Use matrices to determine the coordinates of the vertices of the rotated figure. Then graph the pre-image and the image of the same coordinate grid. (Pictureprovided)

Answer:The coordinates of the vertices of the rotated figure are :U' (1 , -6), V' (-8 , -4), W' (-5 , 7) ⇒ the right answer is figure (d)Step-by-step explanation:* Lets study the matrices of the Rotation by 180°  - When we rotate a point around the origin by 180° clockwise  or anti-clockwise, we change the sign of the x-coordinate and  the y-coordinate of the point- Then matrix of the rotation 180° is  [tex]\left[\begin{array}{ccc}-1&0\\0&-1\end{array}\right][/tex]* Now lets solve the problem - We will multiply the matrix of the rotation  by each point to  find the image of each point - The dimension of the matrix of the rotation  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 U is (-1 , 6)∴ [tex]U'=\left[\begin{array}{ccc}-1&0\\0&-1\end{array}\right]\left[\begin{array}{ccc}-1\\6\end{array}\right]=[/tex]   [tex]\left[\begin{array}{ccc}(-1)(-1)+(0)(6)\\(0)(-1)+(-1)(6)\end{array}\right]=\left[\begin{array}{ccc}1\\-6\end{array}\right][/tex]∴ U' = (1 , -6)∵ Point V is (8 , 4)∴ [tex]V'=\left[\begin{array}{ccc}-1&0\\0&-1\end{array}\right]\left[\begin{array}{ccc}8\\4\end{array}\right]=[/tex]   [tex]\left[\begin{array}{ccc}(-1)(8)+(0)(4)\\(0)(8)+(-1)(4)\end{array}\right]=\left[\begin{array}{ccc}-8\\-4\end{array}\right][/tex]∴ V' = (-8 , -4)∵ Point W is (5 , -7)∴ [tex]W'=\left[\begin{array}{ccc}-1&0\\0&-1\end{array}\right]\left[\begin{array}{ccc}5\\-7\end{array}\right]=[/tex]   [tex]\left[\begin{array}{ccc}(-1)(5)+(0)(-7)\\(0)(5)+(-1)(-7)\end{array}\right]=\left[\begin{array}{ccc}-5\\7\end{array}\right][/tex]∴ W' = (-5 , 7)* Now look to the figures to find the right answer∵ The images of the points are U' (1 , -6), V' (-8 , -4), W' (-5 , 7)∴ The right answer is figure (d)