Question

in triangle $abc$, the measure of $angle a$ is $90^{circ}$, $ab = 10$, 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: Find side $AC$ in $\triangle ABC$

Using the Pythagorean theorem $BC^{2}=AB^{2}+AC^{2}$. Given $AB = 10$ and $BC = 16$, we have $AC=\sqrt{BC^{2}-AB^{2}}=\sqrt{16^{2}-10^{2}}=\sqrt{256 - 100}=\sqrt{156}=2\sqrt{39}$.

Step2: Use similarity of triangles

Since $\triangle DEF\sim\triangle ABC$ and the sides of $\triangle DEF$ are 2 - times the sides of $\triangle ABC$, the angles of the two triangles are equal, i.e., $\angle F=\angle C$.

Step3: Calculate $\sin F$

We know that $\sin C=\frac{AB}{BC}$ in right - triangle $ABC$. Since $\angle F=\angle C$, $\sin F=\sin C$. Substituting $AB = 10$ and $BC = 16$ into the sine formula, we get $\sin F=\frac{10}{16}=\frac{5}{8}$.

Answer:

$\frac{5}{8}$