Point C is on line segment start overline, B, D, end overline
BD
. Given B, C, equals, x, plus, 4, commaBC=x+4, C, D, equals, 5, x, commaCD=5x, and B, D, equals, 5, x, plus, 7, commaBD=5x+7, determine the numerical length of start overline, C, D, end overline, .
CD
.

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.

Learn how to determine the length of segment CD using the segment addition postulate and algebraic expressions. This problem involves solving for variable x and applying geometric concepts. Suitable for high school students.

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