What is the perímeter of the triangle ABC?

Question

I am froze in this question after some hours.

It's a homework (without a detailed solution in the book) that I am trying to do.

I am graduated in high school, but there is some years I didn't study these topics.

I am currently studing about cosin and sin law, and triangles similarity, and I think one of these tools would help to solve the problem, since it's a problem from the chapter of these subjects.

I've froze because $$ \frac{DB}{DA} = \frac{DB}{DC}$$ is resulting in a false equality.

See here:

$$ \frac{4}{x} = \frac{4}{x+6} \\ \text{ } \\ \text{which results in} \\ \text{ } \\ 4x = 4x + 24 \\ \text{ } \\ \text{what led us to: } 0 = 24 $$

Is a clearly false equality, given this, I would like to know what I am doing wrong. The book is: Fundamentos de Matemática, a Brazilian basic math book (In english can be called Mathematics Fundamentals)

Here is the link to the book in amazon: https://a.co/d/9HdJFCj

Answer 1

Since your triangle is drawn in reasonably good proportion I give you

HINTS:

Use similar triangles

$$ DA \cdot DC = 4^2 \to \text {find x}$$ $$ x=2? ~\text {or } x=8 ?$$ Use Pythagoras thm to find $BA,BC..$

Answer 2

I found a solution after a very good tip of @Narasimham, I may thank you for this.

Happens that I forgot to rotate the triangle to compare the similarity in the right way.