Question

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

Explanation:

Step1: Find coordinates of T, U, V, W

From the graph:
- \( T(-6, 2) \)
- \( U(0, 2) \)
- \( V(0, 10) \)
- \( W(-6, 10) \)

Step2: Apply 90° counterclockwise rotation rule

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

For \( T(-6, 2) \):

\( x = -6 \), \( y = 2 \)
New coordinates: \( (-2, -6) \) (Wait, no: Wait, rule is \((x,y) \to (-y, x)\). So \( x=-6 \), \( y=2 \): \( -y = -2 \), \( x = -6 \)? Wait no, wait: Wait, 90° counterclockwise: \((x, y) \to (-y, x)\). Let's recheck.

Wait, correct rule: When rotating 90° counterclockwise about the origin, the transformation is \((x, y) \mapsto (-y, x)\).

So for \( T(-6, 2) \):
\( x = -6 \), \( y = 2 \)
New \( x' = -y = -2 \)
New \( y' = x = -6 \)
So \( T'(-2, -6) \)? Wait no, wait, maybe I mixed up. Wait, let's take a point (1,0). Rotating 90° counterclockwise should be (0,1). Wait, no: (1,0) rotated 90° counterclockwise is (0,1)? Wait no, (x,y) → (-y, x). So (1,0) → (0,1)? Wait, no: (1,0): x=1, y=0. Then -y=0, x=1. So (0,1). But actually, (1,0) rotated 90° counterclockwise is (0,1)? Wait, no, (1,0) is on the x-axis. Rotating 90° counterclockwise around origin would move it to (0,1), yes. So the rule is correct: \((x, y) \to (-y, x)\).

Wait, let's take (0,1). Rotating 90° counterclockwise: (-1, 0). Using the rule: x=0, y=1. -y = -1, x=0. So (-1, 0). Correct.

So back to the points:

- \( T(-6, 2) \): \( x=-6 \), \( y=2 \). So \( T'(-2, -6) \)? Wait, no: Wait, \( x=-6 \), \( y=2 \). Then \( -y = -2 \), \( x = -6 \). So \( T'(-2, -6) \)? Wait, no, wait, maybe I made a mistake. Wait, let's check U(0,2):

- \( U(0, 2) \): \( x=0 \), \( y=2 \). So \( U'(-2, 0) \) (since -y = -2, x=0)
- \( V(0, 10) \): \( x=0 \), \( y=10 \). So \( V'(-10, 0) \) (since -y = -10, x=0)
- \( W(-6, 10) \): \( x=-6 \), \( y=10 \). So \( W'(-10, -6) \) (since -y = -10, x=-6)

Wait, that seems off. Wait, maybe the rule is \((x, y) \to (-y, x)\) is for 90° counterclockwise? Wait, no, actually, the correct rule for 90° counterclockwise rotation about the origin is \((x, y) \mapsto (-y, x)\). Wait, let's take a point (2,3). Rotating 90° counterclockwise should be (-3, 2). Let's check: x=2, y=3. -y = -3, x=2. So (-3,2). Correct. So that's the rule.

So let's re-express the coordinates:

Original points:

- \( T(-6, 2) \): \( x=-6 \), \( y=2 \). So \( T'(-2, -6) \) (since -y = -2, x=-6)
- \( U(0, 2) \): \( x=0 \), \( y=2 \). So \( U'(-2, 0) \) (since -y = -2, x=0)
- \( V(0, 10) \): \( x=0 \), \( y=10 \). So \( V'(-10, 0) \) (since -y = -10, x=0)
- \( W(-6, 10) \): \( x=-6 \), \( y=10 \). So \( W'(-10, -6) \) (since -y = -10, x=-6)

Wait, but when we plot these points, let's see the original rectangle: T(-6,2), U(0,2), V(0,10), W(-6,10). So it's a rectangle with length 6 (from x=-6 to x=0) and height 8 (from y=2 to y=10). After rotating 90° counterclockwise, the length and height should swap, and the coordinates should reflect the rotation.

Alternatively, maybe I mixed up the rule. Wait, another way: 90° counterclockwise rotation: (x,y) → (-y, x). Wait, let's take (1,1). Rotating 90° counterclockwise: (-1,1)? No, (1,1) rotated 90° counterclockwise is (-1,1)? Wait, no, (1,1) is in the first quadrant. Rotating 90° counterclockwise around origin would move it to the second quadrant: (-1,1)? Wait, no, (x,y) → (-y, x). So (1,1) → (-1,1)? Wait, no, (1,1): x=1, y=1. -y = -1, x=1. So (-1,1). But actually, (1,1) rotated 90° counterclockwise is (-1,1)? Wait, no, let's use a graph. (1,1) is at (1,1). Rotating 90° counterclockwise: the x and y swap, and the new x is negative. So (x,y) → (-y, x). So (1,1) → (-1,1). Correct.

So back to the original points:

- T(-6, 2): x=-6, y=2 → (-2, -6)
- U(0, 2): x=0, y=2 → (-2, 0)
- V(0, 10): x=0, y=10 → (-10, 0)
- W(-6, 10): x=-6, y=10 → (-10, -6)

Now, to graph the image, we plot these four points: T'(-2, -6), U'(-2, 0), V'(-10, 0), W'(-10, -6), and connect them in order.

Answer:

The image of rectangle TUVW after a 90° counterclockwise rotation around the origin has vertices at \( T'(-2, -6) \), \( U'(-2, 0) \), \( V'(-10, 0) \), and \( W'(-10, -6) \). Plot these points and connect them to form the rotated rectangle.