Question

find vy.
vy =

Explanation:

Step1: Use the mid - segment theorem

Since \( UW\) is a mid - segment of \(\triangle YXV\) (the red arrows indicate mid - points), by the mid - segment theorem, \( UW\parallel XY\) and \( 2UW = XY\), also \( \frac{XW}{WV}=\frac{YU}{UV}\). Here \( XW = 6\), \( WV=10\), \( UV = 15\). Let \( VY=x\), and we know that \( \frac{XW}{XV}=\frac{YU}{YV}\).

First, \( XV=XW + WV=6 + 10=16\).

By the mid - segment ratio property (Thales' theorem or basic proportionality theorem), if a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides those sides proportionally.

We have \(\frac{XW}{XV}=\frac{UV}{VY}\)

Substitute the known values: \(\frac{6}{16}=\frac{15}{VY}\)

Step2: Solve the proportion for \( VY\)

Cross - multiply: \(6\times VY=16\times15\)

\(6VY = 240\)

Divide both sides by 6: \(VY=\frac{240}{6}\)

Answer:

\(40\)