Question

solve for z.

Explanation:

Step1: Use similarity of triangles

Since \(\triangle ABC\) and \(\triangle ADE\) are similar (by AA similarity, as \(\angle ABC=\angle ADE = 90^{\circ}\) and \(\angle A\) is common). For similar triangles, the ratios of corresponding sides are equal. So, \(\frac{BC}{DE}=\frac{AB}{AD}\).
Given \(BC = 3\), \(AB=4\), \(AD=4 + 4=8\). Let \(DE=z\). Then the proportion is \(\frac{3}{z}=\frac{4}{8}\).

Step2: Solve the proportion

Cross - multiply the proportion \(\frac{3}{z}=\frac{4}{8}\). We get \(4z=3\times8\).
Simplify the right - hand side: \(4z = 24\).
Divide both sides by 4: \(z=\frac{24}{4}\).

Answer:

\(z = 6\)