Math Problem Statement

Point S is on line segment RT. Given ST = 3x - 8, RT = 4x, and RS = 4x - 7, determine the numerical length of RT.

Solution

The problem asks us to determine the numerical length of the segment RTRT. Let's analyze the given information:

  1. ST=3x8ST = 3x - 8
  2. RT=4xRT = 4x
  3. RS=4x7RS = 4x - 7

Approach:

  • Point SS is on the line segment RTRT, so we can use the fact that: RT=ST+RSRT = ST + RS Substituting the given expressions: 4x=(3x8)+(4x7)4x = (3x - 8) + (4x - 7) Simplifying the right-hand side: 4x=3x8+4x74x = 3x - 8 + 4x - 7 Combine like terms: 4x=7x154x = 7x - 15 Now, solve for xx: 4x7x=154x - 7x = -15 3x=15-3x = -15 x=5x = 5

Step 2: Find the length of RTRT

Now that we have x=5x = 5, we can substitute it into the expression for RTRT: RT=4x=4(5)=20RT = 4x = 4(5) = 20

Thus, the numerical length of RTRT is 20\boxed{20}.

Would you like more details or have any questions?

Here are five related questions:

  1. How would you solve for RSRS if x=5x = 5?
  2. What is the length of STST when x=5x = 5?
  3. Can you explain how to check if SS lies on RTRT?
  4. What is the difference between line segments and rays in geometry?
  5. If xx were different, how would the length of RTRT change?

Tip: Always double-check your calculations by plugging the values back into the original equations to ensure correctness.

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Math Problem Analysis

Mathematical Concepts

Algebra
Line Segment
Equation Solving

Formulas

RT = ST + RS
Simplification of linear equations

Theorems

Segment Addition Postulate

Suitable Grade Level

Grades 8-10