*C*

is

a

point

on

segment

*BD*

.

 If 

*BC* = 20*x*

, 

*CD* = 9

, and 

*BD* = 13*x* + 16

, what is *BC*?

*Simplify your answer and write it as a proper fraction, mixed number, or integer.*

Let's revisit the problem with the correct expressions:

C is a point on segment \( \overline{BD} \).

- \( BC = 20x \)
- \( CD = 9 \)
- \( BD = 13x + 16 \)

We need to find \( BC \).

### Step 1: Use the segment addition postulate.
Since \(C\) is a point on segment \( \overline{BD} \), the entire segment \( BD \) is the sum of \( BC \) and \( CD \):
\[
BD = BC + CD
\]

Substituting the given values:
\[
13x + 16 = 20x + 9
\]

### Step 2: Solve for \( x \).
First, subtract \( 20x \) from both sides:
\[
13x + 16 - 20x = 9
\]
\[
-7x + 16 = 9
\]

Next, subtract 16 from both sides:
\[
-7x = 9 - 16
\]
\[
-7x = -7
\]

Now, divide by \(-7\):
\[
x = 1
\]

### Step 3: Find \( BC \).
Since \( BC = 20x \), substitute \( x = 1 \) into the equation:
\[
BC = 20 \times 1 = 20
\]

So, the length of \( BC \) is \( \boxed{20} \).

Would you like further details or have any questions?

Here are five related questions you might consider:
1. What if \( CD \) were given as a function of \( x \) instead of a constant?
2. How would the solution change if \( BD \) were a more complex expression, such as a quadratic function of \( x \)?
3. What if the problem included an additional segment on \( BD \)? How would that affect the equation?
4. Can you confirm the solution by checking if the calculated \( BC \) and \( CD \) add up to \( BD \)?
5. What strategies can you use to simplify similar equations involving segments?

**Tip:** Always make sure to carefully substitute and solve for variables step by step, as small arithmetic errors can lead to incorrect solutions.

Learn how to find the length of segment BC on segment BD using the segment addition postulate. This problem involves solving for a variable length using given segment lengths and algebraic expressions. Suitable for students in Grades 9-12.

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