Question

1. if g is the midpoint of (overline{fh}) and (fh = 6y - 2), find (y). (overline{fg} = 4x + 3), (overline{gh} = 7x - 12) (with a line segment diagram of points f, g, h in order)

Explanation:

Step1: Set FG = GH (midpoint property)

Since \( G \) is the midpoint of \( \overline{FH} \), \( FG = GH \). So we set \( 4x + 3 = 7x - 12 \).

Step2: Solve for x

Subtract \( 4x \) from both sides: \( 3 = 3x - 12 \).
Add 12 to both sides: \( 15 = 3x \).
Divide by 3: \( x = 5 \).

Step3: Find length of FG and GH

Substitute \( x = 5 \) into \( FG = 4x + 3 \): \( FG = 4(5) + 3 = 23 \).
Since \( FG = GH \), \( GH = 23 \).

Step4: Find length of FH

\( FH = FG + GH = 23 + 23 = 46 \).

Step5: Solve for y using FH = 6y - 2

Set \( 6y - 2 = 46 \).
Add 2 to both sides: \( 6y = 48 \).
Divide by 6: \( y = 8 \).

Answer:

\( y = 8 \)