Question

g is the midpoint of \\( \overline{fh} \\). if \\( fg = x + 8 \\) and \\( fh = 5x + 10 \\), what is \\( fg \\)? diagram: segment \\( f---g---h \\) with \\( fg \\) labeled \\( x + 8 \\) and \\( fh \\) labeled \\( 5x + 10 \\) simplify your answer and write it as a proper fraction, mixed number, or integer.

Explanation:

Step1: Use midpoint property

Since \( G \) is the midpoint of \( \overline{FH} \), we know that \( FH = 2 \times FG \). So we can set up the equation:
\( 5x + 10 = 2(x + 8) \)

Step2: Solve for \( x \)

First, expand the right side:
\( 5x + 10 = 2x + 16 \)
Subtract \( 2x \) from both sides:
\( 5x - 2x + 10 = 16 \)
\( 3x + 10 = 16 \)
Subtract 10 from both sides:
\( 3x = 16 - 10 \)
\( 3x = 6 \)
Divide both sides by 3:
\( x = \frac{6}{3} = 2 \)

Step3: Find \( FG \)

Now that we have \( x = 2 \), substitute into \( FG = x + 8 \):
\( FG = 2 + 8 = 10 \)

Answer:

10