Question

a polygon has vertices at (-5, 3), (-1, 3), (1, 0), and (-3, 0). which represents a geometric translation of the given polygon 4 units to the right and 5 units down?

( \begin{bmatrix} -5 & 3 & -1 & 3 \\ 1 & 0 & -3 & 0 end{bmatrix} + \begin{bmatrix} 4 & 4 & 4 & 4 \\ -5 & -5 & -5 & -5 end{bmatrix} )

( \begin{bmatrix} -5 & -1 & 1 & -3 \\ 3 & 3 & 0 & 0 end{bmatrix} + \begin{bmatrix} 4 & 4 & 4 & 4 \\ 5 & 5 & 5 & 5 end{bmatrix} )

( \begin{bmatrix} -5 & -1 & 1 & -3 \\ 3 & 3 & 0 & 0 end{bmatrix} + \begin{bmatrix} 4 & 4 & 4 & 4 \\ -5 & -5 & -5 & -5 end{bmatrix} )

( \begin{bmatrix} -5 & -1 & 1 & -3 \\ 3 & 3 & 0 & 0 end{bmatrix} + \begin{bmatrix} -4 & -4 & -4 & -4 \\ -5 & -5 & -5 & -5 end{bmatrix} )

Explanation:

Step1: Understand the translation rule

For a point \((x,y)\) translated \(a\) units to the right and \(b\) units down, the new point is \((x + a,y - b)\). Here \(a = 4\) and \(b=5\).

Step2: Represent the polygon as a matrix

The vertices of the polygon \((-5,3),(-1,3),(1,0),(-3,0)\) can be represented as a \(2\times4\) matrix \(

$$\begin{bmatrix}-5&-1&1&-3\\3&3&0&0\end{bmatrix}$$

\) (first row for \(x -\) coordinates and second row for \(y -\) coordinates).

Step3: Represent the translation as a matrix

The translation of \(4\) units to the right (add \(4\) to \(x -\) coordinates) and \(5\) units down (add \(- 5\) to \(y -\) coordinates) for \(4\) vertices is represented as a \(2\times4\) matrix \(

$$\begin{bmatrix}4&4&4&4\\-5&-5&-5&-5\end{bmatrix}$$

\)

Step4: Use matrix addition for translation

By the rule of matrix addition for translation \(\text{New Matrix}=\text{Original Matrix}+\text{Translation Matrix}\), we get \(

$$\begin{bmatrix}-5&-1&1&-3\\3&3&0&0\end{bmatrix}$$

+

$$\begin{bmatrix}4&4&4&4\\-5&-5&-5&-5\end{bmatrix}$$

\)

Answer:

C. \(

$$\begin{bmatrix}-5&-1&1&-3\\3&3&0&0\end{bmatrix}$$

+

$$\begin{bmatrix}4&4&4&4\\-5&-5&-5&-5\end{bmatrix}$$

\)