Question

1. if (overline{ef}) bisects (overline{cd}), (cg = 5x - 1), (gd = 7x - 13), (ef = 6x - 4), and (gf = 13), find (eg).

Explanation:

Step1: Use the bisect property to find x

Since \( \overline{EF} \) bisects \( \overline{CD} \), \( CG = GD \). So we set up the equation:
\( 5x - 1 = 7x - 13 \)
Subtract \( 5x \) from both sides:
\( -1 = 2x - 13 \)
Add 13 to both sides:
\( 12 = 2x \)
Divide both sides by 2:
\( x = 6 \)

Step2: Find the length of \( EF \)

Substitute \( x = 6 \) into the expression for \( EF \):
\( EF = 6x - 4 = 6(6) - 4 = 36 - 4 = 32 \)

Step3: Find the length of \( EG \)

We know that \( EF = EG + GF \), and \( GF = 13 \). So we can solve for \( EG \):
\( EG = EF - GF = 32 - 13 = 19 \)

Answer:

\( 19 \)