using the centroid to find the length of a segment
s is the centroid of triangle xyz.
what is the length of $ \overline { sw } $?
5 units
11 units
22 units
28 units

Question

Accepted Answer

# Explanation:
## Step1: Recall centroid property
The centroid of a triangle divides each median into a ratio of $2:1$, with the longer segment being closer to the vertex. Also, in a triangle, the medians intersect at the centroid, so we can use the fact that the lengths related to the centroid and medians can be set equal for some segments (since medians are divided proportionally, but also, the two segments from the centroid to the midpoint and centroid to the vertex have a $1:2$ ratio, and also, the segments from the centroid to the midpoints of the sides can be related. Wait, actually, in the diagram, we can see that $XS$ and $SV$? Wait, no, looking at the labels: $T$ is on $XY$, $W$ is on $YZ$, $V$ is on $XZ$? Wait, actually, the medians are $XT$? No, wait, the centroid $S$ is the intersection of the medians. So the median from $X$ to $W$ (midpoint of $YZ$), median from $Y$ to $V$ (midpoint of $XZ$), and median from $Z$ to $T$ (midpoint of $XY$).

But in the diagram, we have $XT$ is a median? Wait, no, the segments: $2m + 12$ is $XT$? Wait, no, the label $2m + 12$ is from $X$ to $S$? Wait, no, the diagram shows $X$ connected to $S$ with $2m + 12$, $S$ connected to $T$ with $3m - 1$, and $Z$ connected to $S$ with $4m + 8$, and $S$ connected to $W$. Wait, maybe the key is that in a triangle, the centroid divides each median into two parts with a ratio of $2:1$. So, for example, if $XW$ is a median (with $W$ the midpoint of $YZ$), then $XS:SW = 2:1$. But also, maybe the segments $XS$ (from $X$ to centroid) and $SZ$ (from centroid to $Z$)? Wait, no, maybe the medians from $X$ and $Z$ intersect at $S$. Wait, perhaps the two segments $2m + 12$ and $3m - 1$ are parts of the same median? Wait, no, looking at the labels: $T$ is on $XY$, $W$ is on $YZ$, so the median from $Z$ to $T$ (midpoint of $XY$) and median from $X$ to $W$ (midpoint of $YZ$) intersect at $S$.

Wait, another approach: The centroid divides each median into a ratio of $2:1$, so the length from the vertex to the centroid is twice the length from the centroid to the midpoint of the side. Also, in the diagram, we can see that $2m + 12$ and $3m - 1$ might be related? Wait, no, maybe the segments $XS$ (from $X$ to $S$) and $SZ$ (from $S$ to $Z$)? Wait, no, perhaps the two expressions $2m + 12$ and $3m - 1$ are parts of the same median? Wait, no, let's look at the answer choices. Wait, maybe the key is that in a triangle, the centroid divides the median into $2:1$, so the length from the centroid to the midpoint (SW) is half the length from the vertex to the centroid (SZ)? Wait, no, the ratio is vertex to centroid : centroid to midpoint = $2:1$. So if $SZ$ is the segment from $S$ to $Z$, and $SW$ is from $S$ to $W$ (midpoint of $YZ$), then $SZ = 2 \times SW$? Wait, no, the median from $Z$ to $W$ (midpoint of $YZ$)? No, $W$ is on $YZ$, so the median from $X$ to $W$ (midpoint of $YZ$): so $XW$ is the median, with $X$ to $S$ to $W$, so $XS:SW = 2:1$. But also, maybe the median from $Z$ to $T$ (midpoint of $XY$): $ZT$ is the median, with $Z$ to $S$ to $T$, so $ZS:ST = 2:1$.

Looking at the diagram, we have $ST = 3m - 1$ and $XS = 2m + 12$? Wait, no, the label $2m + 12$ is from $X$ to $S$, and $3m - 1$ is from $S$ to $T$. Since $ZT$ is a median ( $T$ is midpoint of $XY$ ), then $ZS:ST = 2:1$, so $ZS = 2 \times ST$. Wait, but $ZS$ is labeled as $4m + 8$? Wait, no, the diagram shows $Z$ connected to $S$ with $4m + 8$, and $S$ connected to $T$ with $3m - 1$. Wait, maybe I got the labels wrong. Let's re-express:

Wait, the centroid $S$ is the intersection of the medians. So the median from $X$ to $W$ (midpoint of $YZ$): $XW$ is the median, so $X$---$S$---$W$, with $XS:SW = 2:1$.

The median from $Z$ to $T$ (midpoint of $XY$): $ZT$ is the median, so $Z$---$S$---$T$, with $ZS:ST = 2:1$.

The median from $Y$ to $V$ (midpoint of $XZ$): $YV$ is the median, so $Y$---$S$---$V$, with $YS:SV = 2:1$.

Now, looking at the diagram, we have:

- From $X$ to $S$: $2m + 12$

- From $S$ to $T$: $3m - 1$

- From $Z$ to $S$: $4m + 8$

Since $ZT$ is a median, $ZS:ST = 2:1$, so $ZS = 2 \times ST$. Therefore:

$4m + 8 = 2(3m - 1)$

Let's solve for $m$:

## Step2: Solve for $m$
$4m + 8 = 2(3m - 1)$

Expand the right side: $4m + 8 = 6m - 2$

Subtract $4m$ from both sides: $8 = 2m - 2$

Add 2 to both sides: $10 = 2m$

Divide by 2: $m = 5$

## Step3: Find $SW$
Now, we need to find $SW$. Wait, but we need to relate $SW$ to other segments. Wait, maybe $SW$ is related to $XS$ or $ZS$. Wait, alternatively, since $XW$ is a median, and $XS:SW = 2:1$, and $XS = 2m + 12$. Let's compute $XS$ when $m = 5$:

$XS = 2(5) + 12 = 10 + 12 = 22$

Since $XS:SW = 2:1$, then $SW = \frac{XS}{2} = \frac{22}{2} = 11$

Wait, let's check with $ZS$. $ZS = 4m + 8 = 4(5) + 8 = 20 + 8 = 28$. And $ST = 3m - 1 = 3(5) - 1 = 15 - 1 = 14$. Then $ZS:ST = 28:14 = 2:1$, which matches the centroid ratio. So that's correct. Then $XS = 22$, so $SW = \frac{XS}{2} = 11$, which is one of the options.

## Step1: Recall centroid ratio
The centroid divides each median into $2:1$ (vertex to centroid : centroid to midpoint). For median $ZT$ ( $T$ midpoint of $XY$ ), $ZS:ST = 2:1$, so $ZS = 2 \times ST$.

## Step2: Set up equation
Given $ZS = 4m + 8$ and $ST = 3m - 1$, use $ZS = 2 \times ST$:
$4m + 8 = 2(3m - 1)$

## Step3: Solve for $m$
Expand: $4m + 8 = 6m - 2$
Subtract $4m$: $8 = 2m - 2$
Add 2: $10 = 2m$
Divide: $m = 5$

## Step4: Find $XS$ (vertex to centroid)
For median $XW$ ( $W$ midpoint of $YZ$ ), $XS = 2m + 12$. Substitute $m = 5$:
$XS = 2(5) + 12 = 22$

## Step5: Find $SW$ (centroid to midpoint)
Since $XS:SW = 2:1$, $SW = \frac{XS}{2} = \frac{22}{2} = 11$

# Answer: 11 units

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Explanation:

Step1: Recall centroid property

Answer: