Point S is on line segment start overline, R, T, end overline

RT
. Given R, T, equals, 3, x, commaRT=3x, R, S, equals, 3, x, minus, 5, commaRS=3x−5, and S, T, equals, 3, x, minus, 1, commaST=3x−1, determine the numerical length of start overline, R, T, end overline, .
RT
.

1 answer

To find the length of the line segment \( \overline{RT} \), we can use the information given about the segments \( \overline{RS} \), \( \overline{ST} \), and \( \overline{RT} \).

According to the properties of a line segment, the length of \( \overline{RT} \) is equal to the sum of the lengths of \( \overline{RS} \) and \( \overline{ST} \):

\[ RT = RS + ST \]

Using the given expressions:

  • \( RT = 3x \)
  • \( RS = 3x - 5 \)
  • \( ST = 3x - 1 \)

We can substitute the values of \( RS \) and \( ST \) into the equation:

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

Now, simplify the right-hand side:

\[ 3x = 3x - 5 + 3x - 1 \] \[ 3x = 6x - 6 \]

Next, we can isolate \( x \) by moving \( 6x \) to the left side:

\[ 3x - 6x = -6 \] \[ -3x = -6 \]

Dividing both sides by -3 gives us:

\[ x = 2 \]

Now that we have \( x \), we can find the length of \( \overline{RT} \):

\[ RT = 3x \] \[ RT = 3(2) = 6 \]

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

\[ \boxed{6}. \]