Question

find the value of y.

Explanation:

Step1: Use the Angle - Bisector Theorem

The Angle - Bisector Theorem states that if a ray bisects an angle of a triangle, then it divides the opposite side into segments proportional to the adjacent sides.
In \(\triangle XWZ\), \(WY\) is the angle - bisector of \(\angle XWZ\). So, \(\frac{XY}{YZ}=\frac{XW}{ZW}\).
We know that \(XY = 30\), \(YZ=y\), \(XW = 45 - 30=15\) (since the length of the side from \(X\) to the end of the non - triangle segment is \(45\) and \(XY = 30\)), and \(ZW = 20\).
Substituting the values into the proportion \(\frac{30}{y}=\frac{45}{20}\).

Step2: Cross - multiply to solve for \(y\)

Cross - multiplying gives us \(45y=30\times20\).
First, calculate \(30\times20 = 600\). So, \(45y = 600\).
Then, \(y=\frac{600}{45}=\frac{40}{3}\approx13.33\).

Answer:

\(y = \frac{40}{3}\)