To determine the perimeter of triangle DEF, we need to find the ratio between the corresponding sides of triangle ABC and triangle DEF.
Let's calculate the ratio of the lengths of the corresponding sides:
Ratio = Length of corresponding sides in ABC / Length of corresponding sides in DEF
Ratio = AB / DE
Ratio = 15 / 12
Ratio = 5/4
Now, we can use this ratio to find the lengths of the other sides of triangle DEF.
If the perimeter of triangle ABC is 40, we can assume that the corresponding sides of triangle DEF have a perimeter of x.
Using the ratio, we can set up the equation:
Perimeter of triangle ABC / Perimeter of triangle DEF = Length of corresponding sides in ABC / Length of corresponding sides in DEF
40 / x = 15 / 12
Next, we can cross-multiply:
15x = 40 * 12
15x = 480
Now, divide both sides by 15:
x = 480 / 15
x = 32
Therefore, the perimeter of triangle DEF is 32.