Answer:
The coordinates of the vertices of the reflected figure are :
R' is (-3 , -7), S' is (5 , -3), T' is (6 , 5) ⇒the right answer is figure (a)
Step-by-step explanation:
* Lets study the matrices of the reflection
- The matrix of the reflection across the x-axis is
[tex]\left[\begin{array}{cc}1&0\\0&-1\end{array}\right][/tex]
- Because when we reflect any point across the x-axis we
change the sign of the y-coordinate
- The matrix of the reflection across the y-axis is
[tex]\left[\begin{array}{cc}-1&0\\0&1\end{array}\right][/tex]
- Because when we reflect any point across the y-axis we
change the sign of the x-coordinate
* Now lets solve the problem
- We will multiply the matrix of the reflection across the x-axis
by each point to find the image of each point
- The dimension of the matrix of the reflection across the x-axis
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
∵ Point R is (-3 , 7)
∴ [tex]R'=\left[\begin{array}{cc}1&0\\0&-1\end{array}\right]\left[\begin{array}{c}-3\\7\end{array}\right]=[/tex]
[tex]\left[\begin{array}{c}(1)(-3)+(0)(7)\\(0)(-3)+(-1)(7)\end{array}\right]=\left[\begin{array}{c}-3\\-7\end{array}\right][/tex]
∴ R' is (-3 , -7)
∵ Point S is (5 , 3)
∴ [tex]S'=\left[\begin{array}{cc}1&0\\0&-1\end{array}\right]\left[\begin{array}{c}5\\3\end{array}\right]=[/tex]
[tex]\left[\begin{array}{c}(1)(5)+(0)(3)\\(0)(5)+(-1)(3)\end{array}\right]=\left[\begin{array}{c}5\\-3\end{array}\right][/tex]
∴ S' is (5 , -3)
∵ Point T is (6 , -5)
∴ [tex]T'=\left[\begin{array}{cc}1&0\\0&-1\end{array}\right]\left[\begin{array}{c}6\\-5\end{array}\right]=[/tex]
[tex]\left[\begin{array}{c}(1)(6)+(0)(-5)\\(0)(6)+(-1)(-5)\end{array}\right]=\left[\begin{array}{c\pi }6\\5\end{array}\right][/tex]
∴ T' is (6 , 5)
* Look to the answer and find the correct figure
- In figure (d) R' is (-3 , -7), S' is (5 , -3), T' is (6 , 5)
∴ The right answer is figure (a)
