Question

The diagram above shows rectangle ABCD.

Which points are in the image of rectangle ABCD under the transformation T = [3 0] over [ 0 3 ] but they’re in the same brackets i can’t type it correctly /:

the options are pictured above

Answer 1

Find the picture below.

Answer 2

The images of the resulting rectangle are $A'(x,y) = (3,9)$, $B'(x,y) = (12,9)$, $C'(x,y) = (12,3)$ and $D'(x,y) = (3,3)$, respectively. (Correct choice: A)

How to determine the resulting quadrillateral by rotation matrix

The resulting points are found by the following vectorial expression:

$\left[\begin{array}{cccc}A'_{x}&B'_{x}&C'_{x}&D'_{x}\\A'_{y}&B'_{y}&C'_{y}&D'_{y}\end{array}\right] = \left[\begin{array}{ccc}3&0\\0&3\end{array}\right] \cdot \left[\begin{array}{cccc}A_{x}&B_{x}&C_{x}&D_{x}\\A_{y}&B_{y}&C_{y}&D_{y}\end{array}\right]$ (1)

If we know that $A(x,y) = (1,3)$, $B(x,y) = (4,3)$, $C(x,y) = (4,1)$ and $D(x,y) = (1,1)$, then the resulting points are:

$\left[\begin{array}{cccc}A'_{x}&B'_{x}&C'_{x}& D'_{x}\\A'_{y}&B'_{y}&C'_{y}& D'_{y}\end{array}\right] = \left[\begin{array}{cc}3&0\\0&3\end{array}\right] \cdot \left[\begin{array}{cccc}1&4&4&1\\3&3&1&1\end{array}\right]$

$\left[\begin{array}{cccc}A'_{x}&B'_{x}&C'_{x}&D'_{x}\\A'_{y}&B'_{y}&C'_{y}&D'_{y}\end{array}\right] = \left[\begin{array}{cccc}3&12&12&3\\9&9&3&3\end{array}\right]$

The images of the resulting rectangle are $A'(x,y) = (3,9)$, $B'(x,y) = (12,9)$, $C'(x,y) = (12,3)$ and $D'(x,y) = (3,3)$, respectively. (Correct choice: A) $\blacksquare$

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