Given:
A figure of dilation and the center of dilation is O.
To find:
The value of y.
Solution:
The distance between center and vertex of image (d') is proportional to the distance between center and the corresponding vertex of original figure (d).
[tex]d'\propto d[/tex]
[tex]d'=kd[/tex]
[tex]\dfrac{d'}{d}=k[/tex]
Where, k is the scale factor or constant of proportionality.
So, the scale factor for the given dilation is
[tex]k=\dfrac{OB'}{OB}[/tex]
[tex]k=\dfrac{36}{16}[/tex]
[tex]k=\dfrac{9}{4}[/tex]
We know that,
[tex]k=\dfrac{\text{Side length of image}}{\text{Corresponding side length of original figure}}[/tex]
[tex]\dfrac{9}{4}=\dfrac{A'C'}{AC}[/tex]
[tex]\dfrac{9}{4}=\dfrac{27}{y}[/tex]
[tex]9\times y=27\times 4[/tex]
Divide both sides by 9.
[tex]y=\dfrac{27\times 4}{9}[/tex]
[tex]y=3\times 4[/tex]
[tex]y=12[/tex]
Therefore, the value of y is 12.