Question

v is the midpoint of \\(\overline{uw}\\). if \\(uv = x + 8\\) and \\(vw = 6x\\), what is \\(uv\\)?
u
v
w
simplify your answer and write it as a proper fraction, mixed number, or integer.

Explanation:

Step1: Use midpoint property

Since \( V \) is the midpoint of \( \overline{UW} \), \( UV = VW \). So we set up the equation \( x + 8 = 6x \).

Step2: Solve for \( x \)

Subtract \( x \) from both sides: \( 8 = 6x - x \), which simplifies to \( 8 = 5x \). Then divide both sides by 5: \( x=\frac{8}{5} \).

Step3: Find \( UV \)

Substitute \( x = \frac{8}{5} \) into \( UV = x + 8 \). So \( UV=\frac{8}{5}+8=\frac{8}{5}+\frac{40}{5}=\frac{48}{5}=9.6 \) (or as a fraction \( \frac{48}{5} \) or mixed number \( 9\frac{3}{5} \)). Wait, wait, no, wait. Wait, let's check the equation again. Wait, if \( V \) is the midpoint, \( UV = VW \), so \( x + 8 = 6x \). Then \( 8 = 5x \), so \( x=\frac{8}{5} \)? Wait, no, that would make \( UV=\frac{8}{5}+8=\frac{8 + 40}{5}=\frac{48}{5}=9.6 \), but let's check the arithmetic again. Wait, \( 6x - x = 5x \), so \( 5x = 8 \), so \( x=\frac{8}{5} \). Then \( UV = x + 8=\frac{8}{5}+8=\frac{8 + 40}{5}=\frac{48}{5}=9.6 \), which is \( 9\frac{3}{5} \) or \( \frac{48}{5} \). Wait, but maybe I made a mistake. Wait, let's re - do the equation:

Starting over:

Since \( V \) is the midpoint of \( \overline{UW} \), \( UV = VW \).

Given \( UV=x + 8 \) and \( VW = 6x \), so:

\( x+8=6x \)

Subtract \( x \) from both sides:

\( 8=6x - x\)

\( 8 = 5x\)

Divide both sides by 5:

\( x=\frac{8}{5}=1.6 \)

Then \( UV=x + 8=1.6+8 = 9.6=\frac{48}{5}=9\frac{3}{5} \)

Wait, but maybe I messed up the equation. Wait, is \( UV = VW \) correct? Yes, because the midpoint divides the segment into two equal parts, so \( UV=VW \).

Alternatively, maybe I made a mistake in the sign. Let's check again.

\( UV=x + 8 \), \( VW = 6x \), \( UV = VW \)

\( x + 8=6x \)

\( 8=6x - x\)

\( 8 = 5x\)

\( x=\frac{8}{5} \)

Then \( UV=\frac{8}{5}+8=\frac{8 + 40}{5}=\frac{48}{5}=9.6 \)

Yes, that's correct.

Answer:

\( \frac{48}{5} \) (or \( 9\frac{3}{5} \) or \( 9.6 \))