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 \(DE\), \(EF\), and \(FG\) by substituting \(x = 13\):

• \(DE = 4(13)+ 1 = 52 + 1 = 53\)
• \(EF = 3(13)+ 41 = 39 + 41 = 80\)
• \(FG = 7(13)- 11 = 91 - 11 = 80\) (which checks out with the midpoint property)

Step4: Calculate DG

\(DG = DE + EF + FG\)
Substitute the values:
\(DG = 53 + 80 + 80 = 213\)

Answer:

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