Question

reflect the given triangle over the line y = -x.
\\(\

$$\begin{bmatrix}0&7&-1\\\\0&-3&-4\\end{bmatrix}$$

\\)
\\(\

$$\begin{bmatrix}0&3&4\\\\0&?&\\end{bmatrix}$$

\\)

Explanation:

Step1: Recall the reflection formula

When reflecting a point \((x,y)\) over the line \(y = -x\), the transformation rule is \((x,y)\to(-y,-x)\).

Step2: Apply the formula to the given points

For the point \((7,-3)\):
Using the rule \((x = 7,y=-3)\to(-y,-x)\), we substitute \(x = 7\) and \(y=-3\).
First, \(-y=-(-3)=3\), and \(-x=-7\).

For the point \((-1,-4)\):
Using the rule \((x=-1,y = -4)\to(-y,-x)\), we substitute \(x=-1\) and \(y=-4\).
First, \(-y=-(-4) = 4\), and \(-x=-(-1)=1\).

Answer:

The matrix after reflection is \(

$$\begin{bmatrix}0&3&4\\0&-7&1\end{bmatrix}$$

\). So the green box (for the second - column, second - row) is \(-7\) and the gray box (for the third - column, second - row) is \(1\).