Question

1. de = 4x + 1, ef = 3x + 41, fg = 7x − 11. point f is the midpoint of \\(\overline{ge}\\). solve for x. find dg.

Explanation:

Step1: Use midpoint property

Since \( F \) is the midpoint of \( \overline{GE} \), \( EF = FG \). So we set up the equation:
\( 3x + 41 = 7x - 11 \)

Step2: Solve for \( x \)

Subtract \( 3x \) from both sides:
\( 41 = 4x - 11 \)
Add 11 to both sides:
\( 52 = 4x \)
Divide both sides by 4:
\( x = 13 \)

Step3: Find lengths of segments

First, find \( DE \): \( DE = 4x + 1 = 4(13)+1 = 53 \)
Find \( EF \): \( EF = 3x + 41 = 3(13)+41 = 80 \)
Find \( FG \): \( FG = 7x - 11 = 7(13)-11 = 80 \) (which checks out with midpoint)

Step4: Calculate \( DG \)

\( DG = DE + EF + FG = 53 + 80 + 80 = 213 \)

Answer:

\( x = 13 \), \( DG = 213 \)