Question

m is the midpoint of $overline{ln}$. if $lm = 4x + 4$ and $ln = 6x + 13$, what is $ln$?
simplify your answer and write it as a proper fraction, mixed number, or integer.

Explanation:

Step1: Use mid - point property

Since $M$ is the mid - point of $\overline{LN}$, we know that $LM=\frac{1}{2}LN$. So, $2LM = LN$.

Step2: Substitute the given expressions

Substitute $LM = 4x + 4$ and $LN=6x + 13$ into $2LM = LN$. We get $2(4x + 4)=6x + 13$.

Step3: Expand the left - hand side

Expand $2(4x + 4)$ using the distributive property: $8x+8 = 6x + 13$.

Step4: Solve for $x$

Subtract $6x$ from both sides: $8x-6x + 8=6x-6x + 13$, which simplifies to $2x+8 = 13$. Then subtract 8 from both sides: $2x+8 - 8=13 - 8$, so $2x=5$. Divide both sides by 2: $x=\frac{5}{2}$.

Step5: Find the value of $LN$

Substitute $x = \frac{5}{2}$ into the expression for $LN$: $LN=6x + 13=6\times\frac{5}{2}+13$. First, $6\times\frac{5}{2}=15$. Then $LN=15 + 13=28$.

Answer:

28