Question

windmill: an architect is designing a new windmill with four sails. in her sketch, the sails center of rotation is the origin, (0,0), and the tip of one of the sails, point q, has coordinates (-2,6). she wants to make another sketch that shows the windmill after the sails have rotated 270° about their center of rotation. what would be the coordinates of q? the coordinates of q would be \boxed{} (type an ordered pair.)

Explanation:

To solve the problem of finding the coordinates of point \( Q(-2, 9) \) after a \( 270^\circ \) rotation about the origin, we use the rotation rule for a \( 270^\circ \) counterclockwise (or \( 90^\circ \) clockwise) rotation.

Step 1: Recall the Rotation Rule

For a \( 270^\circ \) counterclockwise rotation about the origin, the transformation rule for a point \( (x, y) \) is:
\[
(x, y) \xrightarrow{270^\circ \text{ rotation}} (y, -x)
\]

Step 2: Apply the Rule to Point \( Q(-2, 9) \)

For \( Q(-2, 9) \), we identify \( x = -2 \) and \( y = 9 \). Substitute these into the rotation rule:

• The new \( x \)-coordinate is \( y = 9 \).
• The new \( y \)-coordinate is \( -x = -(-2) = 2 \).

Thus, the coordinates of \( Q \) after a \( 270^\circ \) rotation about the origin are \( (9, 2) \).

Answer:

\((9, 2)\)