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User Pranta Saha
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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.


\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}

Perimeter of Triangle is express as the sum of the length of all the sides of traingle


\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}

The scale Factor :


\begin{gathered} \Delta ABC\text{ }\approx\Delta DEF \\ So, \\ (AB)/(DE)=(BC)/(EF)=(CA)/(FD) \\ \text{ Substitute the values and find the ratio} \\ (10.81)/(16.2)=(6)/(EF)=(9)/(FD) \\ 0.66=(6)/(EF)=(9)/(FD) \\ \text{ So, the scale factor = 0.66} \end{gathered}

Since the ratio of perimeter of triangle ABC and DEF are same as the ratio of thier corresponding sides

So,


\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=(25.81)/(0.667) \\ Perimeter\text{ of }\Delta DEF=38.69\text{ unit} \end{gathered}

So, Perimeter of triangle DEF = 38.69 unit

User Chirag Vidani
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