Question

in the diagram, \\( \triangle abc \sim \triangle rst \\).
what is the unknown side length \\( y \\)?
units

Explanation:

Step1: Find the similarity ratio

Since \(\triangle ABC\sim\triangle RST\), the ratio of corresponding sides is equal. The hypotenuse of \(\triangle ABC\) is \(AB = 5\) (using Pythagorean theorem \(AB=\sqrt{3^{2}+4^{2}}=\sqrt{9 + 16}=\sqrt{25} = 5\)), and the hypotenuse of \(\triangle RST\) is \(RS = 8\). The similarity ratio is \(\frac{AB}{RS}=\frac{5}{8}\).

Step2: Set up the proportion for the sides

We know that \(\frac{BC}{ST}=\frac{AB}{RS}\). Substituting \(BC = 4\), \(ST=y\), \(AB = 5\), \(RS = 8\) into the proportion \(\frac{4}{y}=\frac{5}{8}\).

Step3: Solve for \(y\)

Cross - multiply: \(5y=4\times8\). So \(5y = 32\), then \(y=\frac{32}{5}=6.4\).

Answer:

\(6.4\)