Question

m is the midpoint of (overline{ln}). if (lm = 4x) and (mn = x + 10), what is (lm)? simplify your answer and write it as a proper fraction, mixed number, or integer.

Explanation:

Step1: Use midpoint property

Since \( M \) is the midpoint of \( \overline{LN} \), \( LM = MN \). So we set up the equation \( 4x = x + 10 \).

Step2: Solve for \( x \)

Subtract \( x \) from both sides: \( 4x - x = x + 10 - x \), which simplifies to \( 3x = 10 \). Then divide both sides by 3: \( x=\frac{10}{3} \).

Step3: Find \( LM \)

We know \( LM = 4x \), substitute \( x = \frac{10}{3} \) into it: \( LM = 4\times\frac{10}{3}=\frac{40}{3}=13\frac{1}{3} \). Wait, no, wait, let's check again. Wait, \( 4x = x + 10 \), so \( 4x - x = 10 \), \( 3x = 10 \), \( x=\frac{10}{3} \)? Wait, no, that can't be. Wait, maybe I made a mistake. Wait, \( 4x = x + 10 \), subtract \( x \): \( 3x = 10 \), so \( x=\frac{10}{3} \)? But then \( LM = 4x = 4\times\frac{10}{3}=\frac{40}{3}\approx13.33 \). But maybe I misread the problem. Wait, the problem says \( LM = 4x \) and \( MN = x + 10 \). Since \( M \) is the midpoint, \( LM = MN \), so \( 4x = x + 10 \). Then \( 3x = 10 \), \( x=\frac{10}{3} \). Then \( LM = 4\times\frac{10}{3}=\frac{40}{3} \)? Wait, but maybe the problem is \( LM = 4x \) and \( MN = x + 10 \), so solving \( 4x = x + 10 \), \( 3x = 10 \), \( x = \frac{10}{3} \), then \( LM = 4\times\frac{10}{3}=\frac{40}{3} \). But let's check again. Wait, maybe the original problem has a typo, but according to the given, that's the solution. Wait, no, maybe I messed up. Wait, \( 4x = x + 10 \), so \( 4x - x = 10 \), \( 3x = 10 \), \( x = \frac{10}{3} \), so \( LM = 4x = \frac{40}{3} \), which is \( 13\frac{1}{3} \). But let's confirm. If \( x = \frac{10}{3} \), then \( MN = \frac{10}{3} + 10 = \frac{10}{3} + \frac{30}{3} = \frac{40}{3} \), and \( LM = 4\times\frac{10}{3} = \frac{40}{3} \), so they are equal, so that's correct.

Answer:

\(\frac{40}{3}\) (or \(13\frac{1}{3}\))