Question

graph the image of rectangle qrst after a rotation 90° counterclockwise around the origin.

Explanation:

Step1: Identify coordinates of Q, R, S, T

First, find the coordinates of each vertex of rectangle QRST. From the graph:
- \( T(-10, 0) \)
- \( S(-4, 0) \)
- \( R(-4, -8) \)
- \( Q(-10, -8) \)

Step2: Apply 90° counterclockwise rotation rule

The rule for a 90° counterclockwise rotation around the origin \((x, y) \to (-y, x)\). Apply this to each vertex:
- For \( T(-10, 0) \): New coordinates \( T'(-0, -10) = (0, -10) \)? Wait, no, wait: Wait, the rule is \((x, y) \to (-y, x)\). So:
- \( T(-10, 0) \): \( x = -10 \), \( y = 0 \). So \( -y = 0 \), \( x = -10 \)? Wait, no, wait: Wait, 90° counterclockwise: \((x, y) \to (-y, x)\). So:
- \( T(-10, 0) \): \( -y = 0 \), \( x = -10 \)? Wait, no, wait: Wait, \( x=-10 \), \( y=0 \). So \( -y = 0 \), \( x = -10 \)? Wait, no, that can't be. Wait, maybe I mixed up. Wait, 90° counterclockwise rotation: the formula is \((x, y) \to (-y, x)\). Let's check:
- For a point \((a, b)\), rotating 90° counterclockwise around origin gives \((-b, a)\). So:
- \( T(-10, 0) \): \( a = -10 \), \( b = 0 \). So new point \( T'(-0, -10) = (0, -10) \)? Wait, no, \( -b = 0 \), \( a = -10 \)? Wait, no, \( a=-10 \), \( b=0 \). So \( -b = 0 \), \( a = -10 \)? Wait, that would be (0, -10)? Wait, no, maybe I made a mistake. Wait, let's take another point. Let's take \( S(-4, 0) \): \( x=-4 \), \( y=0 \). So \( -y = 0 \), \( x = -4 \)? No, wait, no: Wait, 90° counterclockwise: the rotation matrix is \( \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \), so multiplying by \( \begin{pmatrix} x \\ y \end{pmatrix} \) gives \( \begin{pmatrix} -y \\ x \end{pmatrix} \). So:
- \( T(-10, 0) \): \( \begin{pmatrix} -0 \\ -10 \end{pmatrix} = (0, -10) \)? Wait, no, \( -y = 0 \), \( x = -10 \)? Wait, no, \( x=-10 \), \( y=0 \). So \( -y = 0 \), \( x = -10 \)? Wait, that's (0, -10)? Wait, no, maybe I messed up the sign. Wait, let's take a point (1, 0). Rotating 90° counterclockwise should be (0, 1). Using the formula: \( (x, y) \to (-y, x) \). So (1, 0) becomes (0, 1). Correct. So ( -10, 0 ): \( x=-10 \), \( y=0 \). So \( -y = 0 \), \( x = -10 \)? Wait, no, \( x=-10 \), \( y=0 \). So \( -y = 0 \), \( x = -10 \)? So (0, -10)? Wait, no, (1,0) becomes (0,1). So ( -10, 0 ) should become (0, -10)? Wait, no, (1,0) is (x=1, y=0). So -y=0, x=1. So (0,1). Correct. So (-10, 0): -y=0, x=-10. So (0, -10). Wait, but that seems off. Wait, maybe I got the rule reversed. Wait, 90° clockwise is (x,y)→(y, -x). 90° counterclockwise is (x,y)→(-y, x). Let's confirm with (0,1). 90° counterclockwise should be (-1, 0). Using the rule: (0,1)→(-1, 0). Correct. So yes, the rule is (x,y)→(-y, x).

