Question

find the value of $y$.

Explanation:

Step1: Use the Angle - Angle Similarity (AA)

Since the two triangles share an angle at \(X\) and the other pair of angles (the ones marked equal) are congruent, \(\triangle XYW\sim\triangle XZW\) by AA similarity.

Step2: Set up the proportion

For similar triangles \(\triangle XYW\) and \(\triangle XZW\), the ratios of corresponding sides are equal. So, \(\frac{XY}{XZ}=\frac{y}{20}\).
We know that \(XY = 30\), \(XZ=30 + 45=75\).
Substitute the values into the proportion: \(\frac{30}{75}=\frac{y}{20}\).

Step3: Cross - multiply

Cross - multiplying gives \(75y=30\times20\).
So, \(75y = 600\).

Step4: Solve for \(y\)

Divide both sides by \(75\): \(y=\frac{600}{75}\).
\(y = 8\).

Answer:

\(y = 8\)