Question

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

Explanation:

Step1: Identify Coordinates

First, find the coordinates of rectangle QRST. From the graph:
- \( Q(-9, -8) \)
- \( R(-4, -8) \)
- \( S(-4, 0) \)
- \( T(-9, 0) \)

Step2: Apply 90° CCW Rotation Rule

The rule for rotating a point \((x, y)\) 90° counterclockwise around the origin is \((x, y) \to (-y, x)\).

- For \( Q(-9, -8) \):
New coordinates: \((-(-8), -9) = (8, -9)\)? Wait, no: Wait, rule is \((x,y) \to (-y, x)\). So \( x=-9, y=-8 \): \(-y = 8\), \( x = -9 \)? Wait, no: Wait, 90° CCW rotation: \((x, y) \mapsto (-y, x)\). So:

- \( Q(-9, -8) \): \( -y = -(-8) = 8 \), \( x = -9 \)? Wait, no, wait: \((x,y) \to (-y, x)\). So \( x=-9 \), \( y=-8 \): new \( x = -y = -(-8) = 8 \), new \( y = x = -9 \)? Wait, no, that can't be. Wait, maybe I mixed up. Wait, 90° counterclockwise rotation: \((x, y) \to (-y, x)\). Let's check with a point (1,0): rotate 90° CCW, it becomes (0,1). Using the rule: \( x=1, y=0 \): \(-y=0\), \( x=1 \)? No, that's (0,1)? Wait, no: (1,0) rotated 90° CCW is (0,1). So the rule should be \((x, y) \to (-y, x)\)? Wait, (1,0): \( -y = 0 \), \( x = 1 \)? No, that's (0,1)? Wait, no: (x,y) → (-y, x). So (1,0): -y = 0, x=1 → (0,1). Yes! Correct. So (1,0) → (0,1). Another example: (0,1) rotated 90° CCW is (-1, 0). Using rule: (0,1) → (-1, 0). Correct. So the rule is correct.

So let's re-calculate:

- \( Q(-9, -8) \): \( x=-9 \), \( y=-8 \). New \( x = -y = -(-8) = 8 \), new \( y = x = -9 \). Wait, no: Wait, (x,y) → (-y, x). So \( x \) becomes \(-y\), \( y \) becomes \( x \). So \( Q(-9, -8) \): \(-y = -(-8) = 8\), \( x = -9 \)? Wait, no: \( y=-8 \), so \(-y = 8\); \( x=-9 \), so new \( y = x = -9 \). So \( Q' (8, -9) \)? Wait, no, that seems off. Wait, maybe I made a mistake in coordinates. Wait, original points:

Looking at the graph:

- \( T \) is at (-9, 0) (since it's on x-axis, x=-9, y=0)
- \( S \) is at (-4, 0) (x=-4, y=0)
- \( R \) is at (-4, -8) (x=-4, y=-8)
- \( Q \) is at (-9, -8) (x=-9, y=-8)

Yes, that's correct. So:

- \( T(-9, 0) \): apply rule \((x,y) \to (-y, x)\): \( -y = -0 = 0 \), \( x = -9 \)? Wait, no: (x,y)=(-9,0) → (-y, x) = (0, -9)? Wait, no: (x,y) → (-y, x). So x=-9, y=0: -y = 0, x=-9. So (0, -9)? Wait, no, (1,0) → (0,1). So (x,y) → (-y, x). So ( -9, 0 ): -y = 0, x = -9? No, that would be (0, -9). Wait, (1,0) → (0,1): x=1, y=0 → -y=0, x=1 → (0,1). Correct. So ( -9, 0 ) → (0, -9)? Wait, no, ( -9, 0 ): x=-9, y=0. So -y=0, x=-9 → (0, -9). But (1,0) → (0,1), so ( -1, 0 ) → (0, -1). Yes, that makes sense. So:

- \( T(-9, 0) \): (0, -9)
- \( S(-4, 0) \): (0, -4)
- \( R(-4, -8) \): \( -y = -(-8) = 8 \), \( x = -4 \) → (8, -4)
- \( Q(-9, -8) \): \( -y = -(-8) = 8 \), \( x = -9 \) → (8, -9)

Wait, no, that can't be. Wait, maybe the rule is (x,y) → (-y, x), but let's check with (0,1) rotated 90° CCW: ( -1, 0 ). So (0,1) → (-1, 0). Using the rule: -y = -1, x=0 → (-1, 0). Correct. So (x,y) → (-y, x). So:

For \( S(-4, 0) \): (x=-4, y=0) → (-y, x) = (0, -4). Wait, but (1,0) → (0,1), so ( -4, 0 ) → (0, -4)? That seems mirrored. Wait, maybe I mixed up clockwise and counterclockwise. Wait, 90° clockwise rotation is (x,y) → (y, -x). 90° counterclockwise is (x,y) → (-y, x). Let's confirm with (1,0):

- 90° CCW: (0,1) → using rule (-y, x): y=0, so -y=0; x=1 → (0,1). Correct.
- 90° CW: (0, -1) → using rule (y, -x): y=0, -x=-1 → (0, -1). Correct.

