Q

QuestionMathematics

The rotation rule is given by (x, y) → (new x, new y). In this problem, we need to find which of the four options provided is the correct rule for a 90° counterclockwise rotation.

In a 90° counterclockwise rotation, the x-coordinate becomes the negative of the original y-coordinate, and the y-coordinate becomes the negative of the original x-coordinate. This can be written as: new x = -y new y = -x

Now, let's compare the given rules with the understood transformation. We need to find which rule matches new x = -y and new y = -x. Option 1: (x, y) → (−x, −y) This option does not match the transformation because it changes the signs of both x and y, but we only need the signs changed for the y-coordinate. Option 2: (x, y) → (−y, x) This option matches the transformation. The new x-coordinate is the negative of the original y-coordinate (−y), and the new y-coordinate is the negative of the original x-coordinate (x). Option 3: (x, y) → (y, −x) This option does not match the transformation because it keeps the same sign for the x-coordinate, but we need the negative of the original x-coordinate. Option 4: (x, y) → (x, −y) This option does not match the transformation because it changes the signs of both x and y, but we only need the signs changed for the y-coordinate.

Based on the comparison in Step 3, the correct rotation rule is option 2: (x, y) → (−y, x).

Now, we can use the correct rotation rule to find the new coordinate of A′. We are given that A is at the coordinate (5, −1). Applying the rotation rule: A′ = (−y, x) = (−(- 1), 5) = (1, 5)