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, when we solve \( 4x = x + 10 \), subtract \( x \): \( 3x = 10 \), so \( x=\frac{10}{3} \)? Wait, no, that can't be. Wait, 4x = x + 10, so 4x - x = 10, 3x = 10, x = 10/3? Wait, but then LM is 4(10/3)=40/3≈13.333. But maybe I made a mistake. Wait, no, let's do it again. Midpoint means LM = MN. So LM is 4x, MN is x + 10. So 4x = x + 10. Subtract x: 3x = 10. So x = 10/3. Then LM is 4(10/3)=40/3, which is 13 and 1/3. Wait, but maybe the problem is written differently? Wait, the original problem: LM = 4x, MN = x + 10. So yes, midpoint so LM = MN. So 4x = x + 10. So 3x = 10. x = 10/3. Then LM = 4*(10/3)=40/3. Wait, but maybe I misread the problem. Wait, is it LM = 4x and MN = x + 10? Yes. So that's correct.

Wait, but let's check again. 4x = x + 10. Subtract x: 3x = 10. x = 10/3. Then LM is 4*(10/3)=40/3, which is 13 1/3. But maybe the problem was supposed to be LM = 4x and MN = x + 10, and we need to find LM. So that's the calculation.

Answer:

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