Math Problem Statement

Point G is on line segment start overline, F, H, end overlineFH. Given F, H, equals, 4, x, plus, 8, commaFH=4x+8, F, G, equals, x, commaFG=x, and G, H, equals, 5, x, commaGH=5x, determine the numerical length of start overline, G, H, end overline, .GH.

Solution

We are given the following information:

  • The total length of segment FH=4x+8FH = 4x + 8.
  • The length of segment FG=xFG = x.
  • The length of segment GH=5xGH = 5x.

Since point GG is on the line segment FHFH, we know that:

FG+GH=FHFG + GH = FH

Substitute the given values into the equation:

x+5x=4x+8x + 5x = 4x + 8

Simplify the left-hand side:

6x=4x+86x = 4x + 8

Now, subtract 4x4x from both sides:

2x=82x = 8

Solve for xx:

x=4x = 4

Now that we know x=4x = 4, substitute this value back into the expression for GHGH:

GH=5x=5(4)=20GH = 5x = 5(4) = 20

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

Would you like further details or have any questions?


Here are 5 related questions to expand on this:

  1. How would you determine the length of FGFG using the same method?
  2. What is the total length of FHFH given that x=4x = 4?
  3. If another point JJ was added between GG and HH, how would you calculate the lengths of the new segments?
  4. Can the same method be applied if GG is not between FF and HH?
  5. What happens if the relationship between the segment lengths changes, such as making FG=2xFG = 2x instead?

Tip: Always check if points lie on the same line or segment to avoid overcomplicating problems involving distances!

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Math Problem Analysis

Mathematical Concepts

Algebra
Geometry
Linear Equations

Formulas

Segment addition formula (FG + GH = FH)
Solving linear equations

Theorems

Segment Addition Postulate

Suitable Grade Level

Grades 8-10