Question

a rectangle is plotted on the coordinate plane below: image of coordinate plane with rectangle which image shows a 90° clockwise rotation about the origin? images of options (b), (c), (a) with radio buttons

Explanation:

Step1: Identify original rectangle vertices

Original rectangle (from graph) has vertices: Let's find coordinates. From the first graph, the rectangle is from \( (0,0) \) to \( (6,0) \) to \( (6,2) \) to \( (0,2) \)? Wait, no, looking at the grid: the rectangle is between \( x=0 \) to \( x=6 \), \( y=0 \) to \( y=2 \)? Wait, actually, the original rectangle: left side at \( x=0 \), right at \( x=6 \), bottom at \( y=0 \), top at \( y=2 \)? Wait, no, the first graph: the rectangle is drawn with left at \( x=0 \), right at \( x=6 \), bottom at \( y=0 \), top at \( y=2 \)? Wait, no, the grid: each square is 1 unit. So original rectangle vertices: \( (0,0) \), \( (6,0) \), \( (6,2) \), \( (0,2) \)? Wait, no, looking at the first graph: the rectangle is from \( x=0 \) to \( x=6 \), \( y=0 \) to \( y=2 \)? Wait, the first graph: the rectangle is black, from \( (0,0) \) to \( (6,0) \) to \( (6,2) \) to \( (0,2) \)? Wait, no, the y-axis: the top of the rectangle is at \( y=2 \), bottom at \( y=0 \), left at \( x=0 \), right at \( x=6 \). So vertices: \( A(0,0) \), \( B(6,0) \), \( C(6,2) \), \( D(0,2) \).

Step2: Apply 90° clockwise rotation rule

The rule for 90° clockwise rotation about the origin is \( (x,y) \to (y, -x) \).

Let's apply to each vertex:

- \( A(0,0) \to (0, -0) = (0,0) \)
- \( B(6,0) \to (0, -6) \)
- \( C(6,2) \to (2, -6) \)
- \( D(0,2) \to (2, -0) = (2,0) \)

Wait, that can't be right. Wait, maybe I misidentified the original vertices. Wait, maybe the original rectangle is from \( (0,2) \) to \( (6,2) \) to \( (6,0) \) to \( (0,0) \)? Wait, no, the first graph: the rectangle is above the x-axis, from \( x=0 \) to \( x=6 \), \( y=0 \) to \( y=2 \)? Wait, no, the y-axis: the top of the rectangle is at \( y=2 \), bottom at \( y=0 \), left at \( x=0 \), right at \( x=6 \). So vertices: \( (0,0) \), \( (6,0) \), \( (6,2) \), \( (0,2) \).

Wait, maybe I made a mistake in the rotation rule. Wait, 90° clockwise rotation: the formula is \( (x,y) \mapsto (y, -x) \). Wait, let's check with a point. For example, a point \( (1,0) \) rotated 90° clockwise about origin: becomes \( (0, -1) \). A point \( (0,1) \) rotated 90° clockwise: becomes \( (1, 0) \). Wait, maybe the correct rule is \( (x,y) \to (y, -x) \). Wait, let's take the original rectangle. Wait, maybe the original rectangle is from \( (0,0) \) to \( (6,0) \) to \( (6,2) \) to \( (0,2) \). So applying rotation:

- \( (0,0) \to (0,0) \)
- \( (6,0) \to (0, -6) \)
- \( (6,2) \to (2, -6) \)
- \( (0,2) \to (2, 0) \)

So the rotated rectangle would have vertices at \( (0,0) \), \( (0, -6) \), \( (2, -6) \), \( (2, 0) \). Wait, that's a vertical rectangle? Wait, looking at the options:

Option (a): A rectangle from \( (0,0) \) to \( (2,0) \) to \( (2, -6) \) to \( (0, -6) \)? Wait, no, option (a) shows a rectangle from \( (0,0) \) down to \( (0, -6) \) and right to \( (2, -6) \)? Wait, let's look at the options:

- Option (a): The rectangle is from \( x=0 \) to \( x=2 \), \( y=0 \) down to \( y=-6 \). So vertices \( (0,0) \), \( (2,0) \), \( (2, -6) \), \( (0, -6) \). Which matches the rotated vertices we calculated: \( (0,0) \), \( (2,0) \), \( (2, -6) \), \( (0, -6) \) (wait, earlier I had \( (0, -6) \) but maybe I mixed up x and y). Wait, maybe the original rectangle is \( (0,0) \), \( (6,0) \), \( (6,2) \), \( (0,2) \). So width 6, height 2. After 90° clockwise rotation, the width becomes height and height becomes width, and direction changes. So the rotated rectangle should have width 2, height 6, and positioned in the fourth quadrant? Wait, no, 90° clockwise rotation: the rectangle will be along the positive y to negative y? Wait, maybe I messed up the rotation rule. Let's recall: 90° clockwise rotation about origin: \( (x,y) \to (y, -x) \). So for a point \( (x,y) \), new x is y, new y is -x.

Original rectangle vertices: Let's re-express. Let's take the original rectangle: from \( x=0 \) to \( x=6 \), \( y=0 \) to \( y=2 \). So the four corners are:

1. \( (0, 0) \): bottom-left
2. \( (6, 0) \): bottom-right
3. \( (6, 2) \): top-right
4. \( (0, 2) \): top-left

Applying rotation:

1. \( (0,0) \to (0, 0) \)
2. \( (6,0) \to (0, -6) \)
3. \( (6,2) \to (2, -6) \)
4. \( (0,2) \to (2, 0) \)

So the new vertices are \( (0,0) \), \( (0, -6) \), \( (2, -6) \), \( (2, 0) \). So this is a rectangle with length 2 (from x=0 to x=2) and height 6 (from y=0 to y=-6), located in the fourth quadrant (since y is negative). Looking at the options:

- Option (a): The rectangle is from \( x=0 \) to \( x=2 \), \( y=0 \) down to \( y=-6 \). So that's the rotated rectangle.

- Option (b): Rectangle is in the second quadrant, from \( x=-6 \) to \( x=0 \), \( y=0 \) to \( y=2 \). That's a 180° rotation or 90° counterclockwise?

- Option (c): Rectangle is in the third quadrant, from \( x=-6 \) to \( x=0 \), \( y=0 \) down to \( y=-2 \). Not matching.

So the correct image is (a).

Answer:

(a)