Problem 18 - Algebra 2

Question

Write the vertex matrix and the reflection matrix for each figure. Then find the coordinates of the image after the reflection. Graph the preimage and the image on a coordinate plane. The vertices of \(\triangle X Y Z\) are \(X(1,-1), Y(2,-4),\) and \(Z(7,-1) .\) The triangle is reflected over the line \(y=x .\)

Step-by-Step Solution

Answer

The image vertices after reflection are \((-1, 1), (-4, 2), (-1, 7)\).

Step 1: Write the Vertex Matrix

The vertex matrix of a triangle is a 3x2 matrix that includes the coordinates of its vertices. For \(\triangle XYZ\), the vertices are \(X(1, -1), Y(2, -4), Z(7, -1)\). The vertex matrix is written as: \[ V = \begin{bmatrix} 1 & -1 \ 2 & -4 \ 7 & -1 \end{bmatrix} \] where each row corresponds to the coordinates of points \(X\), \(Y\), and \(Z\).

Step 2: Write the Reflection Matrix

To reflect a figure over the line \(y = x\), we use the reflection matrix \[ R = \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix} \]. This matrix swaps the \(x\)- and \(y\)-coordinates of each point.

Step 3: Find the Image Coordinates

To find the coordinates of the image, multiply the vertex matrix by the reflection matrix. Perform the matrix multiplication: \[ V \times R = \begin{bmatrix} 1 & -1 \ 2 & -4 \ 7 & -1 \end{bmatrix} \times \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix} = \begin{bmatrix} -1 & 1 \ -4 & 2 \ -1 & 7 \end{bmatrix} \]. Thus, the image vertices are \( (-1, 1), (-4, 2), (-1, 7) \).

Step 4: Graph the Preimage and Image

On a coordinate plane, plot the original points \((1, -1), (2, -4), (7, -1)\) to form \(\triangle XYZ\). Then, reflect these points over \(y = x\) to obtain the new points \((-1, 1), (-4, 2), (-1, 7)\) forming the reflected triangle.

Key Concepts

Vertex Matrix

In coordinate geometry, a vertex matrix is a practical way to represent the vertices of a polygon using matrix notation. This is especially useful when dealing with transformations such as reflections or rotations. For a triangle like \(\triangle XYZ\), the vertex matrix is a 3x2 matrix. It is organized so that each row corresponds to a vertex of the triangle, listing the x-coordinate and y-coordinate.

For instance, the coordinates of \(X(1, -1), Y(2, -4), Z(7, -1)\) create the vertex matrix:

• First row: coordinates for \(X\), which are \(1\) for x and \(-1\) for y.
• Second row: coordinates for \(Y\), which are \(2\) for x and \(-4\) for y.
• Third row: coordinates for \(Z\), which are \(7\) for x and \(-1\) for y.

The matrix looks like this:\[V = \begin{bmatrix} 1 & -1 \ 2 & -4 \ 7 & -1 \end{bmatrix}\] This format is a concise way to store and manipulate the points of the triangle systematically.

Reflection Matrix

A reflection matrix is an essential tool for mirroring shapes over specific lines on the coordinate plane. When you reflect a figure over the line \(y = x\), the reflection matrix used is:\[R = \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix}\]This matrix has a unique property of swapping the x- and y-coordinates of any point it is applied to.

To apply the reflection, multiply the vertex matrix of the shape by this reflection matrix. This operation will give you the new coordinates of the reflected image, effectively transforming the shape by mirroring it over the line \(y = x\).

• Each original coordinate \((x, y)\) will become \((y, x)\) after transformation.
• This transformation is vital in various computer graphics applications and analyses.

Coordinate Geometry

Coordinate geometry, or analytic geometry, allows us to solve geometric problems algebraically using the coordinate plane. It links algebra and geometry, making it easier to understand shapes and their transformations.

In this exercise, coordinate geometry helps us describe the original and transformed positions of the triangle \(\triangle XYZ\). By representing points as ordered pairs of numbers, we can easily perform operations like reflections or rotations.

• Each point in the plane has clear coordinates \((x, y)\).
• The transformations are ruled by algebraic equations or matrices like the reflection matrix.

Coordinate geometry is very useful in identifying the properties of geometric shapes, checking symmetry, and proving concepts such as congruence and similarity in transformation processes like triangle reflections.

Triangle Reflection

Reflecting triangles over lines on a coordinate plane is a frequent task in geometry. To perform a triangle reflection, each vertex of the triangle changes its position based on the selected line of reflection, such as \(y = x\). In this exercise, the reflection results in exchanging the coordinates of each point.

Let's consider the original triangle \(\triangle XYZ\) with vertices at \(X(1, -1), Y(2, -4), Z(7, -1)\). After reflection over the line \(y = x\), the coordinates transform to \((-1, 1), (-4, 2), (-1, 7)\). This swapped each point's x- and y-coordinate, forming a new triangle that is a mirror image of the original.

Understanding triangle reflection is crucial because:

• It helps us visualize spatial transformations.
• It's applicable in diverse fields like physics, architecture, and art.
• It's foundational for learning more advanced geometric and algebraic concepts.

Through such exercises, students understand reflection-based transformations better, supporting their wider mathematical comprehension.