identifying necessary information in the side - splitter theorem
based on the side - splitter theorem, which side length would complete the proportion?
$\frac{jm}{mk} = \frac{jn}{?}$
$\circ$ jl
$\circ$ kl
$\circ$ nk
$\circ$ nm

Question

Accepted Answer

# Explanation:
## Step1: Recall Side - Splitter Theorem
The Side - Splitter Theorem states that if a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally. In triangle $JKL$ (assuming the large triangle is $JKL$ with $M$ on $JK$ and $N$ on $KL$ and $MN\parallel JL$), we have $\frac{JM}{MK}=\frac{LN}{NK}$. Wait, no, let's look at the segments. The line $MN$ is parallel to $JL$ (from the markings). So for the sides $JK$ (divided into $JM$ and $MK$) and $KL$ (divided into $LN$ and $NK$), the proportion from the Side - Splitter Theorem is $\frac{JM}{MK}=\frac{LN}{NK}$? Wait, no, let's re - examine the proportion given: $\frac{JM}{MK}=\frac{LN}{?}$.

Looking at the triangle, $JK$ is one side, $KL$ is another side. $M$ is on $JK$, $N$ is on $KL$, and $MN\parallel JL$. By the Side - Splitter Theorem, $\frac{JM}{MK}=\frac{LN}{NK}$? Wait, no, maybe I mixed up. Wait, the Side - Splitter Theorem: If a line parallel to $JL$ (the top side) intersects $JK$ at $M$ and $KL$ at $N$, then $\frac{JM}{MK}=\frac{LN}{NK}$? Wait, no, let's label the triangle properly. Let the large triangle be $JKL$, with $J$ at the top left, $K$ at the bottom left, and $L$ at the bottom right? Wait, no, the diagram shows $J$, $L'$ (maybe a typo for $L$), $K$, with $M$ on $JK$ and $N$ on $KL$, and $MN$ parallel to $JL$. So $JK$ is the left side, $KL$ is the right side, $JL$ is the top side. Then by the Side - Splitter Theorem, $\frac{JM}{MK}=\frac{LN}{NK}$. Wait, but the proportion given is $\frac{JM}{MK}=\frac{LN}{?}$. So we need to find the segment corresponding to $MK$ on the other side. $JM$ and $MK$ are parts of $JK$, so $LN$ and $NK$ should be parts of $KL$. Wait, no, maybe the triangle is $JKL$ with $M$ on $JK$ and $N$ on $KL$, and $MN\parallel JL$. Then the theorem says $\frac{JM}{MK}=\frac{LN}{NK}$. Wait, but let's check the options. The options are $JL$, $KL$, $NK$, $NM$.

Wait, maybe I made a mistake. Let's re - express the Side - Splitter Theorem. The Side - Splitter Theorem: In $\triangle JKL$, if $MN\parallel JL$, where $M\in JK$ and $N\in KL$, then $\frac{JM}{MK}=\frac{LN}{NK}$. So in the proportion $\frac{JM}{MK}=\frac{LN}{?}$, the missing term should be $NK$? Wait, no, the options have $NK$ as an option. Wait, but let's check again. Wait, maybe the triangle is labeled as $J$, $K$, $L$ with $J$ at the top, $K$ at the bottom, and $L$ at the right. $M$ is on $JK$, $N$ is on $KL$, $MN\parallel JL$. Then $JK$ is divided into $JM$ and $MK$, and $KL$ is divided into $LN$ and $NK$. So the proportion is $\frac{JM}{MK}=\frac{LN}{NK}$. So the missing term is $NK$? Wait, no, the options are $JL$, $KL$, $NK$, $NM$. Wait, maybe I mislabeled the triangle. Let's look at the proportion again: $\frac{JM}{MK}=\frac{LN}{?}$. So the numerator on the left is a segment of $JK$ (from $J$ to $M$), the denominator is from $M$ to $K$ (also on $JK$). On the right, the numerator is from $L$ to $N$ (on $KL$), so the denominator should be from $N$ to $K$ (on $KL$), which is $NK$. Wait, but let's check the options. Option C is $NK$. Wait, but let's confirm the Side - Splitter Theorem. The Side - Splitter Theorem states that if a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally. So if $MN\parallel JL$, then $\frac{JM}{MK}=\frac{LN}{NK}$. So the missing side length is $NK$? Wait, no, maybe the triangle is $JKL$ with $J$, $K$, $L$, and $MN$ parallel to $JL$, so $JK$ is split into $JM$ and $MK$, $KL$ is split into $LN$ and $NK$, so the proportion is $\frac{JM}{MK}=\frac{LN}{NK}$. So the answer should be $NK$? Wait, but let's check the options. The options are:

- $JL$: This is a side, not a segment of $KL$.
- $KL$: This is the whole side, not a segment.
- $NK$: This is a segment of $KL$, from $N$ to $K$.
- $NM$: This is the parallel line, not a segment of $KL$.

So according to the Side - Splitter Theorem, $\frac{JM}{MK}=\frac{LN}{NK}$, so the missing term is $NK$. Wait, but let me re - check. Wait, maybe the triangle is $JKL$ with $J$ at the top, $K$ at the bottom left, $L$ at the bottom right. $M$ is on $JK$, $N$ is on $KL$, $MN\parallel JL$. Then $JM$ and $MK$ are parts of $JK$, $LN$ and $NK$ are parts of $KL$. So the proportion is $\frac{JM}{MK}=\frac{LN}{NK}$. So the answer is $NK$.

## Step2: Match with Options
Looking at the options, the option for $NK$ is present. So the side length that completes the proportion is $NK$. Wait, but let me check again. Wait, maybe I made a mistake in the theorem. The Side - Splitter Theorem: If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally. So in $\triangle JKL$, with $MN\parallel JL$, $M\in JK$, $N\in KL$, then $\frac{JM}{MK}=\frac{LN}{NK}$. Yes, that's correct. So the missing term is $NK$.

# Answer:
NK (the option corresponding to NK, which is one of the options: the option with "NK" as the text)

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Explanation:

Step1: Recall Side - Splitter Theorem

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