Respuesta :
The given triangle ABC and DEF are similar
From the properties of the similar triangle :
The ratio of the corresponding sides of similar traingle are equal and the ratio of perimeter is also equal to the ratio of corresponding sides
In the triangle ABC, AC=9, BC=6, and the angle B is 90 degree
then apply pythagoras to find the third side of triangle:
Pythagoras Theorem: The square of sum of base and perpendicular is equal to the square of Hypotenuse.
[tex]\begin{gathered} \text{ By Pythagoras :} \\ AB^2=BC^2+CA^2 \\ AB^2=6^2+9^2 \\ AB^2=36+81 \\ AB^2=117 \\ AB=\sqrt[]{117} \\ AB=10.81 \end{gathered}[/tex]Perimeter of Triangle is express as the sum of the length of all the sides of traingle
[tex]\begin{gathered} \text{Perimeter of }\Delta ABC=AB+BC+CA \\ \text{ Perimeter of }\Delta ABC=10.81+6+9 \\ \text{Perimeter of }\Delta ABC=25.81 \end{gathered}[/tex]The scale Factor :
[tex]\begin{gathered} \Delta ABC\text{ }\approx\Delta DEF \\ So, \\ \frac{AB}{DE}=\frac{BC}{EF}=\frac{CA}{FD} \\ \text{ Substitute the values and find the ratio} \\ \frac{10.81}{16.2}=\frac{6}{EF}=\frac{9}{FD} \\ 0.66=\frac{6}{EF}=\frac{9}{FD} \\ \text{ So, the scale factor = 0.66} \end{gathered}[/tex]Since the ratio of perimeter of triangle ABC and DEF are same as the ratio of thier corresponding sides
So,
[tex]\begin{gathered} \frac{Perimeter\text{ of }\Delta ABC}{Perimeter\text{ of }\Delta DEF}=0.667 \\ \text{ Simplify for the perimeter of }\Delta DEF \\ Perimeter\text{ of }\Delta DEF=\frac{Perimeter\text{ of }\Delta ABC}{0.667} \\ \text{ Substitute the value of }Perimeter\text{ of }\Delta ABC=25.81 \\ Perimeter\text{ of }\Delta DEF=\frac{25.81}{0.667} \\ Perimeter\text{ of }\Delta DEF=38.69\text{ unit} \end{gathered}[/tex]So, Perimeter of triangle DEF = 38.69 unit
