Answer:
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)
Step-by-step 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]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D0%261%5C%5C1%260%5Cend%7Barray%7D%5Cright%5D)
- 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]=](https://tex.z-dn.net/?f=R%27%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D0%261%5C%5C1%260%5Cend%7Barray%7D%5Cright%5D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D-2%5C%5C5%5Cend%7Barray%7D%5Cright%5D%3D)
![\left[\begin{array}{c}(0)(-2)+(1)(5)\\(1)(-2)+(0)(5)\end{array}\right]=\left[\begin{array}{c}5\\-2\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D%280%29%28-2%29%2B%281%29%285%29%5C%5C%281%29%28-2%29%2B%280%29%285%29%5Cend%7Barray%7D%5Cright%5D%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D5%5C%5C-2%5Cend%7Barray%7D%5Cright%5D)
∴ 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]=](https://tex.z-dn.net/?f=S%27%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D0%261%5C%5C1%260%5Cend%7Barray%7D%5Cright%5D%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D5%5C%5C3%5Cend%7Barray%7D%5Cright%5D%3D)
![\left[\begin{array}{c}(0)(5)+(1)(3)\\(1)(5)+(0)(3)\end{array}\right]=\left[\begin{array}{c}3\\5\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D%280%29%285%29%2B%281%29%283%29%5C%5C%281%29%285%29%2B%280%29%283%29%5Cend%7Barray%7D%5Cright%5D%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D3%5C%5C5%5Cend%7Barray%7D%5Cright%5D)
∴ 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]=](https://tex.z-dn.net/?f=T%27%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D0%261%5C%5C1%260%5Cend%7Barray%7D%5Cright%5D%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D6%5C%5C-7%5Cend%7Barray%7D%5Cright%5D%3D)
![\left[\begin{array}{c}(0)(6)+(1)(-7)\\(1)(6)+(0)(-7)\end{array}\right]=\left[\begin{array}{c}-7\\6\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D%280%29%286%29%2B%281%29%28-7%29%5C%5C%281%29%286%29%2B%280%29%28-7%29%5Cend%7Barray%7D%5Cright%5D%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D-7%5C%5C6%5Cend%7Barray%7D%5Cright%5D)
∴ 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)
