BC = x + 4, CD = 2x, and BD = 7, what is BC

To solve for \( BC \), we can use the relationship between the segments \( BC \), \( CD \), and \( BD \).

Given:
- \( BC = x + 4 \)
- \( CD = 2x \)
- \( BD = 7 \)

Since \( BD \) is the total length from \( B \) to \( D \), it is the sum of \( BC \) and \( CD \):

\[
BD = BC + CD
\]

Substituting the given values:

\[
7 = (x + 4) + 2x
\]

Now, combine like terms:

\[
7 = 3x + 4
\]

Next, solve for \( x \):

\[
3x = 7 - 4
\]
\[
3x = 3
\]
\[
x = 1
\]

Now, substitute \( x = 1 \) back into the expression for \( BC \):

\[
BC = x + 4 = 1 + 4 = 5
\]

So, \( BC = 5 \).

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

Here are five related questions to consider:
1. What is the value of \( CD \) in the problem?
2. How would the problem change if \( BD \) was a different value?
3. Can \( BC \) and \( CD \) be equal in this scenario?
4. What if \( BC \) were expressed as a different function of \( x \)?
5. How do we verify the accuracy of our solution?

**Tip:** Always double-check your calculations by substituting the found values back into the original equations.

Learn how to calculate segment lengths using geometric equations with a step-by-step solution. This problem involves determining the length of segment BC based on the given relationships with CD and BD. Suitable for students in Grades 7-9.

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