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)
