Question

in triangle abc, the measure of ∠a is 90°, ab = 10, and bc = 16. triangle def is similar to triangle abc, where vertices d, e, and f correspond to vertices a, b, and c, respectively, and each side of triangle def is 2 times the length of the corresponding side of triangle abc. what is the value of sin f?

Explanation:

Step1: Use the property of similar triangles

Since \(\triangle DEF\sim\triangle ABC\), \(\angle F=\angle C\). So \(\sin F = \sin C\).

Step2: Recall the definition of sine in a right - triangle

In right - triangle \(ABC\) with \(\angle A = 90^{\circ}\), by the definition of sine, \(\sin C=\frac{AB}{BC}\).

Step3: Substitute the given values

Given \(AB = 10\) and \(BC=16\), then \(\sin C=\frac{10}{16}=\frac{5}{8}\).

Answer:

\(\frac{5}{8}\)