$ \triangle \mathrm{ABC} \sim \triangle \mathrm{EFD} $. If $ \mathrm{AB}: \mathrm{BC}: \mathrm{CA}=4: 3: 5 $ and the perimeter of $ \triangle \mathrm{DEF} $ is $ 36 \mathrm{~cm} $, find all the sides of $ \triangle \mathrm{DEF} $.

Given:

\( \triangle \mathrm{ABC} \sim \triangle \mathrm{EFD} \).

\( \mathrm{AB}: \mathrm{BC}: \mathrm{CA}=4: 3: 5 \) and the perimeter of \( \triangle \mathrm{DEF} \) is \( 36 \mathrm{~cm} \).

To do:

We have to find all the sides of \( \triangle \mathrm{DEF} \).

Solution:

\( \triangle \mathrm{ABC} \sim \triangle \mathrm{EFD} \)

The ratio of the perimeters of two similar triangles is same as the ratio of their corresponding sides.

Let $AB=4x, BC=3x$ and $CA=5x$ and let $DE=4y, EF=3y$ and $DF=5y$

This implies,

$4y+3y+5y=36$

$12y=36$

$y=3\ cm$

$4y=4(3)=12\ cm$

$3y=3(3)=9\ cm$

$5y=5(3)=15\ cm$

Hence, the sides of the \( \triangle \mathrm{DEF} \) are $DE=12\ cm, EF=9\ cm$ and $DF=15\ cm$.