Question

Use matrices to determine the coordinates of the vertices of the reflected figure. Then graph the pre-image and the image on the same coordinate grid. (Picture provided)

Answer 1

The coordinates of the vertices of the reflected figure are :

R' is (5 , -2) , S' is (3 , 5) , T' is (-7 , 6) ⇒ the right answer is (d)

Explanation:

* When you reflect a point across the line y = x, the x-coordinate

and y-coordinate change their places.

- If the point is (x , y) then its image is (y , x)

* If you reflect over the line y = -x, the x-coordinate and y-coordinate

change their places and their signs

- If the point is (x , y) then its image is (-y , -x)

* Lets study the matrix of the reflection about the line y = x

- The matrix of the reflection about the line y = x is

`\left[\begin{array}{cc}0&1\\1&0\end{array}\right]`

- Because the x-coordinate and y-coordinate change places.

* Now lets solve the problem

- We will multiply the matrix of the reflection about y = x

by each point to find the image of each point

- The dimension of the matrix of the reflection about y = x

is 2×2 and the dimension of the matrix of each point is 2×1,

then the dimension of the matrix of each image is 2×1

∵ The point R is (-2 , 5)

∴
`R'=\left[\begin{array}{cc}0&1\\1&0\end{array}\right]\left[\begin{array}{cc}-2\\5\end{array}\right]=`

`\left[\begin{array}{c}(0)(-2)+(1)(5)\\(1)(-2)+(0)(5)\end{array}\right]=\left[\begin{array}{c}5\\-2\end{array}\right]`

∴ R' is (5 , -2)

∵ The point S is (5 , 3)

∴
`S'=\left[\begin{array}{cc}0&1\\1&0\end{array}\right]\left[\begin{array}{c}5\\3\end{array}\right]=`

`\left[\begin{array}{c}(0)(5)+(1)(3)\\(1)(5)+(0)(3)\end{array}\right]=\left[\begin{array}{c}3\\5\end{array}\right]`

∴ S' is (3 , 5)

∵ The point T is (6 , -7)

∴
`T'=\left[\begin{array}{cc}0&1\\1&0\end{array}\right]\left[\begin{array}{c}6\\-7\end{array}\right]=`

`\left[\begin{array}{c}(0)(6)+(1)(-7)\\(1)(6)+(0)(-7)\end{array}\right]=\left[\begin{array}{c}-7\\6\end{array}\right]`

∴ T' is (-7 , 6)

* Lets look to the figures to find the right answer

∵ The R' is (5 , -2) , S' is (3 , 5) , T' is (-7 , 6)

∴ The right answer is (d)