In the given diagram, triangle $ABC$ is shown with segment $BD$ perpendicular to both $AB$ and $DC$, forming two right triangles. We are provided the following information:

• $BD = 2x + 4$
• $DC = 8x - 20$

Since $BD$ is perpendicular to both $AB$ and $DC$, we can assume that the triangles share certain geometric properties, such as being similar. However, there is not enough direct information to use triangle similarity for this specific solution. Instead, it appears that the question is simply asking us to solve for $x$ using the relationship between the lengths provided.

Solving for $x$:

We are not given any equation directly relating $BD$ and $DC$, but a typical assumption in these types of questions is that $BD = DC$. This gives us the equation:

$2x + 4 = 8x - 20$

Now, solve for $x$:

1. Subtract $2x$ from both sides: $4 = 6x - 20$

2. Add 20 to both sides: $24 = 6x$

3. Divide both sides by 6: $x = 4$

Thus, the value of $x$ is $4$.

Answer:

The correct option is 4.

Would you like more details or have any questions? Here are five related questions:

1. Why do we assume $BD = DC$ in this problem?
2. How do you determine if two triangles are similar or congruent?
3. What are other ways to solve for unknowns in right triangles?
4. How do perpendicular lines help in geometry problems?
5. Can we use the Pythagorean theorem here?

Tip: When solving for unknown variables, always check if any geometric relationships or properties (like similarity or congruence) can simplify the problem.