https://file.aiquickdraw.com/mathbot/file/17268326518948jqcrgb7.jpg

Let's solve the problem in the image step by step.

The problem gives:
- \( RS = 5x + 17 \)
- \( ST = 8x - 31 \)
- \( RT = 194 \)

We are asked to determine if point \( S \) is the midpoint of \( RT \), meaning we need to check if \( RS = ST \).

### Step 1: Use the fact that \( RT = RS + ST \)
Since \( R \), \( S \), and \( T \) are collinear, and \( RT \) is the sum of \( RS \) and \( ST \), we have:
\[
RS + ST = RT
\]
Substitute the given values:
\[
(5x + 17) + (8x - 31) = 194
\]

### Step 2: Solve for \( x \)
Simplify the left-hand side:
\[
5x + 17 + 8x - 31 = 194
\]
\[
13x - 14 = 194
\]
Add 14 to both sides:
\[
13x = 208
\]
Now divide by 13:
\[
x = \frac{208}{13} = 16
\]

### Step 3: Calculate \( RS \) and \( ST \)
Now that we have \( x = 16 \), we can find \( RS \) and \( ST \).

For \( RS \):
\[
RS = 5x + 17 = 5(16) + 17 = 80 + 17 = 97
\]

For \( ST \):
\[
ST = 8x - 31 = 8(16) - 31 = 128 - 31 = 97
\]

### Step 4: Conclusion
Since \( RS = ST = 97 \), point \( S \) is indeed the midpoint of \( RT \).

Would you like further details or clarification?

Here are some related questions you can consider:
1. What is the general condition for a point to be the midpoint between two points?
2. How does collinearity of points affect the relationships between distances?
3. How would the answer change if \( RT \) were larger or smaller than 194?
4. What is the significance of simplifying and solving linear equations in this context?
5. Can you find the midpoint formula for a number line or coordinate plane?

**Tip**: Always double-check calculations by substituting the found value of \( x \) back into the original equations to ensure accuracy.

If points R, S, and T are collinear with point S between R and T, given RS = 5x + 17, ST = 8x - 31, and RT = 194, determine if point S is the midpoint of RT. Justify your answer.

Solve a problem to determine if point S is the midpoint of RT, using given expressions for RS, ST, and RT. This algebra problem requires solving linear equations and understanding midpoints in geometry. Suitable for students in Grades 8-10.

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