quadrant I.
wrong
I apologize for the incorrect response.
Since the rotation is 90 degrees clockwise, Darnell needs to perform a reflection across the x-axis to align the rotated rectangle with the original rectangle in Quadrant I. This is because a 90-degree clockwise rotation is equivalent to a 90-degree counterclockwise rotation, followed by a reflection across the x-axis.
To find the vertices of the reflected rectangle, we need to change the sign of the y-coordinates while keeping the x-coordinates the same:
(-7, 3), (-7, 1), (-2, 3), (-2, 1)
Now, Darnell needs to translate the reflected rectangle to align it with the original rectangle in Quadrant I. The translation involves moving the rectangle 6 units to the right and 4 units down.
Adding 6 to the x-coordinate and subtracting 4 from the y-coordinate for each vertex, we get:
(-1, -1), (-1, 1), (-6, -1), (-6, 1)
Therefore, Darnell's next step would be to translate the reflected rectangle to the coordinates (-1, -1), (-1, 1), (-6, -1), (-6, 1) to align it with the original rectangle in Quadrant I.
the rectangles vertex's are actually
(2, 3) (2, 5) (7, 5) (7, 3)
I apologize for the confusion. If the vertices of the rectangle in Quadrant I are (2, 3), (2, 5), (7, 5), and (7, 3), and the rectangle in Quadrant III is congruent to it by rotating it 90 degrees clockwise into Quadrant II, the new vertices after the rotation would be:
(-3, 2), (-3, 7), (-2, 2), and (-2, 7).
To align the rotated rectangle with the original rectangle in Quadrant I, you would need to perform a reflection across the y-axis, instead of translating it.
The reflected coordinates would be:
(3, 2), (3, 7), (2, 2), and (2, 7).
Therefore, Darnell's next step would be to reflect the rotated rectangle across the y-axis to get the vertices (3, 2), (3, 7), (2, 2), and (2, 7) to align it with the original rectangle in Quadrant I.