Question

If two similar triangles have ratio of their areas as 16:25, then the ratio of their perimeters will be

• A.
  9:25
• B.
  3:5
• C.
  4:5
• D.
  16:25

Answer 1

The correct option is C

4:5
The ratio of perimeters of two similar triangles is equal to the ratio of their correcponding sides.

$\therefore \frac{\text{Perimeter of}\Delta ABC}{\text{Perimeter of}\Delta DEF}=\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}\frac{(Perimeterof\Delta ABC{)}^2}{(Perimeterof\Delta DEF{)}^2}=\frac{A{B}^2}{D{E}^2}=\frac{B{C}^2}{E{F}^2}=\frac{A{C}^2}{D{F}^2}...\left(1\right)$

And The ratio of the areas of two similar triangles is equal to the ratio of the square of their correponding sides.

$\frac{ar\left(\Delta ABC\right)}{ar\left(\Delta DEF\right)}=\frac{A{B}^2}{D{E}^2}=\frac{B{C}^2}{E{F}^2}=\frac{A{C}^2}{D{F}^2}...\left(2\right)$

From (1) and (2) we get:

$\frac{ar\left(\Delta ABC\right)}{ar\left(\Delta DEF\right)}=\frac{(Perimeterof\Delta ABC{)}^2}{(Perimeterof\Delta DEF{)}^2}\Rightarrow \frac{16}{25}=\frac{(Perimeterof\Delta ABC{)}^2}{(Perimeterof\Delta DEF{)}^2}\Rightarrow \sqrt{\frac{16}{25}}=\frac{\left(Perimeterof\Delta ABC\right)}{\left(Perimeterof\Delta DEF\right)}\Rightarrow \frac{4}{5}=\frac{Perimeterof\Delta ABC}{Perimeterof\Delta DEF}$

$\therefore \text{The Ratio of perimeters is 4:5}$

So, the option c is correct.