Since rectangle ABCD is similar to HGFE, the ratios of the lengths of their corresponding sides are equal. We can infer form our picture that AD is corresponding to EH and DC is corresponding to HG, so lets find the ratios of those corresponding sides and establish a proportion to find the length of HG:
[tex] \frac{AD}{EH} = \frac{DC}{HG} [/tex]
We know that [tex] AD=45[/tex], [tex]EH=27[/tex], and [tex]DC=15[/tex], so lets replace those values in our proportion:
[tex] \frac{45}{27} = \frac{15}{HG} [/tex]
[tex]HG= \frac{27*15}{45} [/tex]
[tex]HG= \frac{405}{45} [/tex]
[tex]HG=9[/tex]
We can conclude that the length of the segment HG is 9.
