# The lengths of the sides of triangle ABC are 3 cm, 4cm, and 6 cm. How do you determine the least possible perimeter of a triangle similar to triangle ABC which has one side of length 12 cm?

Sep 5, 2017

26cm

#### Explanation:

we want a triangle with shorter sides (smaller perimeter) and we got 2 similar triangles , since triangles are similar the corresponding sides would be in ratio.

To get triangle of shorter perimeter we have to use the longest side of $\triangle A B C$ put 6cm side corresponding to 12cm side.
Let triangle ABC ~ triangle DEF

6cm side corresponding to 12 cm side .
therefore, $\frac{A B}{D E} = \frac{B C}{E F} = \frac{C A}{F D} = \frac{1}{2}$
So the perimeter of ABC is half of the perimeter of DEF .
perimeter of DEF = 2×(3+4+6)=2×13=26cm

Sep 5, 2017

$26 c m$

#### Explanation:

Similar triangles have the same shape because they have the same angles.

They are of different sizes, but their sides are in the same ratio.

In $\Delta A B C ,$ the sides are $\text{ "3" ":" "4" ":" } 6$

For the smallest perimeter of the other triangle, the longest side must be $12$cm. The sides will therefore all be twice as long.

$\Delta A B C : \text{ "3" ":" "4" ":" } 6$
New $\Delta : \text{ "6" ":" "8" ":" } 12$

The perimeter of $\Delta A B C = 6 + 4 + 3 = 13 c m$

The perimeter of the second triangle will be $13 \times 2 = 26 c m$

This can be confirmed by adding the sides:

$6 + 8 + 12 = 26 c m$