Question

Find the sequence of transformations for the given figure.

• A. Dilation of $ABCD$ about the origin with a scale factor of $2$; Dilation of $EFGH$ about the origin with a scale factor of $0.5$, rotation of ${180}^{\circ }$ clockwise about the origin
  
• B. Dilation of $ABCD$ about the origin with a scale factor of $2$, rotation of ${180}^{\circ }$ clockwise about the origin; Dilation of $EFGH$ about the origin with a scale factor of $0.5$, rotation of ${180}^{\circ }$ clockwise about the origin
  
• C. Dilation of $ABCD$ about the origin with a scale factor of $2,$ rotation of ${180}^{\circ }$ clockwise about the origin; Dilation of $EFGH$ about the origin with a scale factor of $0.5$
  
• D. Dilation of $ABCD$ about the origin with a scale factor of $0.5$; Dilation of $EFGH$ about origin with a scale factor of $2$, rotation ${180}^{\circ }$ clockwise about the origin
  

Answer 1

The correct option is C Dilation of $ABCD$ about the origin with a scale factor of $2,$ rotation of ${180}^{\circ }$ clockwise about the origin; Dilation of $EFGH$ about the origin with a scale factor of $0.5$


-Scale factor of a dilation is the ratio of the length in the image to the corresponding length in the original figure.
For $ABCD$,e
$AB=2\text{units}$ and $A^{\prime }{B}^{\prime }=4\text{units}$
$\text{Scale factor}=\frac{\text{Length of A'B'}}{\text{Length of AB}}$
$\Rightarrow \frac{4}{2}=2$

Similarly for sides $BC,CD,$ and $DA,$ we get the scale factor as $2$.

For $EFGH,$
$\text{Scale factor}=\frac{\text{Length of E'F'}}{\text{Length of EF}}$

$EF=4\text{units}$ and $E^{\prime }{F}^{\prime }=2\text{units}$
$\Rightarrow \frac{2}{4}=0.5$
Similarly, for sides $FG,GH,$ and $HE$ we get the scale factor as $0.5$.

Hence, we can see we have to dilate figure $ABCD$ with a scale factor of $2$, and figure $EFGH$ with a scale factor of $0.5$. By using this information, we can eliminate some of the options.

In option A, the dilation of figure $ABCD$ is with the scale factor of $0.5$, which will produce a smaller figure, so we can say option A is incorrect. Also, we can see the resulting figure after the instructions in option A.
Resultant image:


Hence, option A is incorrect.

In option B:
Dilation of $ABCD$ about the origin with a scale factor of $2$, rotation ${180}^{\circ }$ clockwise about the origin.
Dilation of $EFGH$ about the origin with a scale factor of $0.5$, but we don’t need to rotate ${180}^{\circ }$ clockwise about the origin.
Resultant image:


Hence, option B is incorrect.

In option C:
Dilation of $ABCD$ about the origin with a scale factor of $2$, we also need to change the orientation of $ABCD$.
Dilation about the origin with a scale factor of $0.5$, but we don’t need to rotate ${180}^{\circ }$ clockwise about the origin.
Resultant image:



Hence, option C is incorrect.

Dilation of $ABCD$ about the origin with a scale factor of 2, and rotation ${180}^{\circ }$ clockwise about the origin will produce the following:



Dilation of $EFGH$ about the origin with a scale factor of $0.5$ will produce the following:


We get the following image after combining the above two transformation results:


Hence, the correct answer is option D.