Point S is on line segment start overline, R, T, end overline 
RT
 . Given S, T, equals, 2, x, minus, 10, commaST=2x−10, R, S, equals, 2, x, minus, 5, commaRS=2x−5, and R, T, equals, 3, x, plus, 1, commaRT=3x+1, determine the numerical length of start overline, R, T, end overline, . 
RT
 .


To determine the numerical length of \( \overline{RT} \), we can use the fact that point \( S \) lies on the line segment \( \overline{RT} \). This implies that the sum of the lengths of \( \overline{RS} \) and \( \overline{ST} \) should be equal to the length of \( \overline{RT} \). In other words, we have the equation:

\[
RS + ST = RT
\]

We are given the following expressions for the lengths:
- \( RS = 2x - 5 \)
- \( ST = 2x - 10 \)
- \( RT = 3x + 1 \)

Now, substitute these into the equation \( RS + ST = RT \):

\[
(2x - 5) + (2x - 10) = 3x + 1
\]

Simplify the left-hand side:

\[
4x - 15 = 3x + 1
\]

Now, solve for \( x \):

\[
4x - 3x = 1 + 15
\]
\[
x = 16
\]

Now that we have \( x = 16 \), substitute it back into the expression for \( RT \):

\[
RT = 3x + 1 = 3(16) + 1 = 48 + 1 = 49
\]

Thus, the numerical length of \( \overline{RT} \) is:

\[
\boxed{49}
\]

Would you like further details on this solution, or do you have any other questions?

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Here are some related questions to expand your understanding:
1. How would the solution change if point \( S \) were not on the line segment \( \overline{RT} \)?
2. How can we generalize the method used here to other line segments?
3. What if the expressions for \( RS \), \( ST \), and \( RT \) involved quadratic terms instead of linear ones?
4. How can we verify that the point \( S \) divides \( \overline{RT} \) in a consistent manner?
5. What would happen if the coordinates of points \( R \), \( S \), and \( T \) were given explicitly?

**Tip:** When solving for unknowns in geometric problems, always check whether the relationships between segments are consistent with the geometric setup (e.g., sum of parts equals the whole).

Point S is on line segment \( \overline{RT} \). Given \( ST = 2x - 10 \), \( RS = 2x - 5 \), and \( RT = 3x + 1 \), determine the numerical length of \( \overline{RT} \).

Determine the length of line segment RT using the Segment Addition Postulate and linear equations. The problem involves algebraic manipulation of expressions for RS, ST, and RT to solve for x and find the length of RT. Suitable for students in Grades 8-10.

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