Question

which diagram shows quadrilateral vwxy rotated 90° counterclockwise about w?

Explanation:

Step1: Recall Rotation Rules

A 90° counterclockwise rotation about a point \( W \) means for each point \( P \) (other than \( W \)) in the quadrilateral, we rotate the segment \( WP \) 90° counterclockwise. The rule for a 90° counterclockwise rotation about a point \((x_0,y_0)\) for a point \((x,y)\) is \((x - x_0, y - y_0)\) becomes \((-(y - y_0), x - x_0)\) (relative to \( W \) as the origin). For a square (quadrilateral \( VWXY \) is a square, so sides are equal and angles 90°), rotating \( VW \) 90° counterclockwise about \( W \): vector \( \overrightarrow{WV} \) (from \( W \) to \( V \)) should rotate 90° counterclockwise. If \( WV \) is a vertical or horizontal segment, rotating 90° counterclockwise changes its direction (e.g., right becomes up, down becomes right, etc.).

Step2: Analyze Each Diagram

- **First Diagram**: Check the rotated vertices. The blue quadrilateral has \( W \) as the center. The segment \( WV \) rotated 90° counterclockwise: if \( WV \) is, say, downward (from \( W \) to \( V \) in original), rotating 90° counterclockwise would make \( WV' \) to the right? Wait, no—wait, original quadrilateral \( VWXY \): \( W \) is a vertex, \( V \) is below \( W \), \( X \) is to the right of \( W \), \( Y \) is below \( X \) (assuming square). Rotating 90° counterclockwise about \( W \): \( V \) (below \( W \)) should go to a position such that \( WV' \) is 90° counterclockwise from \( WV \). If \( WV \) is vertical down, 90° counterclockwise rotation would make \( WV' \) horizontal left? Wait, no—vector from \( W \) to \( V \): if \( W \) is (0,0), \( V \) is (0,-1), then 90° counterclockwise rotation is (1,0) (since \((0,-1)\) rotated 90° counterclockwise is (1,0)). Wait, maybe better to look at the shape. The first diagram's blue quadrilateral: \( WX' \) should be 90° counterclockwise from \( WX \). Original \( WX \) is, say, left (from \( W \) to \( X \) in original? Wait, original quadrilateral: \( X \) is top-left, \( W \) is top-right, \( V \) is bottom-right, \( Y \) is bottom-left (square). So \( WX \) is left (from \( W \) to \( X \)), \( WV \) is down (from \( W \) to \( V \)). Rotating 90° counterclockwise about \( W \): \( X \) (left of \( W \)) should go to a position above \( W \) (since 90° counterclockwise rotation of left (vector \((-a,0)\)) is \((0,a)\) (upward)), and \( V \) (down from \( W \), vector \((0,-b)\)) should go to a position right (vector \((b,0)\)). Wait, maybe the first diagram's blue quadrilateral has \( X' \) to the right of \( W \) and \( V' \) to the right? No, wait, let's check the original: \( VWXY \) is a square with \( X \) connected to \( W \), \( W \) to \( V \), \( V \) to \( Y \), \( Y \) to \( X \). Rotating 90° counterclockwise about \( W \): the side \( WX \) (from \( W \) to \( X \)) should rotate 90° counterclockwise, so \( WX' \) is perpendicular to \( WX \) counterclockwise. If \( WX \) is horizontal left (from \( W \) to \( X \) in original), 90° counterclockwise rotation is vertical up. Wait, maybe the first diagram is correct? Wait, no—wait the second diagram has \( W' \), which is wrong because \( W \) should be the center (not moving, since rotation about \( W \), so \( W \) is fixed). So diagrams with \( W' \) (second diagram) are invalid because \( W \) is the center of rotation, so \( W \) should not move. So second diagram is out (has \( W' \), meaning \( W \) moved, which is wrong for rotation about \( W \)). Third diagram: check the rotated vertices—does \( W \) stay? Yes, but the shape: the rotated quadrilateral's sides—original \( WX \) is, say, top, rotated \( WX' \) (or \( WV' \))—maybe not. Fourth diagram: \( W \) is connected, but the shape looks off. Wait, the first diagram: \( W \) is fixed, the blue quadrilateral is the rotation. The original black quadrilateral and blue quadrilateral: \( W \) is common. The vector \( WV \) (black) and \( WV' \) (blue) should be 90° counterclockwise. In the first diagram, when you rotate \( VWXY \) 90° counterclockwise about \( W \), the new quadrilateral (blue) should have \( W \) as a vertex, and the sides rotated 90° counterclockwise. The first diagram shows this: the blue quadrilateral is rotated 90° counterclockwise, with \( W \) as the center, and the vertices match the rotation (e.g., \( V \) to \( V' \), \( X \) to \( X' \), \( Y \) to \( Y' \)) with \( W \) fixed. The second diagram has \( W \) moved (labeled \( W' \)), so invalid. Third and fourth: check the direction of rotation. The first diagram's blue quadrilateral is rotated 90° counterclockwise (e.g., the angle between \( WV \) and \( WV' \) is 90° counterclockwise, and the side lengths are equal, so it's a square rotated correctly about \( W \)).

Answer:

The first diagram (the top - most diagram among the four options, with the blue quadrilateral and \( W \) as the center of rotation)