Point H is on line segment start overline, G, I, end overline

GI
. Given G, I, equals, 3, x, commaGI=3x, H, I, equals, 2, commaHI=2, and G, H, equals, 5, x, minus, 6, commaGH=5x−6, determine the numerical length of start overline, G, H, end overline, .
GH

1 answer

To determine the length of line segment \( GH \), we have the following lengths based on the problem statement:

  1. \( GI = 3x \)
  2. \( HI = 2 \)
  3. \( GH = 5x - 6 \)

Given that point \( H \) lies on line segment \( GI \), we can set up an equation based on the relationship between these lengths. Since \( G \), \( H \), and \( I \) are collinear, we have:

\[ GH + HI = GI \]

Substituting the known values into the equation gives us:

\[ (5x - 6) + 2 = 3x \]

Now, we can simplify and solve for \( x \):

\[ 5x - 6 + 2 = 3x \]

This simplifies to:

\[ 5x - 4 = 3x \]

Next, we isolate \( x \) by subtracting \( 3x \) from both sides:

\[ 5x - 3x - 4 = 0 \] \[ 2x - 4 = 0 \]

Adding \( 4 \) to both sides gives:

\[ 2x = 4 \]

Dividing both sides by \( 2 \) results in:

\[ x = 2 \]

Now that we have \( x \), we can calculate the length of \( GH \):

\[ GH = 5x - 6 \]

Substituting \( x = 2 \):

\[ GH = 5(2) - 6 = 10 - 6 = 4 \]

Thus, the numerical length of \( GH \) is:

\[ \boxed{4} \]