Problem 21 - Algebra 2

Question

Write the vertex matrix and the rotation matrix for each figure. Then find the coordinates of the image after the rotation. Graph the preimage and the image on a coordinate plane. \(\triangle M N O\) with \(M(-2,-6), N(1,4),\) and \(O(3,-4)\) is rotated \(180^{\circ}\) counterclockwise about the origin.

Step-by-Step Solution

Answer

M' = (2,6), N' = (-1,-4), O' = (-3,4).

Step 1: Understand the Problem

We are given a triangle with vertices \(M(-2,-6)\), \(N(1,4)\), and \(O(3,-4)\). We need to find the coordinates after rotating this triangle 180° counterclockwise about the origin.

Step 2: Write the Vertex Matrix

We start by writing the vertex matrix \(V\) for the triangle \(\triangle MNO\) as follows:\[V = \begin{bmatrix} -2 & 1 & 3 \ -6 & 4 & -4 \end{bmatrix}\]

Step 3: Write the Rotation Matrix

For a 180° counterclockwise rotation, the rotation matrix \(R\) is given by:\[R = \begin{bmatrix} -1 & 0 \ 0 & -1 \end{bmatrix}\]

Step 4: Multiply the Vertex Matrix by the Rotation Matrix

To find the coordinates of the triangle after rotation, multiply the rotation matrix \(R\) by the vertex matrix \(V\):\[RV = \begin{bmatrix} -1 & 0 \ 0 & -1 \end{bmatrix} \begin{bmatrix} -2 & 1 & 3 \ -6 & 4 & -4 \end{bmatrix} = \begin{bmatrix} 2 & -1 & -3 \ 6 & -4 & 4 \end{bmatrix}\]This results in the new coordinates: \(M'(2,6)\), \(N'(-1,-4)\), \(O'(-3,4)\).

Step 5: Graph the Preimage and Image

On a coordinate plane, graph the original triangle \(M(-2,-6), N(1,4), O(3,-4)\) and the new triangle \(M'(2,6), N'(-1,-4), O'(-3,4)\). The newly plotted triangle represents the image after the rotation.

Key Concepts

rotation matrix

A rotation matrix is a mathematical tool used to rotate points in the coordinate plane. It is particularly handy when dealing with geometric transformations like rotating triangles. For a rotation by 180° counterclockwise about the origin, the rotation matrix is:

• \[R = \begin{bmatrix} -1 & 0 \ 0 & -1 \end{bmatrix}\]

This matrix changes the sign of both the x and y coordinates of any point, effectively rotating the point around the origin. In general, a rotation matrix for a certain angle \(\theta\) is given by:

• \[R(\theta) = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \ \sin(\theta) & \cos(\theta) \end{bmatrix}\]

It's important to note that cosine and sine define the matrix’s ability to rotate points by any given angle.

coordinate plane

The coordinate plane is composed of two axes, the x-axis and y-axis, which intersect at the origin \(0,0\). This plane is vital for graphing and analyzing shapes and points in geometry. Every point on this plane can be represented as an ordered pair \(x, y\), where \(x\) is the horizontal value, and \(y\) is the vertical value.
The coordinate plane allows us to visualize geometric transformations. For example, a figure such as a triangle can be rotated around the origin, and the position of each vertex is determined by the rotation matrix's effect on the coordinate values.
As we graph triangles like \(\triangle MNO\) and its image \(\triangle M'N'O'\) after rotation, we can see how these transformations change the figure's orientation but not its shape or size.

triangle rotation

Triangle rotation involves turning a triangle around a point, often the origin, through a specified angle. In this exercise, the triangle \(\triangle MNO\) has vertices \(M(-2,-6)\), \(N(1,4)\), and \(O(3,-4)\), and is rotated 180° counterclockwise. The effect of this transformation on each vertex is a direct application of the rotation matrix.
Using the rotation matrix for 180°:

• \[R = \begin{bmatrix} -1 & 0 \ 0 & -1 \end{bmatrix}\]

By multiplying this matrix by each vertex's coordinates, we find the new positions:

• \(M(2,6)\)
• \(N(-1,-4)\)
• \(O(-3,4)\)

This transformation rotates the whole triangle, maintaining equal side lengths and angles, ensuring congruence between the preimage and image.

matrix multiplication

Matrix multiplication is a crucial operation in transformations involving rotation matrices. When multiplying two matrices, such as a rotation matrix and a vertex matrix, we derive a new matrix representing the rotated coordinates.
For a given vertex matrix \(V\) of a triangle, such as:

• \[V = \begin{bmatrix} -2 & 1 & 3 \ -6 & 4 & -4 \end{bmatrix}\]

The multiplication involves aligning the rotation matrix with each column of the vertex matrix and performing dot products for each relevant row and column combination. For the 180° rotation matrix, it works out like this:

• \[\begin{bmatrix} -1 & 0 \ 0 & -1 \end{bmatrix} \times \begin{bmatrix} -2 & 1 & 3 \ -6 & 4 & -4 \end{bmatrix} = \begin{bmatrix} 2 & -1 & -3 \ 6 & -4 & 4 \end{bmatrix}\]

Through matrix multiplication, we effectively transform the preimage coordinates into the image coordinates with ease.