In the accompanying diagram, triangle ABCis similar to triangle DEF, AC = 6, AB = BC = 12, and DF = 8. Find the perimeter of triangle DEF.  

(8/6)(12+12+6)

To find the perimeter of triangle DEF, we need to determine the lengths of DE and EF.

Since triangle ABC is similar to triangle DEF, corresponding sides are in proportion. In other words, the ratio of the lengths of corresponding sides in the two triangles is the same.

We can use this proportion to find the length of DF.

In triangle ABC, AB = BC = 12. Since AB = BC, triangle ABC is an isosceles triangle, which means that angle BAC = angle BCA.

Similarly, in triangle DEF, corresponding angles are congruent. Therefore, angle DEF = angle DFE.

Using the Side-Angle-Side (SAS) similarity property, we can conclude that triangles ABC and DEF are similar.

So, the proportion between the corresponding sides is:

DE / AB = EF / BC

We can substitute the known values into this proportion:

DE / 12 = EF / 12

Since AB = BC = 12 in triangle ABC and AB = BC = 12 in triangle DEF.

Simplifying this proportion equation, we get:

DE = EF

Let's call this common length x.

So, x = DE = EF.

Now, we can find the length of DF using the Pythagorean theorem. In right triangle DEF, we have:

DF^2 = DE^2 + EF^2

Substituting the values, we get:

8^2 = x^2 + x^2

64 = 2x^2

Dividing both sides by 2, we get:

32 = x^2

Taking the square root of both sides, we get:

x = √32

Simplifying, we get:

x = 4√2

Hence, the length of DE = EF = 4√2.

Finally, to find the perimeter of DEF, we add the lengths of all three sides:

Perimeter of triangle DEF = DE + EF + DF

= 4√2 + 4√2 + 8

= 8√2 + 8

Therefore, the perimeter of triangle DEF is 8√2 + 8.