We are given that $C$ is between $D$ and $E$, and we need to use the Segment Addition Postulate to solve for $v$.

Segment Addition Postulate:

The Segment Addition Postulate states that if a point $C$ is between points $D$ and $E$, then the sum of the lengths of segments $DC$ and $CE$ is equal to the length of segment $DE$. In equation form:

$DC + CE = DE$

Step 1: Set up the equation

We know the following expressions for the lengths of the segments:

• $DC = 3v - 30$
• $CE = 6v - 15$
• $DE = 27$

Using the Segment Addition Postulate, we can write the equation:

$(3v - 30) + (6v - 15) = 27$

Step 2: Simplify the equation

Now, combine like terms on the left-hand side:

$3v + 6v - 30 - 15 = 27$

$9v - 45 = 27$

Step 3: Solve for $v$

To isolate $v$, first add 45 to both sides of the equation:

$9v = 27 + 45$

$9v = 72$

Now, divide both sides by 9:

$v = \frac{72}{9}$

$v = 8$

Final Answer:

The value of $v$ is $8$.

Would you like further details or have any other questions? Here's a quick set of related questions to explore more:

1. What is the length of segment $DC$ when $v = 8$?
2. How can you apply the Segment Addition Postulate in three-dimensional geometry?
3. How does the Segment Addition Postulate relate to coordinate geometry?
4. What are other types of postulates used in geometry?
5. How do you solve equations with more than one variable in segment length problems?

Tip: When solving for variables in segment length problems, always start by combining like terms and simplifying the equation before solving for the unknown.