Question

Given below are two rectangles ABCD and EFGH drawn on the coordinate plane with coordinate values shown at all the vertices. Which of the following sets of transformations has been applied to get rectangle EFGH from ABCD?

• A. Counterclockwise rotation of ${90}^o$, translation of 6 units up, and dilation about (1, 4) with a scale factor of 2.5
• B. Translation of 6 units right, translation of 6 units up, and dilation about (1, 4) with a scale factor of 2.5
• C. Clockwise rotation of ${270}^o$, translation of 6 units right, and dilation about (1, 4) with scale factor of 2.25
• D. Reflection along the x-axis, dilation with a scale factor of 4.5, and reflection along the y-axis

Answer 1

The correct option is B Translation of 6 units right, translation of 6 units up, and dilation about (1, 4) with a scale factor of 2.5

From the given figure, we can observe that:
The two figures ABCD and EFGH are varying in length, which means ABCD would certainly have gone through dilation by some scale factor. To calculate the scale factor, we need to calculate the length of any two corresponding sides of ABCD and EFGH and then take their ratio.

Consider the sides AD and EH:
The length of AB will be the difference between the x-coordinates of A and B
Since $A=(-5,-2)$ and $D=(-2,-2)$,
$\Rightarrow$ x-coordinate of A is −5 and x-coordinate of D is −2
$\Rightarrow$ Length of $AD=-2-(-5)=-2+5=3$
Similarly, length of $EH=8.5-1=7.5(E=(1,4)$ and $H=(8.5,4))$

Now, scale factor $=\frac{7.5}{3}=2.5$ (As scale factor $=\frac{\text{Side length of image}}{\text{Corresponding side length of preimage}})$

This shows that there is a dilation by a scale factor of 2.5.
Clearly, ABCD has not been rotated, because orientation of the corresponding vertices are the same in both the figures.

Now, let’s check the options one-by-one to get the correct sequence of transformations

In option A: Counterclockwise rotation of ${90}^o$, translation of 6 units up, and dilation with scale factor of 2.5