To determine the numerical length of $\overline{CD}$, we will start by using the given information and apply the segment addition postulate.

Given:

• $BC = x + 4$
• $CD = 5x$
• $BD = 5x + 7$

Step 1: Apply the segment addition postulate

The segment addition postulate states that the length of segment $\overline{BD}$ is the sum of the lengths of segments $\overline{BC}$ and $\overline{CD}$. Therefore: $BD = BC + CD$

Step 2: Substitute the given values

Substituting the given expressions for $BD$, $BC$, and $CD$: $5x + 7 = (x + 4) + 5x$

Step 3: Simplify the equation

Now, simplify the equation: $5x + 7 = x + 4 + 5x$ Combine like terms on the right side: $5x + 7 = 6x + 4$

Step 4: Solve for $x$

To isolate $x$, subtract $5x$ from both sides: $7 = x + 4$ Next, subtract 4 from both sides to solve for $x$: $x = 3$

Step 5: Determine the length of $\overline{CD}$

Substitute $x = 3$ into the expression for $CD$: $CD = 5x = 5(3) = 15$

Final Answer:

The numerical length of $\overline{CD}$ is $\boxed{15}$.

Would you like further details on any step, or do you have any other questions?

Here are 8 related questions:

1. How do you find the length of a segment when given different expressions for its parts?
2. What is the segment addition postulate?
3. What would happen if $x$ were negative in this problem?
4. How can you check your answer for correctness in geometry problems?
5. What if $BD$ had been given as $6x + 8$? How would the problem change?
6. Can you find the length of $\overline{BC}$ with the given value of $x$?
7. How would you solve a similar problem if $\overline{BD}$ had different segments?
8. What is the importance of solving for a variable before calculating segment lengths?

Tip: When solving problems with segments and variables, always ensure that all parts add up correctly according to the postulate or theorem you’re using.