Question

which statements about a 90 - degree counterclockwise rotation are correct?
□ the new x - coordinate is the original y - coordinate.
□ the new y - coordinate is the original x - coordinate.
□ the rotated point is in the same quadrant as the original point.
□ the new x - coordinate is the opposite of the original y - coordinate.
□ the new y - coordinate is the opposite of the original x - coordinate.
□ the original horizontal line segment of the hook forms a square corner with its rotation.

Explanation:

Step1: Recall rotation rule

For a 90 - degree counter - clockwise rotation of a point $(x,y)$ about the origin, the new coordinates $(x',y')$ are given by $(x',y')=(-y,x)$.

Step2: Analyze each statement

• For the statement "The new x - coordinate is the original y - coordinate": From $(x',y') = (-y,x)$, the new $x$ - coordinate $x'=-y$, so this is false.
• For the statement "The new y - coordinate is the original x - coordinate": From $(x',y')=(-y,x)$, the new $y$ - coordinate $y' = x$, so this is true.
• For the statement "The rotated point is in the same quadrant as the original point": A 90 - degree counter - clockwise rotation moves a point to a different quadrant, so this is false.
• For the statement "The new x - coordinate is the opposite of the original y - coordinate": From $(x',y')=(-y,x)$, $x'=-y$, so this is true.
• For the statement "The new y - coordinate is the opposite of the original x - coordinate": From $(x',y')=(-y,x)$, $y'=x$, not $-x$, so this is false.
• For the statement "The original horizontal line segment of the hook forms a square corner with its rotation": A 90 - degree rotation results in perpendicular lines, so this is true.

Answer:

The new y - coordinate is the original x - coordinate; The new x - coordinate is the opposite of the original y - coordinate; The original horizontal line segment of the hook forms a square corner with its rotation.