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

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

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

Answer

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.)