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.