Question

the diagram below shows rectangle abcd.
which points are in the image of rectangle abcd under the transformation: $r = \

$$\begin{bmatrix} 3 & 0 \\\\ 0 & 3 \\end{bmatrix}$$

$?
$a(3, 9), b(12, 9), c(12, 3), d(3, 3)$
$a(3, 3), b(12, 3), c(12, 9), d(3, 12)$
$a(4, 7), b(7, 8), c(7, 4), d(4, 4)$
$a(9, 3), b(9, 12), c(3, 12), d(3, 3)$

Explanation:

Step1: Find coordinates of rectangle ABCD

Assume \(A(1,3)\), \(B(4,3)\), \(C(4,1)\), \(D(1,1)\) (from the graph, assuming grid - based coordinates).
For a point \((x,y)\) under the transformation \(T=

$$\begin{bmatrix}3&0\\0&3\end{bmatrix}$$

\), the transformation formula is \(

$$\begin{bmatrix}x'\\y'\end{bmatrix}$$

=

$$\begin{bmatrix}3&0\\0&3\end{bmatrix}$$

$$\begin{bmatrix}x\\y\end{bmatrix}$$

=

$$\begin{bmatrix}3x\\3y\end{bmatrix}$$

\).

Step2: Transform point A

For \(A(1,3)\), \(x = 1,y = 3\). Then \(x'=3\times1 = 3\), \(y'=3\times3=9\).

Step3: Transform point B

For \(B(4,3)\), \(x = 4,y = 3\). Then \(x'=3\times4 = 12\), \(y'=3\times3 = 9\).

Step4: Transform point C

For \(C(4,1)\), \(x = 4,y = 1\). Then \(x'=3\times4=12\), \(y'=3\times1 = 3\).

Step5: Transform point D

For \(D(1,1)\), \(x = 1,y = 1\). Then \(x'=3\times1=3\), \(y'=3\times1 = 3\).

Answer:

\(A'(3,9)\), \(B'(12,9)\), \(C'(12,3)\), \(D'(3,3)\)