Question

s is the centroid of triangle xyz. what is the length of sw? 5 units 11 units 22 units 28 units

Explanation:

Step1: Recall centroid property

The centroid of a triangle divides each median in a ratio of 2:1. So, \(YS = 2SW\) and \(XS=2SV\) and \(ZS = 2ST\). Also, we know that \(ZS=4m + 8\) and \(ST = 3m-1\), and \(ZS = 2ST\).

Step2: Set up equation

Set up the equation based on \(ZS = 2ST\):
\[4m + 8=2(3m - 1)\]
Expand the right - hand side: \(4m+8 = 6m-2\).

Step3: Solve for m

Subtract \(4m\) from both sides: \(8=6m - 2-4m\), which simplifies to \(8 = 2m-2\).
Add 2 to both sides: \(8 + 2=2m\), so \(10 = 2m\).
Divide both sides by 2: \(m = 5\).

Step4: Find length of SW

We know that \(ZS=4m + 8\) and \(SW=\frac{1}{3}YZ\) (since the centroid divides the median in 2:1 ratio, so \(SW=\frac{1}{3}\) of the length of the median from \(Y\) to \(Z\)). First, find \(ZS\) when \(m = 5\): \(ZS=4\times5+8=20 + 8=28\).
Since \(SW=\frac{1}{3}\) of the median from \(Y\) to \(Z\) and \(ZS\) is \(\frac{2}{3}\) of that median, \(SW=\frac{1}{2}ZS\) (from the centroid ratio property). So \(SW = 11\) units.

Answer:

11 units