Question

a star is shown above point q, with three additional stars r, s, and t. which figure shows where the black star will be if it is rotated 90 degrees counterclockwise about q? choose the correct answer. a) figure r b) figure s c) figure t

Explanation:

Step1: Understand rotation direction

Counterclockwise rotation about point \( Q \) by \( 90^\circ \) means we rotate the figure (the black star, though here we analyze the star positions) such that each point moves \( 90^\circ \) counterclockwise around \( Q \).

Step2: Analyze star positions

Looking at the stars \( R, S, T \) and the original star (not labeled but the one to rotate). When we rotate a star (or the figure) \( 90^\circ \) counterclockwise about \( Q \), we check the orientation and position. The star \( S \) is in a position that, after \( 90^\circ \) counterclockwise rotation about \( Q \), would align with the rotated position. Wait, actually, let's visualize: if we consider the center \( Q \), a \( 90^\circ \) counterclockwise rotation. Let's assume the original star (the one to rotate, maybe the unlabeled one? Wait, no, the question is about the black star (maybe the unlabeled one) rotating 90 degrees counterclockwise about \( Q \). Wait, the options are Figure \( R \), \( S \), \( T \)? Wait, no, the options are A) Figure \( R \), B) Figure \( S \), C) Figure \( T \)? Wait, the image shows stars \( T \), \( S \), \( R \) around \( Q \). Wait, maybe the black star (the one at the position, maybe the unlabeled star) when rotated 90 degrees counterclockwise about \( Q \) lands on the position of star \( S \)? Wait, no, let's think of the angle. A 90-degree counterclockwise rotation: if we have a point, the rotation matrix is \( \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} \) for \( \theta = 90^\circ \), so \( \cos90 = 0 \), \( \sin90 = 1 \), so the matrix is \( \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \). But maybe simpler: look at the positions. The star \( T \) is at the top, \( S \) at the left, \( R \) at the bottom. Rotating a star (say, the one at the right, unlabeled) 90 degrees counterclockwise about \( Q \) would move it to the top? No, wait, maybe the black star is the one we rotate. Wait, the question says "where the black star will be if it is rotated 90 degrees counterclockwise about \( Q \)". So let's assume the black star is at a position, and we rotate it. Let's look at the options: A) Figure \( R \), B) Figure \( S \), C) Figure \( T \). Wait, maybe the original black star is at the position of the unlabeled star (right), rotating 90 degrees counterclockwise about \( Q \) (center) would move it to the position of star \( S \) (left)? No, wait, 90 degrees counterclockwise: from right, rotating 90 degrees counterclockwise would go to top? Wait, no, directions: counterclockwise is left turn. So if you are at the right (east), rotating 90 degrees counterclockwise (left turn) would face north (top). But the star \( T \) is at top. Wait, maybe I got the stars wrong. Wait, the stars are labeled \( T \) (top), then the right star (unlabeled), \( R \) (bottom), \( S \) (left). So the black star (maybe the right one) when rotated 90 degrees counterclockwise about \( Q \) would move to the top (star \( T \))? No, that doesn't match. Wait, maybe the black star is at the position of star \( T \)? No, the question says "a star is shown above Point \( Q \), with three additional stars \( R, S, \) and \( T \)". So the original star is above \( Q \) (maybe \( T \) is above? Wait, the image: \( T \) is top, \( S \) is left, \( R \) is bottom, and a right star. So the original star (above \( Q \)) is \( T \)? No, the question says "a star is shown above Point \( Q \), with three additional stars \( R, S, \) and \( T \)". So the original star is above \( Q \) (let's say at the top, \( T \) is there? Wait, no, the stars are \( T \) (top), right (unlabeled), \( R \) (bottom), \( S \) (left). So the original star (above \( Q \)) is maybe the top one (\( T \))? No, the question is about rotating the black star (maybe the one at the right) 90 degrees counterclockwise about \( Q \). Wait, maybe the correct answer is B) Figure \( S \)? Wait, no, let's re-express. When you rotate a figure 90 degrees counterclockwise about a center, each vertex moves 90 degrees counterclockwise. Let's take the center \( Q \), and a point on the star. If we rotate the star (say, the right star) 90 degrees counterclockwise, its position would be the left star ( \( S \))? Wait, no, 90 degrees counterclockwise from right (east) is north (top), but \( T \) is top. Wait, maybe I messed up the direction. Wait, clockwise 90 degrees is right turn, counterclockwise is left turn. So from east (right), left turn 90 degrees is north (top), right turn 90 degrees is south (bottom). So if the black star is at east (right), rotating counterclockwise 90 degrees would go to north (top, \( T \))? But that's not an option? Wait, the options are A) Figure \( R \), B) Figure \( S \), C) Figure \( T \). Wait, maybe the original black star is at north ( \( T \)), rotating counterclockwise 90 degrees would go to west (left, \( S \))? Yes! Because north (top) rotating counterclockwise 90 degrees (left turn) would face west (left). So the star at north ( \( T \)) when rotated 90 degrees counterclockwise about \( Q \) would move to the left ( \( S \))? Wait, no, the rotation is of the black star. Wait, maybe the black star is at the position of \( T \), and rotating it 90 degrees counterclockwise about \( Q \) would place it at \( S \)'s position. So the correct answer is B) Figure \( S \).

Answer:

B) Figure \( S \)