So applying to each point:
- \( T(-10, 0) \): \( (-0, -10) = (0, -10) \)? Wait, no, \( x=-10 \), \( y=0 \). So \( -y = 0 \), \( x = -10 \)? Wait, no, \( x=-10 \), \( y=0 \). So \( -y = 0 \), \( x = -10 \). So \( T'(0, -10) \)? Wait, no, (x,y)→(-y, x). So x=-10, y=0. So -y=0, x=-10. So (0, -10). Wait, but (1,0) becomes (0,1). So (-10,0) becomes (0, -10). Okay.
- \( S(-4, 0) \): \( x=-4 \), \( y=0 \). So \( -y=0 \), \( x=-4 \). So \( S'(0, -4) \)? Wait, no, (x,y)→(-y, x). So (-4,0)→(0, -4)? Wait, no, (1,0)→(0,1), so (-4,0)→(0, -4). Correct.
- \( R(-4, -8) \): \( x=-4 \), \( y=-8 \). So \( -y = 8 \), \( x = -4 \). So \( R'(8, -4) \)? Wait, no: (x,y)→(-y, x). So x=-4, y=-8. So -y=8, x=-4? Wait, no, x is -4, so the new x is -y=8, new y is x=-4. So \( R'(8, -4) \)? Wait, let's check with a point (1,1). Rotating 90° counterclockwise: (-1,1). Using the rule: (1,1)→(-1,1). Correct. So ( -4, -8 ): x=-4, y=-8. So -y=8, x=-4? Wait, no, new x is -y, new y is x. So new point is (-y, x) = (8, -4). Yes.
- \( Q(-10, -8) \): \( x=-10 \), \( y=-8 \). So \( -y = 8 \), \( x = -10 \). So \( Q'(8, -10) \)? Wait, no: (x,y)→(-y, x). So x=-10, y=-8. So -y=8, x=-10? Wait, no, new x is -y=8, new y is x=-10. So \( Q'(8, -10) \).

Wait, that seems off. Wait, maybe I made a mistake in the rotation rule. Wait, let's check with a point (2,3). Rotating 90° counterclockwise should be (-3,2). Using the rule: (2,3)→(-3,2). Correct. So ( -4, -8 ): x=-4, y=-8. So -y=8, x=-4? Wait, no, (x,y)→(-y, x). So x=-4, y=-8. So -y=8, x=-4. So (8, -4). Correct. ( -10, -8 ): x=-10, y=-8. So -y=8, x=-10? Wait, no, (x,y)→(-y, x). So (-10, -8)→(8, -10). Yes.

Wait, but let's plot these points. Wait, maybe I mixed up the rotation direction. Wait, 90° counterclockwise: the figure should rotate left around the origin. Let's re-express the coordinates:

Original coordinates:
- T(-10, 0)
- S(-4, 0)
- R(-4, -8)
- Q(-10, -8)

After 90° counterclockwise rotation:
- T' (0, -10)
- S' (0, -4)
- R' (8, -4)
- Q' (8, -10)

Wait, but when we plot these, let's see:

Wait, maybe I made a mistake in the rotation rule. Wait, another way: 90° counterclockwise rotation is equivalent to rotating the coordinate system 90° clockwise, so the transformation is (x, y) → (-y, x). Let's confirm with a square with vertices (0,0), (1,0), (1,1), (0,1). Rotating 90° counterclockwise: (0,0), (0,1), (-1,1), (-1,0). Using the rule: (1,0)→(0,1), (1,1)→(-1,1), (0,1)→(-1,0), (0,0)→(0,0). Correct. So the rule is correct.

Wait, but when we apply the rule to T(-10,0), we get (0, -10). S(-4,0)→(0, -4). R(-4,-8)→(8, -4). Q(-10,-8)→(8, -10). Now, let's check the shape. The original rectangle has length 6 (from x=-10 to x=-4) and height 8 (from y=0 to y=-8). After rotation, the length should become the height and vice versa, and the orientation should change.

Wait, maybe I messed up the sign of the y-coordinate. Wait, let's take a point (0,1). Rotating 90° counterclockwise: (-1, 0). Using the rule: (0,1)→(-1, 0). Correct. So (x,y)→(-y, x). So (0,1)→(-1,0). So ( -10, 0 )→(0, -10). ( -4, 0 )→(0, -4). ( -4, -8 )→(8, -4). ( -10, -8 )→(8, -10). Now, let's plot these points:

- \( T'(0, -10) \)
- \( S'(0, -4) \)
- \( R'(8, -4) \)
- \( Q'(8, -10) \)

Wait, but when we connect these, it should be a rectangle. Let's check the distances:

- \( T'S' \): from (0,-10) to (0,-4): vertical distance 6 (correct, original length was 6)
- \( S'R' \): from (0,-4) to (8,-4): horizontal distance 8 (correct, original height was 8)
- \( R'Q' \): from (8,-4) to (8,-10): vertical distance 6
- \( Q'T' \): from (8,-10) to (0,-10): horizontal distance 8

Yes, that's a rectangle. So the rotated rectangle has vertices at (0, -10), (0, -4), (8, -4), (8, -10).

Step3: Graph the new points

Plot the new vertices \( T'(0, -10) \), \( S'(0, -4) \), \( R'(8, -4) \), \( Q'(8, -10) \) and connect them to form the rectangle.

Answer:

The image of rectangle QRST after a 90° counterclockwise rotation around the origin has vertices at \( T'(0, -10) \), \( S'(0, -4) \), \( R'(8, -4) \), and \( Q'(8, -10) \). (To graph, plot these points and connect them in order.)