Problem 29 - Algebra 2

Question

For Exercises \(28-31,\) use rectangle \(A B C D\) with vertices \(A(-4,4), B(4,4),\) \(C(4,-4),\) and \(D(-4,-4) .\) Find the coordinates of the image in matrix form after a \(180^{\circ}\) rotation about the origin.

Step-by-Step Solution

Answer

The coordinates after a 180° rotation are \((4, -4), (-4, -4), (-4, 4), (4, 4)\).

Step 1: Understand the Rotation Matrix

A matrix that rotates a point by an angle \( \theta \) around the origin can be represented as: \[ R = \begin{bmatrix} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{bmatrix} \]For a \( 180^{\circ} \) rotation, \( \cos 180^{\circ} = -1 \) and \( \sin 180^{\circ} = 0 \). This gives us the rotation matrix:\[ R = \begin{bmatrix} -1 & 0 \ 0 & -1 \end{bmatrix} \]

Step 2: Apply the Rotation Matrix to Vertex A

The original coordinates of vertex \( A \) are \((-4, 4)\). To find the new coordinates, multiply by the rotation matrix:\[ \begin{bmatrix} -1 & 0 \ 0 & -1 \end{bmatrix} \begin{bmatrix} -4 \ 4 \end{bmatrix} = \begin{bmatrix} (-1)(-4) + 0(4) \ 0(-4) + (-1)(4) \end{bmatrix} = \begin{bmatrix} 4 \ -4 \end{bmatrix} \]Thus, the new coordinates of \( A' \) are \((4, -4)\).

Step 3: Apply the Rotation Matrix to Vertex B

The original coordinates of vertex \( B \) are \((4, 4)\). Multiply by the rotation matrix:\[ \begin{bmatrix} -1 & 0 \ 0 & -1 \end{bmatrix} \begin{bmatrix} 4 \ 4 \end{bmatrix} = \begin{bmatrix} (-1)(4) + 0(4) \ 0(4) + (-1)(4) \end{bmatrix} = \begin{bmatrix} -4 \ -4 \end{bmatrix} \]So, the new coordinates of \( B' \) are \((-4, -4)\).

Step 4: Apply the Rotation Matrix to Vertex C

The original coordinates of vertex \( C \) are \((4, -4)\). Multiply by the rotation matrix:\[ \begin{bmatrix} -1 & 0 \ 0 & -1 \end{bmatrix} \begin{bmatrix} 4 \ -4 \end{bmatrix} = \begin{bmatrix} (-1)(4) + 0(-4) \ 0(4) + (-1)(-4) \end{bmatrix} = \begin{bmatrix} -4 \ 4 \end{bmatrix} \]Thus, the new coordinates of \( C' \) are \((-4, 4)\).

Step 5: Apply the Rotation Matrix to Vertex D

The original coordinates of vertex \( D \) are \((-4, -4)\). Multiply by the rotation matrix:\[ \begin{bmatrix} -1 & 0 \ 0 & -1 \end{bmatrix} \begin{bmatrix} -4 \ -4 \end{bmatrix} = \begin{bmatrix} (-1)(-4) + 0(-4) \ 0(-4) + (-1)(-4) \end{bmatrix} = \begin{bmatrix} 4 \ 4 \end{bmatrix} \]So, the new coordinates of \( D' \) are \((4, 4)\).

Step 6: Final Step: Present the Coordinates in Matrix Form

The coordinates of the rectangle after a \( 180^{\circ} \) rotation are:\[ A'(4, -4), B'(-4, -4), C'(-4, 4), D'(4, 4) \]In matrix form, this can be written as:\[ \begin{bmatrix} 4 & -4 \ -4 & -4 \ -4 & 4 \ 4 & 4 \end{bmatrix} \]

Key Concepts

Rectangle Coordinates

Understanding rectangle coordinates involves identifying the position of each vertex on a two-dimensional plane. Each vertex of a rectangle has a specific coordinate, expressed in the form \(x, y\). These coordinate pairs tell us exactly where each corner of the rectangle is located in the plane.

In the exercise, the vertices of our rectangle are given as:

• Vertex \(A\) at \((-4, 4)\)
• Vertex \(B\) at \( (4, 4)\)
• Vertex \(C\) at \( (4, -4)\)
• Vertex \(D\) at \((-4, -4)\)

These coordinates define the four corners of rectangle \(ABCD\). When we discuss transformations like rotations, we are essentially talking about how these coordinates change or move within the plane.

180-Degree Rotation

A 180-degree rotation is a specific type of geometric transformation. It involves turning a figure around a center point, which in this exercise is the origin \((0, 0)\), by 180 degrees. This means every point of the rectangle will be flipped to the opposite side of the origin.

When you perform a 180-degree rotation, there's a simple rule for calculating the new coordinates:

• For any point \( (x, y)\), the new coordinates after rotation will be \((-x, -y)\).

Therefore, a 180-degree rotation is basically "flipping" each coordinate to its opposite counterpart on the plane. This transformation is useful in many mathematical and practical applications where symmetry or alignment around a central point is needed.

Matrices in Geometry

Matrices are powerful tools in geometry to perform transformations. They are especially useful in operations such as rotations, translations, and scaling. Using matrices, we can easily compute the result of such transformations on points or shapes.

For rotations, the rotation matrix simplifies the calculations. The general form of a rotation matrix for an angle \( \theta \) is:
\[ R = \begin{bmatrix} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{bmatrix} \]
In our specific case with a 180-degree rotation, \( \cos 180^\circ = -1 \) and \( \sin 180^\circ = 0 \). Thus, the rotation matrix becomes:
\[ R = \begin{bmatrix} -1 & 0 \ 0 & -1 \end{bmatrix} \]
By multiplying this matrix with the coordinates of each vertex of the rectangle, we can determine the new position of each point after the rotation.

Vertex Transformation

Vertex transformation involves changing the coordinates of a vertex through mathematical operations. In our exercise, each vertex undergoes transformation via a rotation matrix.

Here's how the process works for each vertex of rectangle \(ABCD\):

• Start with the original coordinates.
• Multiply the coordinates by the rotation matrix for a 180-degree rotation \(R = \begin{bmatrix} -1 & 0 \ 0 & -1 \end{bmatrix} \).
• The result gives us the new coordinates of the vertex after rotation.

For example, for vertex \(A:\)
Original: \((-4, 4)\)
New: \((4, -4)\)
This process is repeated for each vertex \(A, B, C,\) and \(D\), leading to the transformed rectangle's vertices. This systematic application of the rotation matrix ensures each vertex is correctly repositioned in the plane after rotation.