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(D) Given the similarity $\Delta ABC \sim \Delta DEF$ with the correspondence $ABC \leftrightarrow EFD$,the ratio of corresponding sides is equal. Thu

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(D) Given the similarity $\Delta ABC \sim \Delta DEF$ with the correspondence $ABC \leftrightarrow EFD$,the ratio of corresponding sides is equal.
Thus,$\frac{AB}{EF} = \frac{BC}{FD} = \frac{CA}{DE} = k$.
Given $AB : BC : CA = 4 : 3 : 5$,let $AB = 4x, BC = 3x, CA = 5x$.
From the correspondence $ABC \leftrightarrow EFD$,we have $EF = BC = 3x$,$FD = CA = 5x$,and $DE = AB = 4x$.
The perimeter of $\Delta DEF = EF + FD + DE = 3x + 5x + 4x = 12x$.
Given the perimeter is $36$,we have $12x = 36$,which implies $x = 3$.
Therefore,the sides are $EF = 3(3) = 9$,$FD = 5(3) = 15$,and $DE = 4(3) = 12$.
Thus,the sides of $\Delta DEF$ are $12, 9, 15$.

$\Delta ABC \sim \Delta DEF$ for the correspondence $ABC \leftrightarrow EFD$. $AB : BC : CA = 4 : 3 : 5$. If the perimeter of $\Delta DEF$ is $36$,find the measures of all the sides of $\Delta DEF$.

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