Question

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 1

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

 $\left[\begin{array}{ccc}-1&0\\0&-1\end{array}\right]$

* 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)

∴ $U'=\left[\begin{array}{ccc}-1&0\\0&-1\end{array}\right]\left[\begin{array}{ccc}-1\\6\end{array}\right]=$

  $\left[\begin{array}{ccc}(-1)(-1)+(0)(6)\\(0)(-1)+(-1)(6)\end{array}\right]=\left[\begin{array}{ccc}1\\-6\end{array}\right]$

∴ U' = (1 , -6)

∵ Point V is (8 , 4)

∴ $V'=\left[\begin{array}{ccc}-1&0\\0&-1\end{array}\right]\left[\begin{array}{ccc}8\\4\end{array}\right]=$

  $\left[\begin{array}{ccc}(-1)(8)+(0)(4)\\(0)(8)+(-1)(4)\end{array}\right]=\left[\begin{array}{ccc}-8\\-4\end{array}\right]$

∴ V' = (-8 , -4)

∵ Point W is (5 , -7)

∴ $W'=\left[\begin{array}{ccc}-1&0\\0&-1\end{array}\right]\left[\begin{array}{ccc}5\\-7\end{array}\right]=$

  $\left[\begin{array}{ccc}(-1)(5)+(0)(-7)\\(0)(5)+(-1)(-7)\end{array}\right]=\left[\begin{array}{ccc}-5\\7\end{array}\right]$

∴ 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)