So back to the points:

- \( T(-9, 0) \): (x=-9, y=0) → (-y, x) = (0, -9)
- \( S(-4, 0) \): (x=-4, y=0) → (0, -4)
- \( R(-4, -8) \): (x=-4, y=-8) → (-(-8), -4) = (8, -4)
- \( Q(-9, -8) \): (x=-9, y=-8) → (-(-8), -9) = (8, -9)

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

Original rectangle: from x=-9 to -4 (length 5), y=0 to -8 (height 8). After rotation, it should be a rectangle with length 8 (vertical) and height 5 (horizontal), rotated 90° CCW. Wait, maybe I made a mistake. Wait, let's re-express the coordinates:

Wait, original points:

- \( T(-9, 0) \)
- \( S(-4, 0) \)
- \( R(-4, -8) \)
- \( Q(-9, -8) \)

So the sides: TS is horizontal from (-9,0) to (-4,0) (length 5), SR is vertical from (-4,0) to (-4,-8) (length 8), RQ is horizontal from (-4,-8) to (-9,-8) (length 5), QT is vertical from (-9,-8) to (-9,0) (length 8).

After 90° CCW rotation, the horizontal side (length 5) should become vertical, and the vertical side (length 8) should become horizontal, but in the opposite direction. Wait, maybe the rule is (x,y) → (-y, x). Let's take point T(-9,0):

Applying (x,y) → (-y, x): (-0, -9) = (0, -9). Wait, no, (x,y) → (-y, x). So x=-9, y=0: -y = 0, x=-9 → (0, -9). But (1,0) → (0,1), so ( -1, 0 ) → (0, -1). So that's correct.

Point S(-4,0): (0, -4)

Point R(-4,-8): (-(-8), -4) = (8, -4)

Point Q(-9,-8): (-(-8), -9) = (8, -9)

Now, let's check the new rectangle:

- \( T'(0, -9) \)
- \( S'(0, -4) \)
- \( R'(8, -4) \)
- \( Q'(8, -9) \)

Wait, but when we connect these, it's a rectangle from (0,-9) to (0,-4) (vertical), then to (8,-4) (horizontal), then to (8,-9) (vertical), then back to (0,-9) (horizontal). Wait, but that's a 90° CCW rotation? Wait, maybe I messed up the rule. Wait, maybe the correct rule for 90° CCW is (x,y) → (-y, x). Let's check with a square from (0,0) to (1,0) to (1,1) to (0,1). Rotating 90° CCW should give (0,0) → (0,0), (1,0) → (0,1), (1,1) → (-1,1), (0,1) → (-1,0). Wait, no, that's not right. Wait, no, the correct 90° CCW rotation around origin is (x, y) → (-y, x). Let's take (1,0): (-0,1)=(0,1). Correct. (1,1): (-1,1). Wait, no, (1,1) rotated 90° CCW is (-1,1)? No, (1,1) rotated 90° CCW should be (-1,1)? Wait, no, using the right-hand rule: rotate (1,1) 90° CCW around origin: the x-coordinate becomes -y, y becomes x. So (1,1) → (-1,1). Wait, but plotting that, (1,1) to (-1,1) to (-1,-1) to (1,-1)? No, maybe I'm confused.

Wait, maybe the original rectangle is in the third quadrant (x negative, y negative or zero). Rotating 90° CCW around origin should move it to the fourth quadrant? Wait, no, 90° CCW from third quadrant (x-, y-) would go to fourth quadrant (x+, y-)? Wait, no:

Wait, let's take a point in third quadrant: (-a, -b) where a,b >0. Rotating 90° CCW: (-(-b), -a) = (b, -a). So (b, -a) is in fourth quadrant (x+, y-). So that's correct.

So for Q(-9,-8) (a=9, b=8): (8, -9) → fourth quadrant.

S(-4,0): (0, -4) → on y-axis, negative.

T(-9,0): (0, -9) → on y-axis, negative.

R(-4,-8): (8, -4) → fourth quadrant.

So the new rectangle has vertices at (0,-9), (0,-4), (8,-4), (8,-9). To graph this, plot these points and connect them.

Answer:

The image of rectangle QRST after 90° counterclockwise rotation around the origin has vertices at \( Q'(8, -9) \), \( R'(8, -4) \), \( S'(0, -4) \), and \( T'(0, -9) \). (Graph by plotting these points and connecting them in order.)