Answer: The required scale factor that could be used to dilate the original rectangle to final rectangle is 2.
Step-by-step explanation: Given that a rectangle is dilated using the scale factor of 6. The image is dilated using the scale factor of [tex]\dfrac{1}{3}.[/tex]
We are to find the scale factor that could be used to dilate the original rectangle to final rectangle.
Let a, a' and a'' be the the lengths of the corresponding sides of the original rectangle, first dilated rectangle and the final dilated rectangle respectively.
Also, let s and s' be the first scale factor and the second scale factor respectively.
Then, according to the given information, we have
[tex]s=6,~~s'=\dfrac{1}{3}.[/tex]
From the definition of a scale factor, we have
[tex]s=\dfrac{a'}{a}\\\\\\\Rightarrow 6=\dfrac{a'}{a}\\\\\Rightarrow a'=6a~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]
and
[tex]s''=\dfrac{a''}{a'}\\\\\\\Rightarrow \dfrac{1}{3}=\dfrac{a''}{a'}\\\\\\\Rightarrow \dfrac{1}{3}=\dfrac{a''}{6a}~~~~~~~~~~~~[\textup{Using equation (i)}]\\\\\\\Rightarrow \dfrac{a''}{a}=\dfrac{6}{3}\\\\\\\Rightarrow \dfrac{a''}{a}=2\\\\\\\Rightarrow S=2.[/tex]
Thus, the required scale factor that could be used to dilate the original rectangle to final rectangle is 2.
