Question

part a
graph rectangle (abcd). then draw a rotation of the rectangle (90^circ) counterclockwise about the origin.
use the polygon tool to graph the rectangle and the rotated image.

Explanation:

Step1: Identify Rectangle Vertices

First, find the coordinates of the vertices of Rectangle \(ABCD\) from the graph. Let's assume the vertices are:

• \(A(1, 5)\)
• \(B(3, 6)\)
• \(C(3, 1)\)
• \(D(5, 2)\) (Wait, actually, let's correctly identify. Looking at the graph, the blue points: let's list them properly. Let's check the grid:

First point: \((1, 5)\) (x=1, y=5)
Second: \((3, 6)\) (x=3, y=6)
Third: \((3, 1)\) (x=3, y=1)
Fourth: \((5, 2)\) (x=5, y=2) Wait, no, a rectangle has opposite sides equal and all angles 90 degrees. Wait, maybe I misread. Let's re-express:

Wait, the four blue points: Let's check their coordinates:

1. (1, 5)
2. (3, 6)
3. (3, 1)
4. (5, 2)

Wait, maybe the correct vertices (since it's a rectangle) should have two pairs of equal x or y. Wait, maybe I made a mistake. Let's instead use the rotation rule for 90° counterclockwise about the origin: \((x, y) \to (-y, x)\).

So first, let's correctly identify the vertices of \(ABCD\). Let's look at the graph again:

• Point 1: (1, 5)
• Point 2: (3, 6)
• Point 3: (3, 1)
• Point 4: (5, 2)

Wait, maybe the rectangle is formed by these four points. Now, to rotate 90° counterclockwise about the origin, we apply the transformation \((x, y) \to (-y, x)\) to each vertex.

Step2: Apply Rotation Rule to Each Vertex

For each vertex \((x, y)\), the new coordinates after 90° counterclockwise rotation about the origin are \((-y, x)\).

Let's take each vertex:

1. Vertex \(A(1, 5)\):

New coordinates: \((-5, 1)\)

1. Vertex \(B(3, 6)\):

New coordinates: \((-6, 3)\)

1. Vertex \(C(3, 1)\):

New coordinates: \((-1, 3)\)

1. Vertex \(D(5, 2)\):

New coordinates: \((-2, 5)\)

Now, we can plot these new points \((-5, 1)\), \((-6, 3)\), \((-1, 3)\), \((-2, 5)\) and connect them to form the rotated rectangle.

Answer:

To solve the problem, follow these steps:

1. Identify Original Vertices: From the graph, the vertices of Rectangle \(ABCD\) are approximately \((1, 5)\), \((3, 6)\), \((3, 1)\), and \((5, 2)\).
2. Apply 90° Counterclockwise Rotation Rule: For a point \((x, y)\), the rotation about the origin is \((x, y) \to (-y, x)\).

• For \((1, 5)\): \((-5, 1)\)
• For \((3, 6)\): \((-6, 3)\)
• For \((3, 1)\): \((-1, 3)\)
• For \((5, 2)\): \((-2, 5)\)

1. Graph the Rotated Rectangle: Plot the new vertices \((-5, 1)\), \((-6, 3)\), \((-1, 3)\), \((-2, 5)\) and connect them using the Polygon Tool.

(Note: The exact coordinates may vary slightly based on the graph’s precision, but the rotation rule \((x, y) \to (-y, x)\) is the key for 90° counterclockwise rotation about the origin.)