To solve for $x$ in the given diagram, let's analyze the information provided:

Step 1: Key observations

1. $CB$ is the midsegment of triangle $RST$.

   • By the Midsegment Theorem, the midsegment of a triangle is parallel to one side of the triangle and equal to half the length of that side.
   • Therefore, $CB \parallel RT$ and $CB = \frac{1}{2} RT$.

2. The given lengths are:

   • $CB = x + 19$
   • $RT = x + 29$.

Step 2: Apply the Midsegment Theorem

Since $CB = \frac{1}{2} RT$, we can write the equation:

$x + 19 = \frac{1}{2} (x + 29).$

Step 3: Solve for $x$

1. Multiply through by 2 to eliminate the fraction:

$2(x + 19) = x + 29.$

2. Distribute:

$2x + 38 = x + 29.$

3. Subtract $x$ from both sides:

$x + 38 = 29.$

4. Subtract 38 from both sides:

$x = -9.$

Final Answer:

The value of $x$ is -9.

Verification:

1. If $x = -9$:
   • $CB = x + 19 = -9 + 19 = 10$.
   • $RT = x + 29 = -9 + 29 = 20$.
2. Check the Midsegment Theorem:
   • $CB = \frac{1}{2} RT \implies 10 = \frac{1}{2}(20)$, which is correct.

Thus, $x = -9$ satisfies the conditions.

Would you like further clarification or a detailed explanation of any step?

Here are some related questions to explore further:

1. What is the Midsegment Theorem, and how does it apply to triangles?
2. How do you prove that a midsegment is parallel to one side of a triangle?
3. What happens to the lengths when a midsegment divides a triangle?
4. How can this property of midsegments be used in coordinate geometry?
5. Can midsegments be used to find the area of triangles?

Tip: The Midsegment Theorem is a powerful tool in geometry that simplifies the analysis of triangles and helps solve for unknown lengths.