Question

1. if $overline{ef}$ bisects $overline{cd}$, $cg = 5x - 1$, $gd = 7x - 13$, $ef = 6x - 4$, and $gf = 13$, find $eg$. 15. if $r$ is the midpoint of $overline{qs}$, $rs = 2x - 4$, $st = 4x - 1$, and $rt = 8x - 43$, find $qs$.

Explanation:

Problem 14

Step1: Since \( \overline{EF} \) bisects \( \overline{CD} \), \( CG = GD \).

Set \( 5x - 1 = 7x - 13 \)

Step2: Solve for \( x \).

\( 5x - 1 = 7x - 13 \)
Subtract \( 5x \) from both sides: \( -1 = 2x - 13 \)
Add 13 to both sides: \( 12 = 2x \)
Divide by 2: \( x = 6 \)

Step3: Find \( EF \) using \( x = 6 \).

\( EF = 6x - 4 = 6(6) - 4 = 36 - 4 = 32 \)

Step4: Since \( EF = EG + GF \), solve for \( EG \).

\( 32 = EG + 13 \)
Subtract 13: \( EG = 32 - 13 = 19 \)

Step1: Since \( R \) is the midpoint of \( \overline{QS} \), \( QR = RS \) and \( QS = 2RS \). Also, \( RT = RS + ST \).

Set \( 8x - 43 = (2x - 4) + (4x - 1) \)

Step2: Solve for \( x \).

\( 8x - 43 = 6x - 5 \)
Subtract \( 6x \): \( 2x - 43 = -5 \)
Add 43: \( 2x = 38 \)
Divide by 2: \( x = 19 \)

Step3: Find \( RS \) using \( x = 19 \).

\( RS = 2x - 4 = 2(19) - 4 = 38 - 4 = 34 \)

Step4: Find \( QS \) (since \( QS = 2RS \)).

\( QS = 2 \times 34 = 68 \)

Answer:

\( EG = 19 \)

Problem 15